SU\G(𝔸)/K·U
Sign Up for free Log In

Author Description

Login to generate an author description

Ask a Question About This Mathematician

Yoneda Lemma for Simplicial Spaces

2023-07-11
Nima Rasekh
Abstract We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and studying their associated model structure, the covariant model structure … Abstract We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and studying their associated model structure, the covariant model structure . In particular, we prove a recognition principle for covariant equivalences over an arbitrary simplicial space and invariance of the covariant model structure with respect to complete Segal space equivalences.

Cartesian fibrations and representability

2022-01-01
Nima Rasekh
We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in … We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in $\infty$-category theory. In particular, we give a construction of representable Cartesian fibrations using over-categories and prove the Yoneda lemma for representable Cartesian fibration, which generalizes the established Yoneda lemma for right fibrations. We then use the theory of Cartesian fibrations to study complete Segal objects internal to an $\infty$-category. Concretely, we prove the {\it fundamental theorem of complete Segal objects}, which characterizes equivalences of complete Segal objects. Finally we give two application of the results. First, we present a method to construct Segal objects and second we study the representability of the universal coCartesian fibration.
View PDF Chat

A theory of elementary higher toposes

2018-06-29
Nima Rasekh
We define an elementary higher topos that simultaneously generalizes an elementary topos and higher topos. Then we show it satisfies classical topos theoretic properties, such being locally Cartesian closed and … We define an elementary higher topos that simultaneously generalizes an elementary topos and higher topos. Then we show it satisfies classical topos theoretic properties, such being locally Cartesian closed and descent. Finally we show we can classify univalent maps in an elementary higher topos.

Cartesian Fibrations of Complete Segal Spaces

2023-05-21
Nima Rasekh
Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in (∞, 1)-category theory to study presheaves valued in (∞, 1)-categories.In this work we … Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in (∞, 1)-category theory to study presheaves valued in (∞, 1)-categories.In this work we define and study fibrations modeling presheaves valued in simplicial spaces and their localizations.This includes defining a model structure for these fibrations and giving effective tools to recognize its fibrations and weak equivalences.This in particular gives us a new method to construct Cartesian fibrations via complete Segal spaces.In addition to that, it allows us to define and study fibrations modeling presheaves of Segal spaces.
View PDF Chat

Complete Segal Objects

2018-05-09
Nima Rasekh
We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete … We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete Segal objects. Finally we use Segal objects to define univalence in a locally Cartesian closed category that is not presentable and generalize some previous results to the non-presentable setting.

Introduction to Complete Segal Spaces

2018-01-01
Nima Rasekh
The goal is to review the notion of a complete Segal space and how certain categorical notions behave in this context. In particular, we study functoriality in complete Segal spaces … The goal is to review the notion of a complete Segal space and how certain categorical notions behave in this context. In particular, we study functoriality in complete Segal spaces via fibrations. Then we use it to define limits and adjunctions in a complete Segal space. This is mostly expository and the focus is on examples and intuition.

An Elementary Approach to Truncations

2018-01-01
Nima Rasekh
We study truncated objects using elementary methods. Concretely, we use universes and the resulting natural number object to define internal truncation levels and prove they behave similar to standard truncated … We study truncated objects using elementary methods. Concretely, we use universes and the resulting natural number object to define internal truncation levels and prove they behave similar to standard truncated objects. Moreover, we take an elementary approach to localizations, giving various equivalent conditions that characterize localizations and constructing a localization out of a sub-universe of local objects via an internal right Kan extension. We then use this general approach, as well as an inductive approach, to construct truncation functors. We use the resulting truncation functors to prove classical results about truncations, such as Blakers-Massey theorem, in the elementary setting. We also give examples of non-presentable $(\infty, 1)$-categories where the elementary approach can be used to define and compute truncations. Finally, we turn around and use truncations to study elementary $(\infty, 1)$-toposes and show how they can help us better understand subobject classifiers and universes

Quasi-categories vs. Segal spaces: Cartesian edition

2021-08-20
Nima Rasekh
Abstract We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]), On bisimplicial … Abstract We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]), On bisimplicial spaces (due to deBrito [12]), On bisimplicial sets, On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.
View PDF Chat

Every Elementary Higher Topos has a Natural Number Object.

2018-09-05
Nima Rasekh
We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural … We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural number object out of it. Part of the proof involves showing that various definitions of natural number objects (Lawvere, Freyd and Peano) agree with each other in an elementary $(\infty,1)$-topos. As part of this effort we also study the internal object of contractibility in $(\infty,1)$-categories, which is of independent interest. Finally, we discuss various applications of natural number objects. In particular, we use it to define internal sequential colimits in an elementary $(\infty,1)$-topos.

A Theory of Elementary Higher Toposes

2018-01-01
Nima Rasekh
We define an elementary $\infty$-topos that simultaneously generalizes an elementary topos and Grothendieck $\infty$-topos. We then prove it satisfies the expected topos theoretic properties, such as descent, local Cartesian closure, … We define an elementary $\infty$-topos that simultaneously generalizes an elementary topos and Grothendieck $\infty$-topos. We then prove it satisfies the expected topos theoretic properties, such as descent, local Cartesian closure, locality and classification of univalent morphisms, generalizing results by Lurie and Gepner-Kock. We also define $\infty$-logical functors and show the resulting $\infty$-category is closed under limits and filtered colimits, generalizing the analogous result for elementary toposes and Grothendieck $\infty$-toposes. Moreover, we give an alternative characterization of elementary $\infty$-toposes and their $\infty$-logical functors via their ind-completions. Finally we generalize these results by discussing the case of elementary (n,1)-toposes and give various examples and non-examples.
View PDF Chat

Every Elementary Higher Topos has a Natural Number Object

2018-01-01
Nima Rasekh
We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural … We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural number object out of it. Part of the proof involves showing that various definitions of natural number objects (Lawvere, Freyd and Peano) agree with each other in an elementary $(\infty,1)$-topos. As part of this effort we also study the internal object of contractibility in $(\infty,1)$-categories, which is of independent interest. Finally, we discuss various applications of natural number objects. In particular, we use it to define internal sequential colimits in an elementary $(\infty,1)$-topos.

RGB image-based data analysis via discrete Morse theory and persistent homology

2018-01-01
Chuan Du , Christopher Szul , Adarsh Manawa +3 more
Understanding and comparing images for the purposes of data analysis is currently a very computationally demanding task. A group at Australian National University (ANU) recently developed open-source code that can … Understanding and comparing images for the purposes of data analysis is currently a very computationally demanding task. A group at Australian National University (ANU) recently developed open-source code that can detect fundamental topological features of a grayscale image in a computationally feasible manner. This is made possible by the fact that computers store grayscale images as cubical cellular complexes. These complexes can be studied using the techniques of discrete Morse theory. We expand the functionality of the ANU code by introducing methods and software for analyzing images encoded in red, green, and blue (RGB), because this image encoding is very popular for publicly available data. Our methods allow the extraction of key topological information from RGB images via informative persistence diagrams by introducing novel methods for transforming RGB-to-grayscale. This paradigm allows us to perform data analysis directly on RGB images representing water scarcity variability as well as crime variability. We introduce software enabling a a user to predict future image properties, towards the eventual aim of more rapid image-based data behavior prediction.
View PDF Chat

A Model for the Higher Category of Higher Categories

2018-05-10
Nima Rasekh
We use fibrations to construct four complete Segal spaces: simplicial spaces, Segal spaces and complete Segal spaces and spaces. Each one comes with a universal fibration that classifies Reedy left … We use fibrations to construct four complete Segal spaces: simplicial spaces, Segal spaces and complete Segal spaces and spaces. Each one comes with a universal fibration that classifies Reedy left fibrations, Segal coCartesian fibrations, coCartesian fibrations and left fibrations.

Thom spectra, higher $THH$ and tensors in $\infty$-categories.

2019-11-11
Nima Rasekh , Bruno Stonek , Gabriel Valenzuela
Let $f:G\to \mathrm{Pic}(R)$ be a map of $E_\infty$-groups, where $\mathrm{Pic}(R)$ denotes the Picard space of an $E_\infty$-ring spectrum $R$. We determine the tensor $X\otimes_R Mf$ of the Thom $E_\infty$-$R$-algebra $Mf$ … Let $f:G\to \mathrm{Pic}(R)$ be a map of $E_\infty$-groups, where $\mathrm{Pic}(R)$ denotes the Picard space of an $E_\infty$-ring spectrum $R$. We determine the tensor $X\otimes_R Mf$ of the Thom $E_\infty$-$R$-algebra $Mf$ with a space $X$; when $X$ is the circle, the tensor with $X$ is topological Hochschild homology over $R$. We use the theory of localizations of $\infty$-categories as a technical tool: we contribute to this theory an $\infty$-categorical analogue of Day's reflection theorem about closed symmetric monoidal structures on localizations, and we prove that for a smashing localization $L$ of the $\infty$-category of presentable $\infty$-categories, the free $L$-local presentable $\infty$-category on a small simplicial set $K$ is given by presheaves on $K$ valued on the $L$-localization of the $\infty$-category of spaces. If $X$ is a pointed space, a map $g: A\to B$ of $E_\infty$-ring spectra satisfies $X$-base change if $X\otimes B$ is the pushout of $A\to X\otimes A$ along $g$. Building on a result of Mathew, we prove that if $g$ is etale then it satisfies $X$-base change provided $X$ is connected. We also prove that $g$ satisfies $X$-base change provided the multiplication map of $B$ is an equivalence. Finally, we prove that, under some hypotheses, the Thom isomorphism of Mahowald cannot be an instance of $S^0$-base change.

