Biequivalences in tricategories

Authors

Nick Gurski

Type: Preprint

Publication Date: 2011-01-01

Citations: 59

DOI: https://doi.org/10.48550/arxiv.1102.0979

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Abstract

We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one for equipping a monoidal bicategory with invertible objects with a coherent choice of those inverses.

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The transportation of trifunctors in the tricategory of bicategories

2020-02-04

Andrew P. Smith

Tricategories, as the construction for the most general sort of weak 3-category being given by explicit coherence axioms, are a particularly important structure in the study of low-dimensional higher category … Tricategories, as the construction for the most general sort of weak 3-category being given by explicit coherence axioms, are a particularly important structure in the study of low-dimensional higher category theory. As such the correct notion of a morphism between tricategories, the Trifunctor, is also an important object of interest. Just as many constructions in mathematics can be realised by using functors between appropriate categories, these constructions can be generalised to the 3-dimensional level by using trifunctors between the appropriate tricategories. Of particular interest are trifunctors into the tricategory of bicategories. Given a mathematical structure laid on top of a base object, it can be useful to transport that structure from the original object to a new object across a suitable sort of equivalence. The collection of trifunctors between two tricategories forms a tricategory of its own. So does the collection of functions from the objects of the source tricategory to the objects of the target tricategory, which form the object level of any trifunctor. Therefore in this case the appropriate notion of equivalence is that of biequivalence, and we would hope to be able to transport the structure of a trifunctor across a collection of biequivalences at the object level. While the transport of structure at lower dimensions is achieved using monadic methods, at the general 3-dimensional level these haven't been developed. This thesis aims to provide a method for transporting the structure of a trifunctor into the tricategory of bicategories across object-indexed biequivalences. We do this by working directly from the