Filter quotients and non-presentable (∞,1)-toposes

2021-04-15
Nima Rasekh
We define filter quotients of $(\infty,1)$-categories and prove that filter quotients preserve the structure of an elementary $(\infty,1)$-topos and in particular lift the filter quotient of the underlying elementary topos. … We define filter quotients of $(\infty,1)$-categories and prove that filter quotients preserve the structure of an elementary $(\infty,1)$-topos and in particular lift the filter quotient of the underlying elementary topos. We then specialize to the case of filter products of $(\infty,1)$-categories and prove a characterization theorem for equivalences in a filter product. Then we use filter products to construct a large class of elementary $(\infty,1)$-toposes that are not Grothendieck $(\infty,1)$-toposes. Moreover, we give one detailed example for the interested reader who would like to see how we can construct such an $(\infty,1)$-category, but would prefer to avoid the technicalities regarding filters.
View PDF Chat

Cartesian Fibrations and Representability

2017-01-01
Nima Rasekh
We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in … We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in $\infty$-category theory. In particular, we give a construction of representable Cartesian fibrations using over-categories and prove the Yoneda lemma for representable Cartesian fibration, which generalizes the established Yoneda lemma for right fibrations. We then use the theory of Cartesian fibrations to study complete Segal objects internal to an $\infty$-category. Concretely, we prove the {\it fundamental theorem of complete Segal objects}, which characterizes equivalences of complete Segal objects. Finally we give two application of the results. First, we present a method to construct Segal objects and second we study the representability of the universal coCartesian fibration.
View PDF Chat

Yoneda Lemma for Simplicial Spaces

2017-01-01
Nima Rasekh
We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and and studying its associated model structure, the covariant model structure. … We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and and studying its associated model structure, the covariant model structure. In particular, we prove a recognition principle for covariant equivalences over an arbitrary simplicial space and invariance of the covariant model structure with respect to complete Segal space equivalences.
View PDF Chat

Thom spectra, higher THH and tensors in ∞–categories

2022-10-10
Nima Rasekh , Bruno Stonek , Gabriel Valenzuela
Let f : G → Pic(R) be a map of E ∞ -groups, where Pic(R) denotes the Picard space of an E ∞ -ring spectrum R. We determine the tensor … Let f : G → Pic(R) be a map of E ∞ -groups, where Pic(R) denotes the Picard space of an E ∞ -ring spectrum R. We determine the tensor X ⊗ R M f of the Thom E ∞ -R-algebra M f with a space X; when X is the circle, the tensor with X is topological Hochschild homology over R. We use the theory of localizations of ∞-categories as a technical tool: we contribute to this theory an ∞-categorical analogue of Day's reflection theorem about closed symmetric monoidal structures on localizations, and we prove that for a smashing localization L of the ∞-category of presentable ∞-categories, the free L-local presentable ∞-category on a small simplicial set K is given by presheaves on K valued on the L-localization of the ∞-category of spaces.If X is a pointed space, a map g :Building on a result of Mathew, we prove that if g is étale then it satisfies X-base change provided X is connected.We also prove that g satisfies X-base change provided the multiplication map of B is an equivalence.Finally, we prove that, under some hypotheses, the Thom isomorphism of Mahowald cannot be an instance of S 0 -base change.
View PDF Chat

Yoneda Lemma for Elementary Higher Toposes

2018-01-01
Nima Rasekh
We prove the Yoneda lemma inside an elementary higher topos, generalizing the Yonda lemma for spaces. We prove the Yoneda lemma inside an elementary higher topos, generalizing the Yonda lemma for spaces.

Truncations and Blakers-Massey in an Elementary Higher Topos

2018-12-26
Nima Rasekh
We study truncated and connected objects in an elementary higher topos. In particular, we show they have the same behavior as in spaces, construct a universal truncation functor using natural … We study truncated and connected objects in an elementary higher topos. In particular, we show they have the same behavior as in spaces, construct a universal truncation functor using natural number objects and show all modalities, and in particular connected maps, satisfy the statement of the Blakers-Massey Theorem.

Univalence in Higher Category Theory

2021-01-01
Nima Rasekh
Univalence was first defined in the setting of homotopy type theory by Voevodsky, who also (along with Kapulkin and Lumsdaine) adapted it to a model categorical setting, which was subsequently … Univalence was first defined in the setting of homotopy type theory by Voevodsky, who also (along with Kapulkin and Lumsdaine) adapted it to a model categorical setting, which was subsequently generalized to locally Cartesian closed presentable $\infty$-categories by Gepner and Kock. These definitions were used to characterize various $\infty$-categories as models of type theories. We give a definition for univalent morphisms in finitely complete $\infty$-categories that generalizes the aforementioned definitions and completely focuses on the $\infty$-categorical aspects, characterizing it via representability of certain functors, which should remind the reader of concepts such as adjunctions or limits. We then prove that in a locally Cartesian closed $\infty$-category (that is not necessarily presentable) univalence of a morphism is equivalent to the completeness of a certain Segal object we construct out of the morphism, characterizing univalence via internal $\infty$-categories, which had been considered in a strict setting by Stenzel. We use these results to study the connection between univalence and elementary topos theory. We also study univalent morphisms in the category of groups, the $\infty$-category of $\infty$-categories, and pointed $\infty$-categories.

Constructing coproducts in locally Cartesian closed $\infty$-categories

2023-01-01
Jonas Frey , Nima Rasekh
We prove that every locally Cartesian closed ∞-category with a subobject classifier has a strict initial object and disjoint and universal binary coproducts. We prove that every locally Cartesian closed ∞-category with a subobject classifier has a strict initial object and disjoint and universal binary coproducts.
View PDF Chat

Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

2021-08-13
Nima Rasekh
For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\times\Delta$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant … For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\times\Delta$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized $\mathcal{D}$-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on $\mathcal{D}$, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations and weak equivalences in this model structure and in particular obtain a Yoneda lemma. We apply this general framework to study Cartesian fibrations of $(\infty,n)$-categories, for models of $(\infty,n)$-categories that arise via simplicial presheaves, such as $n$-fold complete Segal spaces. This, in particular, results in the Yoneda lemma and Grothendieck construction for Cartesian fibrations of $(\infty,n)$-categories.

Shadows are Bicategorical Traces

2021-01-01
Kathryn Hess , Nima Rasekh
The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using Berman's extension of THH to … The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using Berman's extension of THH to bicategories, we prove that there is an equivalence between functors out of Hochschild homology of a bicategory and shadows on that bicategory. As part of that proof we give a computational description of pseudo-functors out of the truncated simplex category and a variety thereof, which can be of independent interest. Building on this result we present three applications: (1) We provide a new, conceptual proof that shadows, and as a result the Euler characteristic and trace introduced by Campbell and Ponto, are Morita invariant. (2) We strengthen this result by using an explicit computation of the Hochschild homology of the free adjunction bicategory to show that the construction of the Euler characteristic is homotopically unique. (3) We generalize the construction of $\mathscr{V}$-enriched Hochschild homology, where $\mathscr{V}$ is a presentably symmetric monoidal $\infty$-category, to bimodules, and prove it gives us a shadow.
View PDF Chat

A homotopy coherent nerve for (∞,n)-categories

2024-01-24
Lyne Moser , Nima Rasekh , Martina Rovelli
In the case of (∞,1)-categories, the homotopy coherent nerve gives a right Quillen equivalence between the models of simplicially enriched categories and of quasi-categories. This shows that homotopy coherent diagrams … In the case of (∞,1)-categories, the homotopy coherent nerve gives a right Quillen equivalence between the models of simplicially enriched categories and of quasi-categories. This shows that homotopy coherent diagrams of (∞,1)-categories can equivalently be defined as functors of quasi-categories or as simplicially enriched functors out of the homotopy coherent categorifications. In this paper, we construct a homotopy coherent nerve for (∞,n)-categories. We show that it realizes a right Quillen equivalence between the models of categories strictly enriched in (∞,n−1)-categories and of Segal category objects in (∞,n−1)-categories. This similarly enables us to define homotopy coherent diagrams of (∞,n)-categories equivalently as functors of Segal category objects or as strictly enriched functors out of the homotopy coherent categorifications.
View PDF Chat

The cotangent complex and Thom spectra

2020-12-01
Nima Rasekh , Bruno Stonek
Abstract The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $$E_\infty … Abstract The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -ring spectra in various ways. In this work we first establish, in the context of $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> -categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -ring spectrum and $$\mathrm {Pic}(R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Pic</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> denote its Picard $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -group. Let Mf denote the Thom $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> - R -algebra of a map of $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -groups $$f:G\rightarrow \mathrm {Pic}(R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>G</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Pic</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ; examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of $$R\rightarrow Mf$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>→</mml:mo> <mml:mi>M</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> is equivalent to the smash product of Mf and the connective spectrum associated to G .
View PDF Chat

Shadows are Bicategorical Traces

2021-09-05
Kathryn Hess , Nima Rasekh
The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using Berman's extension of THH to … The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using Berman's extension of THH to bicategories, we prove that there is an equivalence between functors out of THH of a bicategory and shadows on that bicategory. As an application we provide a new, conceptual proof that shadows are Morita invariant.

Constructing Coproducts in locally Cartesian closed $\infty$-Categories

2021-01-01
Jonas Frey , Nima Rasekh
We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts. We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.
View PDF Chat

Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

2021-01-01
Nima Rasekh
For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\times\Delta$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant … For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\times\Delta$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized $\mathcal{D}$-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on $\mathcal{D}$, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations and weak equivalences in this model structure and in particular obtain a Yoneda lemma. We apply this general framework to study Cartesian fibrations of $(\infty,n)$-categories, for models of $(\infty,n)$-categories that arise via simplicial presheaves, such as $n$-fold complete Segal spaces. This, in particular, results in the Yoneda lemma and Grothendieck construction for Cartesian fibrations of $(\infty,n)$-categories.

The cotangent complex and Thom spectra

2020-01-01
Nima Rasekh , Bruno Stonek
The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $E_\infty$-ring spectra … The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $E_\infty$-ring spectra in various ways. In this work we first establish, in the context of $\infty$-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of $E_\infty$-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let $R$ be an $E_\infty$-ring spectrum and $\mathrm{Pic}(R)$ denote its Picard $E_\infty$-group. Let $Mf$ denote the Thom $E_\infty$-$R$-algebra of a map of $E_\infty$-groups $f:G\to \mathrm{Pic}(R)$; examples of $Mf$ are given by various flavors of cobordism spectra. We prove that the cotangent complex of $R\to Mf$ is equivalent to the smash product of $Mf$ and the connective spectrum associated to $G$.
View PDF Chat

A homotopy coherent nerve for $(\infty,n)$-categories

2022-01-01
Lyne Moser , Nima Rasekh , Martina Rovelli
We study a homotopy coherent nerve for categories strictly enriched over $(\infty,n-1)$-categories, and show it realizes a right Quillen equivalence when valued in Segal category objects in $(\infty,n-1)$-categories. The motivating … We study a homotopy coherent nerve for categories strictly enriched over $(\infty,n-1)$-categories, and show it realizes a right Quillen equivalence when valued in Segal category objects in $(\infty,n-1)$-categories. The motivating feature for this specific nerve construction is that it satisfies injective fibrancy, a technical assumption which we plan to exploit in future work towards the study of $(\infty,n)$-limits.
View PDF Chat

A Model for the Higher Category of Higher Categories

2018-01-01
Nima Rasekh
We use fibrations of complete Segal spaces to construct four complete Segal spaces: Reedy fibrant simplicial spaces, Segal spaces, complete Segal spaces, and spaces. Moreover, we show each one comes … We use fibrations of complete Segal spaces to construct four complete Segal spaces: Reedy fibrant simplicial spaces, Segal spaces, complete Segal spaces, and spaces. Moreover, we show each one comes with a universal fibration that classifies Reedy left fibrations, Segal coCartesian fibrations, coCartesian fibrations and left fibrations and prove these are representable fibrations. Finally, we use equivalences between quasi-categories and complete Segal spaces to present analogous constructions using fibrations of quasi-categories.

Complete Segal Objects

2018-01-01
Nima Rasekh
We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete … We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete Segal objects. Finally we use Segal objects to define univalence in a locally Cartesian closed category that is not presentable and generalize some previous results to the non-presentable setting.

Cartesian Fibrations of Complete Segal Spaces

2021-01-01
Nima Rasekh
Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in $(\infty,1)$-category theory to study presheaves valued in $(\infty,1)$-categories. In this work we define … Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in $(\infty,1)$-category theory to study presheaves valued in $(\infty,1)$-categories. In this work we define and study fibrations modeling presheaves valued in simplicial spaces and their localizations. This includes defining a model structure for these fibrations and giving effective tools to recognize its fibrations and weak equivalences. This in particular gives us a new method to construct Cartesian fibrations via complete Segal spaces. In addition to that, it allows us to define and study fibrations modeling presheaves of Segal spaces.
View PDF Chat

An $(\infty,n)$-categorical straightening-unstraightening construction

2023-01-01
Lyne Moser , Nima Rasekh , Martina Rovelli
We provide an $(\infty,n)$-categorical version of the straightening-unstraightening construction, asserting an equivalence between the $(\infty,n)$-category of double $(\infty,n-1)$-right fibrations over an $(\infty,n)$-category $\mathcal{C}$ and that of the $(\infty,n)$-functors from $\mathcal{C}$ … We provide an $(\infty,n)$-categorical version of the straightening-unstraightening construction, asserting an equivalence between the $(\infty,n)$-category of double $(\infty,n-1)$-right fibrations over an $(\infty,n)$-category $\mathcal{C}$ and that of the $(\infty,n)$-functors from $\mathcal{C}$ valued in $(\infty,n-1)$-categories. We realize this in the form of a Quillen equivalence between appropriate model structures; on the one hand, a model structure for double $(\infty,n-1)$-right fibrations over a generic precategory object $W$ in $(\infty,n-1)$-categories and, on the other hand, a model structure for $(\infty,n)$-functors from its homotopy coherent categorification $\mathfrak{C} W$ valued in $(\infty,n-1)$-categories.

Univalent Double Categories

2023-01-01
Niels van der Weide , Nima Rasekh , Benedikt Ahrens +1 more
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of … Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also "morphisms", which capture how different objects interact with each other. Category theory has found many applications in mathematics and in computer science, for example in functional programming. Double categories are a natural generalization of categories which incorporate the data of two separate classes of morphisms, allowing a more nuanced representation of relationships and interactions between objects. Similar to category theory, double categories have been successfully applied to various situations in mathematics and computer science, in which objects naturally exhibit two types of morphisms. Examples include categories themselves, but also lenses, petri nets, and spans. While categories have already been formalized in a variety of proof assistants, double categories have received far less attention. In this paper we remedy this situation by presenting a formalization of double categories via the proof assistant Coq, relying on the Coq UniMath library. As part of this work we present two equivalent formalizations of the definition of a double category, an unfolded explicit definition and a second definition which exhibits excellent formal properties via 2-sided displayed categories. As an application of the formal approach we establish a notion of univalent double category along with a univalence principle: equivalences of univalent double categories coincide with their identities
View PDF Chat

$(\infty,n)$-Limits I: Definition and first consistency results

2023-01-01
Lyne Moser , Nima Rasekh , Martina Rovelli
We define limits for diagrams valued in an $(\infty,n)$-category. As a model of $(\infty,n)$-categories, we use complete Segal objects in $(\infty,n-1)$-categories. We show that this definition is compatible with the … We define limits for diagrams valued in an $(\infty,n)$-category. As a model of $(\infty,n)$-categories, we use complete Segal objects in $(\infty,n-1)$-categories. We show that this definition is compatible with the existing notion of homotopy 2-limits for 2-categories, with the existing notion of $(\infty,1)$-limits for $(\infty,1)$-categories, and with itself across different values of $n$.

Univalent Double Categories

2024-01-09
Niels van der Weide , Nima Rasekh , Benedikt Ahrens +1 more
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of … Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also "morphisms", which capture how different objects interact with each other. Category theory has found many applications in mathematics and in computer science, for example in functional programming.
View PDF Chat

Insights From Univalent Foundations: A Case Study Using Double Categories

2024-02-07
Nima Rasekh , Niels van der Weide , Benedikt Ahrens +1 more
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional … Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional programming, and semantics. Certain objects naturally exhibit two classes of morphisms, leading to the concept of a double category, which has found applications in computing science (e.g., ornaments, profunctor optics, denotational semantics). The emergence of diverse categorical structures motivated a unified framework for category theory. However, unlike other mathematical objects, classification of categorical structures faces challenges due to various relevant equivalences. This poses significant challenges when pursuing the formalization of categories and restricts the applicability of powerful techniques, such as transport along equivalences. This work contends that univalent foundations offers a suitable framework for classifying different categorical structures based on desired notions of equivalences, and remedy the challenges when formalizing categories. The richer notion of equality in univalent foundations makes the equivalence of a categorical structure an inherent part of its structure. We concretely apply this analysis to double categorical structures. We characterize and formalize various definitions in Coq UniMath, including (pseudo) double categories and double bicategories, up to chosen equivalences. We also establish univalence principles, making chosen equivalences part of the double categorical structure, analyzing strict double setcategories (invariant under isomorphisms), pseudo double setcategories (invariant under isomorphisms), univalent pseudo double categories (invariant under vertical equivalences) and univalent double bicategories (invariant under gregarious equivalences).
View PDF Chat

Insights From Univalent Foundations: A Case Study Using Double Categories

2024-02-07
Nima Rasekh , Niels van der Weide , Benedikt Ahrens +1 more
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional … Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional programming, and semantics. Certain objects naturally exhibit two classes of morphisms, leading to the concept of a double category, which has found applications in computing science (e.g., ornaments, profunctor optics, denotational semantics). The emergence of diverse categorical structures motivated a unified framework for category theory. However, unlike other mathematical objects, classification of categorical structures faces challenges due to various relevant equivalences. This poses significant challenges when pursuing the formalization of categories and restricts the applicability of powerful techniques, such as transport along equivalences. This work contends that univalent foundations offers a suitable framework for classifying different categorical structures based on desired notions of equivalences, and remedy the challenges when formalizing categories. The richer notion of equality in univalent foundations makes the equivalence of a categorical structure an inherent part of its structure. We concretely apply this analysis to double categorical structures. We characterize and formalize various definitions in Coq UniMath, including (pseudo) double categories and double bicategories, up to chosen equivalences. We also establish univalence principles, making chosen equivalences part of the double categorical structure, analyzing strict double setcategories (invariant under isomorphisms), pseudo double setcategories (invariant under isomorphisms), univalent pseudo double categories (invariant under vertical equivalences) and univalent double bicategories (invariant under gregarious equivalences).
View PDF Chat

A homotopy coherent nerve for (∞,n)-categories

2024-01-24
Lyne Moser , Nima Rasekh , Martina Rovelli
In the case of (∞,1)-categories, the homotopy coherent nerve gives a right Quillen equivalence between the models of simplicially enriched categories and of quasi-categories. This shows that homotopy coherent diagrams … In the case of (∞,1)-categories, the homotopy coherent nerve gives a right Quillen equivalence between the models of simplicially enriched categories and of quasi-categories. This shows that homotopy coherent diagrams of (∞,1)-categories can equivalently be defined as functors of quasi-categories or as simplicially enriched functors out of the homotopy coherent categorifications. In this paper, we construct a homotopy coherent nerve for (∞,n)-categories. We show that it realizes a right Quillen equivalence between the models of categories strictly enriched in (∞,n−1)-categories and of Segal category objects in (∞,n−1)-categories. This similarly enables us to define homotopy coherent diagrams of (∞,n)-categories equivalently as functors of Segal category objects or as strictly enriched functors out of the homotopy coherent categorifications.
View PDF Chat

Univalent Double Categories

2024-01-09
Niels van der Weide , Nima Rasekh , Benedikt Ahrens +1 more
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of … Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also "morphisms", which capture how different objects interact with each other. Category theory has found many applications in mathematics and in computer science, for example in functional programming.
View PDF Chat

Yoneda Lemma for Simplicial Spaces

2023-07-11
Nima Rasekh
Abstract We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and studying their associated model structure, the covariant model structure … Abstract We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and studying their associated model structure, the covariant model structure . In particular, we prove a recognition principle for covariant equivalences over an arbitrary simplicial space and invariance of the covariant model structure with respect to complete Segal space equivalences.

Cartesian Fibrations of Complete Segal Spaces

2023-05-21
Nima Rasekh
Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in (∞, 1)-category theory to study presheaves valued in (∞, 1)-categories.In this work we … Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in (∞, 1)-category theory to study presheaves valued in (∞, 1)-categories.In this work we define and study fibrations modeling presheaves valued in simplicial spaces and their localizations.This includes defining a model structure for these fibrations and giving effective tools to recognize its fibrations and weak equivalences.This in particular gives us a new method to construct Cartesian fibrations via complete Segal spaces.In addition to that, it allows us to define and study fibrations modeling presheaves of Segal spaces.
View PDF Chat

Constructing coproducts in locally Cartesian closed $\infty$-categories

2023-01-01
Jonas Frey , Nima Rasekh
We prove that every locally Cartesian closed ∞-category with a subobject classifier has a strict initial object and disjoint and universal binary coproducts. We prove that every locally Cartesian closed ∞-category with a subobject classifier has a strict initial object and disjoint and universal binary coproducts.
View PDF Chat

An $(\infty,n)$-categorical straightening-unstraightening construction

2023-01-01
Lyne Moser , Nima Rasekh , Martina Rovelli
We provide an $(\infty,n)$-categorical version of the straightening-unstraightening construction, asserting an equivalence between the $(\infty,n)$-category of double $(\infty,n-1)$-right fibrations over an $(\infty,n)$-category $\mathcal{C}$ and that of the $(\infty,n)$-functors from $\mathcal{C}$ … We provide an $(\infty,n)$-categorical version of the straightening-unstraightening construction, asserting an equivalence between the $(\infty,n)$-category of double $(\infty,n-1)$-right fibrations over an $(\infty,n)$-category $\mathcal{C}$ and that of the $(\infty,n)$-functors from $\mathcal{C}$ valued in $(\infty,n-1)$-categories. We realize this in the form of a Quillen equivalence between appropriate model structures; on the one hand, a model structure for double $(\infty,n-1)$-right fibrations over a generic precategory object $W$ in $(\infty,n-1)$-categories and, on the other hand, a model structure for $(\infty,n)$-functors from its homotopy coherent categorification $\mathfrak{C} W$ valued in $(\infty,n-1)$-categories.

Univalent Double Categories

2023-01-01
Niels van der Weide , Nima Rasekh , Benedikt Ahrens +1 more
Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of … Category theory is a branch of mathematics that provides a formal framework for understanding the relationship between mathematical structures. To this end, a category not only incorporates the data of the desired objects, but also "morphisms", which capture how different objects interact with each other. Category theory has found many applications in mathematics and in computer science, for example in functional programming. Double categories are a natural generalization of categories which incorporate the data of two separate classes of morphisms, allowing a more nuanced representation of relationships and interactions between objects. Similar to category theory, double categories have been successfully applied to various situations in mathematics and computer science, in which objects naturally exhibit two types of morphisms. Examples include categories themselves, but also lenses, petri nets, and spans. While categories have already been formalized in a variety of proof assistants, double categories have received far less attention. In this paper we remedy this situation by presenting a formalization of double categories via the proof assistant Coq, relying on the Coq UniMath library. As part of this work we present two equivalent formalizations of the definition of a double category, an unfolded explicit definition and a second definition which exhibits excellent formal properties via 2-sided displayed categories. As an application of the formal approach we establish a notion of univalent double category along with a univalence principle: equivalences of univalent double categories coincide with their identities
View PDF Chat

$(\infty,n)$-Limits I: Definition and first consistency results

2023-01-01
Lyne Moser , Nima Rasekh , Martina Rovelli
We define limits for diagrams valued in an $(\infty,n)$-category. As a model of $(\infty,n)$-categories, we use complete Segal objects in $(\infty,n-1)$-categories. We show that this definition is compatible with the … We define limits for diagrams valued in an $(\infty,n)$-category. As a model of $(\infty,n)$-categories, we use complete Segal objects in $(\infty,n-1)$-categories. We show that this definition is compatible with the existing notion of homotopy 2-limits for 2-categories, with the existing notion of $(\infty,1)$-limits for $(\infty,1)$-categories, and with itself across different values of $n$.

Thom spectra, higher THH and tensors in ∞–categories

2022-10-10
Nima Rasekh , Bruno Stonek , Gabriel Valenzuela
Let f : G → Pic(R) be a map of E ∞ -groups, where Pic(R) denotes the Picard space of an E ∞ -ring spectrum R. We determine the tensor … Let f : G → Pic(R) be a map of E ∞ -groups, where Pic(R) denotes the Picard space of an E ∞ -ring spectrum R. We determine the tensor X ⊗ R M f of the Thom E ∞ -R-algebra M f with a space X; when X is the circle, the tensor with X is topological Hochschild homology over R. We use the theory of localizations of ∞-categories as a technical tool: we contribute to this theory an ∞-categorical analogue of Day's reflection theorem about closed symmetric monoidal structures on localizations, and we prove that for a smashing localization L of the ∞-category of presentable ∞-categories, the free L-local presentable ∞-category on a small simplicial set K is given by presheaves on K valued on the L-localization of the ∞-category of spaces.If X is a pointed space, a map g :Building on a result of Mathew, we prove that if g is étale then it satisfies X-base change provided X is connected.We also prove that g satisfies X-base change provided the multiplication map of B is an equivalence.Finally, we prove that, under some hypotheses, the Thom isomorphism of Mahowald cannot be an instance of S 0 -base change.
View PDF Chat

Cartesian fibrations and representability

2022-01-01
Nima Rasekh
We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in … We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in $\infty$-category theory. In particular, we give a construction of representable Cartesian fibrations using over-categories and prove the Yoneda lemma for representable Cartesian fibration, which generalizes the established Yoneda lemma for right fibrations. We then use the theory of Cartesian fibrations to study complete Segal objects internal to an $\infty$-category. Concretely, we prove the {\it fundamental theorem of complete Segal objects}, which characterizes equivalences of complete Segal objects. Finally we give two application of the results. First, we present a method to construct Segal objects and second we study the representability of the universal coCartesian fibration.
View PDF Chat

A homotopy coherent nerve for $(\infty,n)$-categories

2022-01-01
Lyne Moser , Nima Rasekh , Martina Rovelli
We study a homotopy coherent nerve for categories strictly enriched over $(\infty,n-1)$-categories, and show it realizes a right Quillen equivalence when valued in Segal category objects in $(\infty,n-1)$-categories. The motivating … We study a homotopy coherent nerve for categories strictly enriched over $(\infty,n-1)$-categories, and show it realizes a right Quillen equivalence when valued in Segal category objects in $(\infty,n-1)$-categories. The motivating feature for this specific nerve construction is that it satisfies injective fibrancy, a technical assumption which we plan to exploit in future work towards the study of $(\infty,n)$-limits.
View PDF Chat

Shadows are Bicategorical Traces

2021-09-05
Kathryn Hess , Nima Rasekh
The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using Berman's extension of THH to … The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using Berman's extension of THH to bicategories, we prove that there is an equivalence between functors out of THH of a bicategory and shadows on that bicategory. As an application we provide a new, conceptual proof that shadows are Morita invariant.

Quasi-categories vs. Segal spaces: Cartesian edition

2021-08-20
Nima Rasekh
Abstract We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]), On bisimplicial … Abstract We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]), On bisimplicial spaces (due to deBrito [12]), On bisimplicial sets, On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.
View PDF Chat

Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

2021-08-13
Nima Rasekh
For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\times\Delta$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant … For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\times\Delta$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized $\mathcal{D}$-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on $\mathcal{D}$, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations and weak equivalences in this model structure and in particular obtain a Yoneda lemma. We apply this general framework to study Cartesian fibrations of $(\infty,n)$-categories, for models of $(\infty,n)$-categories that arise via simplicial presheaves, such as $n$-fold complete Segal spaces. This, in particular, results in the Yoneda lemma and Grothendieck construction for Cartesian fibrations of $(\infty,n)$-categories.

Filter quotients and non-presentable (∞,1)-toposes

2021-04-15
Nima Rasekh
We define filter quotients of $(\infty,1)$-categories and prove that filter quotients preserve the structure of an elementary $(\infty,1)$-topos and in particular lift the filter quotient of the underlying elementary topos. … We define filter quotients of $(\infty,1)$-categories and prove that filter quotients preserve the structure of an elementary $(\infty,1)$-topos and in particular lift the filter quotient of the underlying elementary topos. We then specialize to the case of filter products of $(\infty,1)$-categories and prove a characterization theorem for equivalences in a filter product. Then we use filter products to construct a large class of elementary $(\infty,1)$-toposes that are not Grothendieck $(\infty,1)$-toposes. Moreover, we give one detailed example for the interested reader who would like to see how we can construct such an $(\infty,1)$-category, but would prefer to avoid the technicalities regarding filters.
View PDF Chat

Univalence in Higher Category Theory

2021-01-01
Nima Rasekh
Univalence was first defined in the setting of homotopy type theory by Voevodsky, who also (along with Kapulkin and Lumsdaine) adapted it to a model categorical setting, which was subsequently … Univalence was first defined in the setting of homotopy type theory by Voevodsky, who also (along with Kapulkin and Lumsdaine) adapted it to a model categorical setting, which was subsequently generalized to locally Cartesian closed presentable $\infty$-categories by Gepner and Kock. These definitions were used to characterize various $\infty$-categories as models of type theories. We give a definition for univalent morphisms in finitely complete $\infty$-categories that generalizes the aforementioned definitions and completely focuses on the $\infty$-categorical aspects, characterizing it via representability of certain functors, which should remind the reader of concepts such as adjunctions or limits. We then prove that in a locally Cartesian closed $\infty$-category (that is not necessarily presentable) univalence of a morphism is equivalent to the completeness of a certain Segal object we construct out of the morphism, characterizing univalence via internal $\infty$-categories, which had been considered in a strict setting by Stenzel. We use these results to study the connection between univalence and elementary topos theory. We also study univalent morphisms in the category of groups, the $\infty$-category of $\infty$-categories, and pointed $\infty$-categories.

Constructing Coproducts in locally Cartesian closed $\infty$-Categories

2021-01-01
Jonas Frey , Nima Rasekh
We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts. We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.
View PDF Chat

Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

2021-01-01
Nima Rasekh
For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\times\Delta$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant … For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\times\Delta$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized $\mathcal{D}$-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on $\mathcal{D}$, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations and weak equivalences in this model structure and in particular obtain a Yoneda lemma. We apply this general framework to study Cartesian fibrations of $(\infty,n)$-categories, for models of $(\infty,n)$-categories that arise via simplicial presheaves, such as $n$-fold complete Segal spaces. This, in particular, results in the Yoneda lemma and Grothendieck construction for Cartesian fibrations of $(\infty,n)$-categories.

Shadows are Bicategorical Traces

2021-01-01
Kathryn Hess , Nima Rasekh
The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using Berman's extension of THH to … The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using Berman's extension of THH to bicategories, we prove that there is an equivalence between functors out of Hochschild homology of a bicategory and shadows on that bicategory. As part of that proof we give a computational description of pseudo-functors out of the truncated simplex category and a variety thereof, which can be of independent interest. Building on this result we present three applications: (1) We provide a new, conceptual proof that shadows, and as a result the Euler characteristic and trace introduced by Campbell and Ponto, are Morita invariant. (2) We strengthen this result by using an explicit computation of the Hochschild homology of the free adjunction bicategory to show that the construction of the Euler characteristic is homotopically unique. (3) We generalize the construction of $\mathscr{V}$-enriched Hochschild homology, where $\mathscr{V}$ is a presentably symmetric monoidal $\infty$-category, to bimodules, and prove it gives us a shadow.
View PDF Chat

Cartesian Fibrations of Complete Segal Spaces

2021-01-01
Nima Rasekh
Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in $(\infty,1)$-category theory to study presheaves valued in $(\infty,1)$-categories. In this work we define … Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in $(\infty,1)$-category theory to study presheaves valued in $(\infty,1)$-categories. In this work we define and study fibrations modeling presheaves valued in simplicial spaces and their localizations. This includes defining a model structure for these fibrations and giving effective tools to recognize its fibrations and weak equivalences. This in particular gives us a new method to construct Cartesian fibrations via complete Segal spaces. In addition to that, it allows us to define and study fibrations modeling presheaves of Segal spaces.
View PDF Chat

The cotangent complex and Thom spectra

2020-12-01
Nima Rasekh , Bruno Stonek
Abstract The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $$E_\infty … Abstract The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -ring spectra in various ways. In this work we first establish, in the context of $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> -categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -ring spectrum and $$\mathrm {Pic}(R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Pic</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> denote its Picard $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -group. Let Mf denote the Thom $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> - R -algebra of a map of $$E_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> -groups $$f:G\rightarrow \mathrm {Pic}(R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>G</mml:mi> <mml:mo>→</mml:mo> <mml:mi>Pic</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ; examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of $$R\rightarrow Mf$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>→</mml:mo> <mml:mi>M</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> is equivalent to the smash product of Mf and the connective spectrum associated to G .
View PDF Chat

The cotangent complex and Thom spectra

2020-01-01
Nima Rasekh , Bruno Stonek
The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $E_\infty$-ring spectra … The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of $E_\infty$-ring spectra in various ways. In this work we first establish, in the context of $\infty$-categories and using Goodwillie's calculus of functors, that various definitions of the cotangent complex of a map of $E_\infty$-ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let $R$ be an $E_\infty$-ring spectrum and $\mathrm{Pic}(R)$ denote its Picard $E_\infty$-group. Let $Mf$ denote the Thom $E_\infty$-$R$-algebra of a map of $E_\infty$-groups $f:G\to \mathrm{Pic}(R)$; examples of $Mf$ are given by various flavors of cobordism spectra. We prove that the cotangent complex of $R\to Mf$ is equivalent to the smash product of $Mf$ and the connective spectrum associated to $G$.
View PDF Chat

Thom spectra, higher $THH$ and tensors in $\infty$-categories.

2019-11-11
Nima Rasekh , Bruno Stonek , Gabriel Valenzuela
Let $f:G\to \mathrm{Pic}(R)$ be a map of $E_\infty$-groups, where $\mathrm{Pic}(R)$ denotes the Picard space of an $E_\infty$-ring spectrum $R$. We determine the tensor $X\otimes_R Mf$ of the Thom $E_\infty$-$R$-algebra $Mf$ … Let $f:G\to \mathrm{Pic}(R)$ be a map of $E_\infty$-groups, where $\mathrm{Pic}(R)$ denotes the Picard space of an $E_\infty$-ring spectrum $R$. We determine the tensor $X\otimes_R Mf$ of the Thom $E_\infty$-$R$-algebra $Mf$ with a space $X$; when $X$ is the circle, the tensor with $X$ is topological Hochschild homology over $R$. We use the theory of localizations of $\infty$-categories as a technical tool: we contribute to this theory an $\infty$-categorical analogue of Day's reflection theorem about closed symmetric monoidal structures on localizations, and we prove that for a smashing localization $L$ of the $\infty$-category of presentable $\infty$-categories, the free $L$-local presentable $\infty$-category on a small simplicial set $K$ is given by presheaves on $K$ valued on the $L$-localization of the $\infty$-category of spaces. If $X$ is a pointed space, a map $g: A\to B$ of $E_\infty$-ring spectra satisfies $X$-base change if $X\otimes B$ is the pushout of $A\to X\otimes A$ along $g$. Building on a result of Mathew, we prove that if $g$ is etale then it satisfies $X$-base change provided $X$ is connected. We also prove that $g$ satisfies $X$-base change provided the multiplication map of $B$ is an equivalence. Finally, we prove that, under some hypotheses, the Thom isomorphism of Mahowald cannot be an instance of $S^0$-base change.

Truncations and Blakers-Massey in an Elementary Higher Topos

2018-12-26
Nima Rasekh
We study truncated and connected objects in an elementary higher topos. In particular, we show they have the same behavior as in spaces, construct a universal truncation functor using natural … We study truncated and connected objects in an elementary higher topos. In particular, we show they have the same behavior as in spaces, construct a universal truncation functor using natural number objects and show all modalities, and in particular connected maps, satisfy the statement of the Blakers-Massey Theorem.

Every Elementary Higher Topos has a Natural Number Object.

2018-09-05
Nima Rasekh
We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural … We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural number object out of it. Part of the proof involves showing that various definitions of natural number objects (Lawvere, Freyd and Peano) agree with each other in an elementary $(\infty,1)$-topos. As part of this effort we also study the internal object of contractibility in $(\infty,1)$-categories, which is of independent interest. Finally, we discuss various applications of natural number objects. In particular, we use it to define internal sequential colimits in an elementary $(\infty,1)$-topos.

A theory of elementary higher toposes

2018-06-29
Nima Rasekh
We define an elementary higher topos that simultaneously generalizes an elementary topos and higher topos. Then we show it satisfies classical topos theoretic properties, such being locally Cartesian closed and … We define an elementary higher topos that simultaneously generalizes an elementary topos and higher topos. Then we show it satisfies classical topos theoretic properties, such being locally Cartesian closed and descent. Finally we show we can classify univalent maps in an elementary higher topos.

A Model for the Higher Category of Higher Categories

2018-05-10
Nima Rasekh
We use fibrations to construct four complete Segal spaces: simplicial spaces, Segal spaces and complete Segal spaces and spaces. Each one comes with a universal fibration that classifies Reedy left … We use fibrations to construct four complete Segal spaces: simplicial spaces, Segal spaces and complete Segal spaces and spaces. Each one comes with a universal fibration that classifies Reedy left fibrations, Segal coCartesian fibrations, coCartesian fibrations and left fibrations.

Complete Segal Objects

2018-05-09
Nima Rasekh
We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete … We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete Segal objects. Finally we use Segal objects to define univalence in a locally Cartesian closed category that is not presentable and generalize some previous results to the non-presentable setting.

RGB image-based data analysis via discrete Morse theory and persistent homology

2018-01-01
Chuan Du , Christopher Szul , Adarsh Manawa +3 more
Understanding and comparing images for the purposes of data analysis is currently a very computationally demanding task. A group at Australian National University (ANU) recently developed open-source code that can … Understanding and comparing images for the purposes of data analysis is currently a very computationally demanding task. A group at Australian National University (ANU) recently developed open-source code that can detect fundamental topological features of a grayscale image in a computationally feasible manner. This is made possible by the fact that computers store grayscale images as cubical cellular complexes. These complexes can be studied using the techniques of discrete Morse theory. We expand the functionality of the ANU code by introducing methods and software for analyzing images encoded in red, green, and blue (RGB), because this image encoding is very popular for publicly available data. Our methods allow the extraction of key topological information from RGB images via informative persistence diagrams by introducing novel methods for transforming RGB-to-grayscale. This paradigm allows us to perform data analysis directly on RGB images representing water scarcity variability as well as crime variability. We introduce software enabling a a user to predict future image properties, towards the eventual aim of more rapid image-based data behavior prediction.
View PDF Chat

Introduction to Complete Segal Spaces

2018-01-01
Nima Rasekh
The goal is to review the notion of a complete Segal space and how certain categorical notions behave in this context. In particular, we study functoriality in complete Segal spaces … The goal is to review the notion of a complete Segal space and how certain categorical notions behave in this context. In particular, we study functoriality in complete Segal spaces via fibrations. Then we use it to define limits and adjunctions in a complete Segal space. This is mostly expository and the focus is on examples and intuition.

Yoneda Lemma for Elementary Higher Toposes

2018-01-01
Nima Rasekh
We prove the Yoneda lemma inside an elementary higher topos, generalizing the Yonda lemma for spaces. We prove the Yoneda lemma inside an elementary higher topos, generalizing the Yonda lemma for spaces.

An Elementary Approach to Truncations

2018-01-01
Nima Rasekh
We study truncated objects using elementary methods. Concretely, we use universes and the resulting natural number object to define internal truncation levels and prove they behave similar to standard truncated … We study truncated objects using elementary methods. Concretely, we use universes and the resulting natural number object to define internal truncation levels and prove they behave similar to standard truncated objects. Moreover, we take an elementary approach to localizations, giving various equivalent conditions that characterize localizations and constructing a localization out of a sub-universe of local objects via an internal right Kan extension. We then use this general approach, as well as an inductive approach, to construct truncation functors. We use the resulting truncation functors to prove classical results about truncations, such as Blakers-Massey theorem, in the elementary setting. We also give examples of non-presentable $(\infty, 1)$-categories where the elementary approach can be used to define and compute truncations. Finally, we turn around and use truncations to study elementary $(\infty, 1)$-toposes and show how they can help us better understand subobject classifiers and universes

Every Elementary Higher Topos has a Natural Number Object

2018-01-01
Nima Rasekh
We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural … We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural number object out of it. Part of the proof involves showing that various definitions of natural number objects (Lawvere, Freyd and Peano) agree with each other in an elementary $(\infty,1)$-topos. As part of this effort we also study the internal object of contractibility in $(\infty,1)$-categories, which is of independent interest. Finally, we discuss various applications of natural number objects. In particular, we use it to define internal sequential colimits in an elementary $(\infty,1)$-topos.

A Theory of Elementary Higher Toposes

2018-01-01
Nima Rasekh
We define an elementary $\infty$-topos that simultaneously generalizes an elementary topos and Grothendieck $\infty$-topos. We then prove it satisfies the expected topos theoretic properties, such as descent, local Cartesian closure, … We define an elementary $\infty$-topos that simultaneously generalizes an elementary topos and Grothendieck $\infty$-topos. We then prove it satisfies the expected topos theoretic properties, such as descent, local Cartesian closure, locality and classification of univalent morphisms, generalizing results by Lurie and Gepner-Kock. We also define $\infty$-logical functors and show the resulting $\infty$-category is closed under limits and filtered colimits, generalizing the analogous result for elementary toposes and Grothendieck $\infty$-toposes. Moreover, we give an alternative characterization of elementary $\infty$-toposes and their $\infty$-logical functors via their ind-completions. Finally we generalize these results by discussing the case of elementary (n,1)-toposes and give various examples and non-examples.
View PDF Chat

A Model for the Higher Category of Higher Categories

2018-01-01
Nima Rasekh
We use fibrations of complete Segal spaces to construct four complete Segal spaces: Reedy fibrant simplicial spaces, Segal spaces, complete Segal spaces, and spaces. Moreover, we show each one comes … We use fibrations of complete Segal spaces to construct four complete Segal spaces: Reedy fibrant simplicial spaces, Segal spaces, complete Segal spaces, and spaces. Moreover, we show each one comes with a universal fibration that classifies Reedy left fibrations, Segal coCartesian fibrations, coCartesian fibrations and left fibrations and prove these are representable fibrations. Finally, we use equivalences between quasi-categories and complete Segal spaces to present analogous constructions using fibrations of quasi-categories.

Complete Segal Objects

2018-01-01
Nima Rasekh
We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete … We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete Segal objects. Finally we use Segal objects to define univalence in a locally Cartesian closed category that is not presentable and generalize some previous results to the non-presentable setting.

Cartesian Fibrations and Representability

2017-01-01
Nima Rasekh
We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in … We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in $\infty$-category theory. In particular, we give a construction of representable Cartesian fibrations using over-categories and prove the Yoneda lemma for representable Cartesian fibration, which generalizes the established Yoneda lemma for right fibrations. We then use the theory of Cartesian fibrations to study complete Segal objects internal to an $\infty$-category. Concretely, we prove the {\it fundamental theorem of complete Segal objects}, which characterizes equivalences of complete Segal objects. Finally we give two application of the results. First, we present a method to construct Segal objects and second we study the representability of the universal coCartesian fibration.
View PDF Chat

Yoneda Lemma for Simplicial Spaces

2017-01-01
Nima Rasekh
We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and and studying its associated model structure, the covariant model structure. … We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and and studying its associated model structure, the covariant model structure. In particular, we prove a recognition principle for covariant equivalences over an arbitrary simplicial space and invariance of the covariant model structure with respect to complete Segal space equivalences.
View PDF Chat
Coauthor Papers Together
Bruno Stonek 4
Martina Rovelli 4
Lyne Moser 4
Paige Randall North 3
Benedikt Ahrens 3
Niels van der Weide 3
Jonas Frey 2
Kathryn Hess 2
Gabriel Valenzuela 2
Adarsh Manawa 1
Christopher Szul 1
Rosemary K. Guzman 1
Ruth Davidson 1
Chuan Du 1

A model for the homotopy theory of homotopy theory

2000-06-20
Charles Rezk
We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy … We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, “functors between two homotopy theories form a homotopy theory”, or more precisely that the category of such models has a well-behaved internal hom-object.

Yoneda Lemma for Simplicial Spaces

2023-07-11
Nima Rasekh
Abstract We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and studying their associated model structure, the covariant model structure … Abstract We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and studying their associated model structure, the covariant model structure . In particular, we prove a recognition principle for covariant equivalences over an arbitrary simplicial space and invariance of the covariant model structure with respect to complete Segal space equivalences.

Cartesian fibrations and representability

2022-01-01
Nima Rasekh
We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in … We use the complete Segal approach to the theory of Cartesian fibrations to define and study representable Cartesian fibrations, generalizing representable right fibrations which have played a key role in $\infty$-category theory. In particular, we give a construction of representable Cartesian fibrations using over-categories and prove the Yoneda lemma for representable Cartesian fibration, which generalizes the established Yoneda lemma for right fibrations. We then use the theory of Cartesian fibrations to study complete Segal objects internal to an $\infty$-category. Concretely, we prove the {\it fundamental theorem of complete Segal objects}, which characterizes equivalences of complete Segal objects. Finally we give two application of the results. First, we present a method to construct Segal objects and second we study the representability of the universal coCartesian fibration.
View PDF Chat

Fibrations and Yoneda's lemma in an ∞-cosmos

2016-07-31
Emily Riehl , Dominic Verity
View PDF Chat

A cartesian presentation of weak<i>n</i>–categories

2010-01-02
Charles Rezk
A cartesian presentation of weak n-categories CHARLES REZKWe propose a notion of weak .nCk;n/-category, which we call .nCk;n/-‚-spaces.The .nCk; n/-‚-spaces are precisely the fibrant objects of a certain model category … A cartesian presentation of weak n-categories CHARLES REZKWe propose a notion of weak .nCk;n/-category, which we call .nCk;n/-‚-spaces.The .nCk; n/-‚-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category ‚ n .This notion is a generalization of that of complete Segal spaces (which are precisely the .1;1/-‚-spaces).Our main result is that the above model category is cartesian.18D05; 55U40 k is the full subcategory of fibrant objects in ‚ n Sp k ; equivalences in ‚ n Sp fib k are just levelwise weak equivalences of presheaves.
View PDF Chat

A theory of elementary higher toposes

2018-06-29
Nima Rasekh
We define an elementary higher topos that simultaneously generalizes an elementary topos and higher topos. Then we show it satisfies classical topos theoretic properties, such being locally Cartesian closed and … We define an elementary higher topos that simultaneously generalizes an elementary topos and higher topos. Then we show it satisfies classical topos theoretic properties, such being locally Cartesian closed and descent. Finally we show we can classify univalent maps in an elementary higher topos.

Homotopy Type Theory: Univalent Foundations of Mathematics

2013-01-01
Peter Aczel , Benedikt Ahrens , Thorsten Altenkirch +7 more
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of … Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.
View PDF Chat

The Simplicial Model of Univalent Foundations (after Voevodsky)

2012-11-12
Chris Kapulkin , Peter LeFanu Lumsdaine
We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models … We present Voevodsky's construction of a model of univalent type theory in the category of simplicial sets. To this end, we first give a general technique for constructing categorical models of dependent type theory, using universes to obtain coherence. We then construct a (weakly) universal Kan fibration, and use it to exhibit a model in simplicial sets. Lastly, we introduce the Univalence Axiom, in several equivalent formulations, and show that it holds in our model. As a corollary, we conclude that Martin-Lof type theory with one univalent universe (formulated in terms of contextual categories) is at least as consistent as ZFC with two inaccessible cardinals.

Univalence in locally cartesian closed ∞-categories

2016-11-06
David Gepner , Joachim Kock
Abstract After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> ${\infty}$ -categories, we establish the representability of equivalences and show that … Abstract After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> ${\infty}$ -categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> ${\infty}$ -topos has a hierarchy of “universal” univalent families, indexed by regular cardinals, and that n -topoi have univalent families classifying <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> ${(n-2)}$ -truncated maps. We show that univalent families are preserved (and detected) by right adjoints to locally cartesian localizations, and use this to exhibit certain canonical univalent families in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> $\infty$ -quasitopoi (certain <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> ${\infty}$ -categories of “separated presheaves”, introduced here). We also exhibit some more exotic examples of univalent families, illustrating that a univalent family in an n -topos need not be <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> ${(n-2)}$ -truncated, as well as some univalent families in the Morel–Voevodsky <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> ${\infty}$ -category of motivic spaces, an instance of a locally cartesian closed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> ${\infty}$ -category which is not an n -topos for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mi>∞</m:mi> </m:mrow> </m:math> ${0\leq n\leq\infty}$ . Lastly, we show that any presentable locally cartesian closed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> ${\infty}$ -category is modeled by a combinatorial type-theoretic model category, and conversely that the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> ${\infty}$ -category underlying a combinatorial type-theoretic model category is presentable and locally cartesian closed. Under this correspondence, univalent families in presentable locally cartesian closed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>∞</m:mi> </m:math> ${\infty}$ -categories correspond to univalent fibrations in combinatorial type-theoretic model categories.

Homotopy Invariant Algebraic Structures on Topological Spaces

1973-01-01
J. M. Boardman , R. M. Vogt
View PDF Chat

Enriched ∞-categories via non-symmetric ∞-operads

2015-05-12
David Gepner , Rune Haugseng
View PDF Chat

Introduction to Complete Segal Spaces

2018-01-01
Nima Rasekh
The goal is to review the notion of a complete Segal space and how certain categorical notions behave in this context. In particular, we study functoriality in complete Segal spaces … The goal is to review the notion of a complete Segal space and how certain categorical notions behave in this context. In particular, we study functoriality in complete Segal spaces via fibrations. Then we use it to define limits and adjunctions in a complete Segal space. This is mostly expository and the focus is on examples and intuition.

Higher Topos Theory (AM-170)

2009-12-31
Jacob Lurie
Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are … Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory , Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Cartesian Fibrations of Complete Segal Spaces

2023-05-21
Nima Rasekh
Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in (∞, 1)-category theory to study presheaves valued in (∞, 1)-categories.In this work we … Cartesian fibrations were originally defined by Lurie in the context of quasi-categories and are commonly used in (∞, 1)-category theory to study presheaves valued in (∞, 1)-categories.In this work we define and study fibrations modeling presheaves valued in simplicial spaces and their localizations.This includes defining a model structure for these fibrations and giving effective tools to recognize its fibrations and weak equivalences.This in particular gives us a new method to construct Cartesian fibrations via complete Segal spaces.In addition to that, it allows us to define and study fibrations modeling presheaves of Segal spaces.
View PDF Chat

Quasi-categories vs. Segal spaces: Cartesian edition

2021-08-20
Nima Rasekh
Abstract We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]), On bisimplicial … Abstract We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: On marked simplicial sets (due to Lurie [31]), On bisimplicial spaces (due to deBrito [12]), On bisimplicial sets, On marked simplicial spaces. The main way to prove these equivalences is by using the Quillen equivalences between quasi-categories and complete Segal spaces as defined by Joyal–Tierney and the straightening construction due to Lurie.
View PDF Chat

(Infinity,2)-Categories and the Goodwillie Calculus I

2009-01-01
Jacob Lurie
The bulk of this paper is devoted to the comparison of several models for the theory of (infinity,2)-categories: that is, higher categories in which all k-morphisms are invertible for k … The bulk of this paper is devoted to the comparison of several models for the theory of (infinity,2)-categories: that is, higher categories in which all k-morphisms are invertible for k &gt; 2 (the case of (infinity,n)-categories is also considered). Our ultimate goal is to lay the foundations for a study of Tom Goodwillie's calculus of functors. To this end, we have included some simple applications to the theory of first derivatives.

Covariant Model Structures and Simplicial Localization

2015-01-01
Danny Stevenson
In this paper we prove that for any simplicial set $B$, there is a Quillen equivalence between the covariant model structure on $\mathbf{S}/B$ and a certain localization of the projective … In this paper we prove that for any simplicial set $B$, there is a Quillen equivalence between the covariant model structure on $\mathbf{S}/B$ and a certain localization of the projective model structure on the category of simplicial presheaves on the simplex category $Δ/B$ of $B$. We extend this result to give a new Quillen equivalence between this covariant model structure and the projective model structure on the category of simplicial presheaves on the simplicial category $\mathfrak{C}[B]$. We study the relationship with Lurie's straightening theorem. Along the way we prove some results on localizations of simplicial categories and quasi-categories.

A generalized Blakers–Massey theorem

2020-09-07
Mathieu Anel , Georg Biedermann , Eric Finster +1 more
We prove a generalization of the classical connectivity theorem of Blakers–Massey, valid in an arbitrary higher topos and with respect to an arbitrary modality, that is, a factorization system ( … We prove a generalization of the classical connectivity theorem of Blakers–Massey, valid in an arbitrary higher topos and with respect to an arbitrary modality, that is, a factorization system ( L , R ) in which the left class is stable by base change. We explain how to rederive the classical result, as well as the recent generalization of Chachólski, Scherer and Werndli (Ann. Inst. Fourier 66 (2016) 2641–2665). Our proof is inspired by the one given in homotopy-type theory in Favonia et al. (2016).
View PDF Chat

Quasi-categories vs Segal spaces

2007-01-01
André Joyal , Myles Tierney
We show that complete Segal spaces and Segal categories are Quillen equivalent to quasi-categories. We show that complete Segal spaces and Segal categories are Quillen equivalent to quasi-categories.
View PDF Chat

Higher Quasi-Categories vs Higher Rezk Spaces

2014-12-01
Dimitri Ara
We introduce a notion of n-quasi-categories as fibrant objects of a model category structure on presheaves on Joyal's n-cell category \Theta_n. Our definition comes from an idea of Cisinski and … We introduce a notion of n-quasi-categories as fibrant objects of a model category structure on presheaves on Joyal's n-cell category \Theta_n. Our definition comes from an idea of Cisinski and Joyal. However, we show that this idea has to be slightly modified to get a reasonable notion. We construct two Quillen equivalences between the model category of n-quasi-categories and the model category of Rezk \Theta_n-spaces showing that n-quasi-categories are a model for (\infty, n)-categories. For n = 1, we recover the two Quillen equivalences defined by Joyal and Tierney between quasi-categories and complete Segal spaces.
View PDF Chat

Universality of multiplicative infinite loop space machines

2015-12-31
David Gepner , Moritz Groth , Thomas Nikolaus
We establish a canonical and unique tensor product for commutative monoids and groups in an 1-category C which generalizes the ordinary tensor product of abelian groups.Using this tensor product we … We establish a canonical and unique tensor product for commutative monoids and groups in an 1-category C which generalizes the ordinary tensor product of abelian groups.Using this tensor product we show that E n -(semi)ring objects in C give rise to E n -ring spectrum objects in C .In the case that C is the 1-category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K-theory of rings and ring spectra.The main tool we use to establish these results is the theory of smashing localizations of presentable 1-categories.In particular, we identify preadditive and additive 1-categories as the local objects for certain smashing localizations.A central theme is the stability of algebraic structures under basechange; for example, we show Ring.D ˝C/ ' Ring.D/ ˝C .Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in 1-categories.
View PDF Chat

Axiomatic Sheaf Theory : Some Constructions and Applications

2010-01-01
Myles Tierney

Segal objects and the Grothendieck construction

2018-01-01
Pedro Boavida de Brito
View PDF Chat

Weak complicial sets I. Basic homotopy theory

2008-07-15
Dominic Verity

A note on the (∞,n)–category of cobordisms

2019-03-12
Damien Calaque , Claudia Scheimbauer
In this extended note we give a precise definition of fully extended topological field theories \`a la Lurie. Using complete $n$-fold Segal spaces as a model, we construct an $(\infty,n)$-category … In this extended note we give a precise definition of fully extended topological field theories \`a la Lurie. Using complete $n$-fold Segal spaces as a model, we construct an $(\infty,n)$-category of $n$-dimensional cobordisms, possibly with tangential structure. We endow it with a symmetric monoidal structure and show that we can recover the usual category of cobordisms $n\operatorname{Cob}$ and the cobordism bicategory $n\operatorname{Cob}^{ext}$ from it.
View PDF Chat

Model Categories and Their Localizations

2009-08-24
Philip Hirschhorn
Localization of model category structures: Summary of part 1 Local spaces and localization The localization model category for spaces Localization of model categories Existence of left Bousfield localizations Existence of … Localization of model category structures: Summary of part 1 Local spaces and localization The localization model category for spaces Localization of model categories Existence of left Bousfield localizations Existence of right Bousfield localizations Fiberwise localization Homotopy theory in model categories: Summary of part 2 Model categories Fibrant and cofibrant approximations Simplicial model categories Ordinals, cardinals, and transfinite composition Cofibrantly generated model categories Cellular model categories Proper model categories The classifying space of a small category The Reedy model category structure Cosimplicial and simplicial resolutions Homotopy function complexes Homotopy limits in simplicial model categories Homotopy limits in general model categories Index Bibliography.

Basic concepts of enriched category theory

1982-01-01
G. M. Kelly
Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20. Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.

Every Elementary Higher Topos has a Natural Number Object

2018-01-01
Nima Rasekh
We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural … We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural number object out of it. Part of the proof involves showing that various definitions of natural number objects (Lawvere, Freyd and Peano) agree with each other in an elementary $(\infty,1)$-topos. As part of this effort we also study the internal object of contractibility in $(\infty,1)$-categories, which is of independent interest. Finally, we discuss various applications of natural number objects. In particular, we use it to define internal sequential colimits in an elementary $(\infty,1)$-topos.

Fibrations and lax limits of $(\infty,2)$-categories

2020-01-01
Andrea Gagna , Yonatan Harpaz , Edoardo Lanari
We study four types of (co)cartesian fibrations of $\infty$-bicategories over a given base $\mathcal{B}$, and prove that they encode the four variance flavors of $\mathcal{B}$-indexed diagrams of $\infty$-categories. We then … We study four types of (co)cartesian fibrations of $\infty$-bicategories over a given base $\mathcal{B}$, and prove that they encode the four variance flavors of $\mathcal{B}$-indexed diagrams of $\infty$-categories. We then use this machinery to set up a general theory of 2-(co)limits for diagrams valued in an $\infty$-bicategory, capable of expressing lax, weighted and pseudo limits. When the $\infty$-bicategory at hand arises from a model category tensored over marked simplicial sets, we show that this notion of 2-(co)limit can be calculated as a suitable form of a weighted homotopy limit on the model categorical level, thus showing in particular the existence of these 2-(co)limits in a wide range of examples. We finish by discussing a notion of cofinality appropriate to this setting and use it to deduce the unicity of 2-(co)limits, once exist.

A model category structure on the category of simplicial categories

2006-12-19
Julia E. Bergner
In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence … In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.
View PDF Chat

Comparison of models for (∞<i>,n</i>)–categories, I

2013-07-19
Julia E. Bergner , Charles Rezk
While many different models for $(\infty,1)$-categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as … While many different models for $(\infty,1)$-categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for $(\infty, n)$-categories. In this paper, we establish model structures for some naturally arising categories of objects which should be thought of as $(\infty,n)$-categories. Furthermore, we establish Quillen equivalences between them.
View PDF Chat

Sheaves in Geometry and Logic

1994-01-01
Saunders Mac Lane , Ieke Moerdijk

Comparison of models for (∞,n)‐categories, II

2020-09-07
Julia E. Bergner , Charles Rezk
In this paper we complete a chain of explicit Quillen equivalences between the model category for $\Theta_{n+1}$-spaces and the model category of small categories enriched in $\Theta_n$-spaces. The Quillen equivalences … In this paper we complete a chain of explicit Quillen equivalences between the model category for $\Theta_{n+1}$-spaces and the model category of small categories enriched in $\Theta_n$-spaces. The Quillen equivalences given here connect Segal category objects in $\Theta_n$-spaces, complete Segal objects in $\Theta_n$-spaces, and $\Theta_{n+1}$-spaces.
View PDF Chat

$(\infty,1)$-Categorical Comprehension Schemes

2020-10-19
Raffael Stenzel
In this paper we define and study a notion of comprehension schemes for cartesian fibrations over $(\infty,1)$-categories, generalizing Johnstone's respective notion for ordinary fibered categories. This includes natural generalizations of … In this paper we define and study a notion of comprehension schemes for cartesian fibrations over $(\infty,1)$-categories, generalizing Johnstone's respective notion for ordinary fibered categories. This includes natural generalizations of smallness, local smallness and the notion of definability in the sense of Benabou, as well as of Jacob's comprehension categories. Thereby, among others, we will characterize numerous categorical properties of $\infty$-toposes, the notion of univalence and the externalization of internal $(\infty,1)$-categories via comprehension schemes. An example of particular interest will be the universal cartesian fibration given by externalization of the ''freely walking chain'' in the $(\infty,1)$-category of small $(\infty,1)$-categories. In the end, we take a look at the externalization construction of internal $(\infty,1)$-categories from a model categorical perspective and review various examples from the literature in this light.

An Elementary Approach to Truncations

2018-01-01
Nima Rasekh
We study truncated objects using elementary methods. Concretely, we use universes and the resulting natural number object to define internal truncation levels and prove they behave similar to standard truncated … We study truncated objects using elementary methods. Concretely, we use universes and the resulting natural number object to define internal truncation levels and prove they behave similar to standard truncated objects. Moreover, we take an elementary approach to localizations, giving various equivalent conditions that characterize localizations and constructing a localization out of a sub-universe of local objects via an internal right Kan extension. We then use this general approach, as well as an inductive approach, to construct truncation functors. We use the resulting truncation functors to prove classical results about truncations, such as Blakers-Massey theorem, in the elementary setting. We also give examples of non-presentable $(\infty, 1)$-categories where the elementary approach can be used to define and compute truncations. Finally, we turn around and use truncations to study elementary $(\infty, 1)$-toposes and show how they can help us better understand subobject classifiers and universes

Higher topological Hochschild homology of Thom spectra

2011-01-01
Christian Schlichtkrull
In this paper we analyze the higher topological Hochschild homology of commutative Thom S-algebras. This includes the case of the classical cobordism spectra MO, MSO, MU, etc. We consider the … In this paper we analyze the higher topological Hochschild homology of commutative Thom S-algebras. This includes the case of the classical cobordism spectra MO, MSO, MU, etc. We consider the homotopy orbits of the torus action on iterated topological Hochschild homology and we describe the relationship to topological Andre-Quillen homology.
View PDF Chat

Higher topological Hochschild homology of periodic complex K-theory

2020-06-11
Bruno Stonek
View PDF Chat

An ∞-categorical approach to <i>R</i> -line bundles, <i>R</i> -module Thom spectra, and twisted <i>R</i> -homology

2013-10-25
Matthew Ando , Charles Rezk , Andrew J. Blumberg +2 more
We develop a generalization of the theory of Thom spectra using the language of ∞-categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new … We develop a generalization of the theory of Thom spectra using the language of ∞-categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parameterized spectra, and our definition is motivated by the geometric definition of Thom spectra of May–Sigurdsson. For an A∞-ring spectrum R, we associate a Thom spectrum to a map of ∞-categories from the ∞-groupoid of a space X to the ∞-category of free rank one R-modules, which we show is a model for BGL1R; we show that BGL1R classifies homotopy sheaves of rank one R-modules, which we call R-line bundles. We use our R-module Thom spectrum to define the twisted R-homology and cohomology of R-line bundles over a space classified by a map X→BGL1R, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory.
View PDF Chat

On topological cyclic homology

2018-01-01
Thomas Nikolaus , Peter Scholze
Recall that, if G is a finite group acting on an abelian group M , then the Tate cohomology H i (G, M ) is defined by splicing together cohomology … Recall that, if G is a finite group acting on an abelian group M , then the Tate cohomology H i (G, M ) is defined by splicing together cohomology and homology.More precisely, H i (G, M
View PDF Chat

Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map

2018-12-06
Matthew Ando , Andrew J. Blumberg , David Gepner
We introduce a general theory of parametrized objects in the setting of 1-categories.Although parametrised spaces and spectra are the most familiar examples, we establish our theory in the generality of … We introduce a general theory of parametrized objects in the setting of 1-categories.Although parametrised spaces and spectra are the most familiar examples, we establish our theory in the generality of families of objects of a presentable 1-category parametrized over objects of an 1-topos.We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures.Our main applications are to the study of generalized Thom spectra.We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality.In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we express the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.55P99, 55R70 1. Introduction 3762 2. Background on 1-categories 3771 3. Parametrized spaces and spectra 3775 4. Twisted cohomology theories and the twisted Umkehr map 3781 5. Parametrized 1-category theory 3794 6. Algebraic structures on parametrized objects 3804 7. Picard spaces 3810 8. Multiplicative properties of the Thom spectrum functor 3815 Appendix A. The Brauer group and twisted parametrized spectra 3817 Appendix B. Comparison to the May-Sigurdsson model 3818
View PDF Chat

Algebraic Topology — Homotopy and Homology

1975-01-01
Robert M. Switzer

Left fibrations and homotopy colimits

2014-11-01
Gijs Heuts , Ieke Moerdijk
View PDF Chat

Left fibrations and homotopy colimits II

2016-02-03
Gijs Heuts , Ieke Moerdijk
For a small simplicial category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the homotopy-coherent … For a small simplicial category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the homotopy-coherent nerve of A provides a left Quillen equivalence between the projective model structure on the former category and the covariant model structure on the latter. We compare this Quillen equivalence to the straightening-unstraightening equivalence previously established by Lurie, where the left adjoint goes in the opposite direction. The existence of left Quillen functors in both directions considerably simplifies the proof that these constructions provide Quillen equivalences. The results of this paper generalize those of part I, where A was an ordinary category. The proofs for a simplicial category are more involved and can be read independently.

A Survey of (∞, 1)-Categories

2009-09-16
Julia E. Bergner
In this paper we give a summary of the comparisons between different definitions of so-called (∞, 1)-categories, which are considered to be models for ∞-categories whose n-morphisms are all invertible … In this paper we give a summary of the comparisons between different definitions of so-called (∞, 1)-categories, which are considered to be models for ∞-categories whose n-morphisms are all invertible for n > 1. They are also, from the viewpoint of homotopy theory, models for the homotopy theory of homotopy theories. The four different structures, all of which are equivalent, are simplicial categories, Segal categories, complete Segal spaces, and quasi-categories.

Every Elementary Higher Topos has a Natural Number Object.

2018-09-05
Nima Rasekh
We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural … We prove that every elementary $(\infty,1)$-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural number object out of it. Part of the proof involves showing that various definitions of natural number objects (Lawvere, Freyd and Peano) agree with each other in an elementary $(\infty,1)$-topos. As part of this effort we also study the internal object of contractibility in $(\infty,1)$-categories, which is of independent interest. Finally, we discuss various applications of natural number objects. In particular, we use it to define internal sequential colimits in an elementary $(\infty,1)$-topos.

Localized stable homotopy of some classifying spaces

1981-03-01
Victor Snaith
In this note I will give new, simplified proofs of some of the results announced in (18) and proved in ((19), I, § 4, II, §§ 2, 9).These results mostly … In this note I will give new, simplified proofs of some of the results announced in (18) and proved in ((19), I, § 4, II, §§ 2, 9).These results mostly date from 1975/6 at which time they were proved using my crude stable decomposition of ω n σ n X (20) and differential-geometric techniques with the Becker-Gottlieb transfer. Since then more systematic approaches have been developed towards the stable decompositions ((4); (5); (6); (10); (12)) and towards the transfer ((7), I and II; (8)). Actually, the system-atization of the stable decompositions was already latent in (15) if only I had realized it!

On the Classification of Topological Field Theories

2008-01-01
Jacob Lurie
Our goal in this article is to give an expository account of some recent work on the classification of topological field theories.More specifically, we will outline the proof of a … Our goal in this article is to give an expository account of some recent work on the classification of topological field theories.More specifically, we will outline the proof of a version of the cobordism hypothesis conjectured by Baez and Dolan in [2]. The tangle hypothesis 272 References 279Terminology.Unless otherwise specified, we will use the word manifold to refer to a compact smooth manifold M , possibly with boundary (or with corners).If M is a manifold, we will denote its boundary by ∂ M. We will say that M is closed if the boundary ∂ M is empty.For a brief description of how the ideas of this paper generalize to manifolds which are not smooth, we refer the reader to Remark 2.4.30.Throughout this paper, we will make informal use of the language of higher category theory.We will always use the term n-category to refer to what is sometimes called a weak n-category: that is, a collection of objects {X, Y, Z, . ..} together with an (n -1)-category Map(X, Y ) for every pair of objects X and Y , which are equipped with a notion of composition which is associative up to coherent isomorphism.We refer the reader to 1.3 for an informal discussion and 2.1 for the outline of a more precise definition.If C is a category (or a higher category) equipped with an associative and unital tensor product ⊗, we will let 1 denote the unit object of C.Let V be a finite-dimensional real vector space.By an inner product on V we will mean a symmetric bilinear form b : V × V → R which is positivedefinite (so that b(v, v) > 0 for v = 0).More generally, if X is a topological space and ζ is a real vector bundle on X, then by an inner product on ζ we will mean an inner product on each fiber ζ x , which depends continuously on the point x ∈ X.
View PDF Chat

Complete Segal Objects

2018-05-09
Nima Rasekh
We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete … We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete Segal objects. Finally we use Segal objects to define univalence in a locally Cartesian closed category that is not presentable and generalize some previous results to the non-presentable setting.

A simple universal property of Thom ring spectra

2018-10-31
Omar Antolín Camarena , Tobias Barthel
We give a simple universal property of the multiplicative structure on the Thom spectrum of an n-fold loop map, obtained as a special case of a characterization of the algebra … We give a simple universal property of the multiplicative structure on the Thom spectrum of an n-fold loop map, obtained as a special case of a characterization of the algebra structure on the colimit of a lax O-monoidal functor.This allows us to relate Thom spectra to En-algebras of a given characteristic in the sense of Szymik.As applications, we recover the Hopkins-Mahowald theorem realizing HFp and HZ as Thom spectra, and compute the topological Hochschild homology and the cotangent complex of various Thom spectra.ring spectrum.Finally, we describe the topological Hochschild homology and the E n -cotangent complex of the E n -algebras considered above.
View PDF Chat

Simplicial Homotopy Theory

1999-01-01
Paul G. Goerss , John F. Jardine
View PDF Chat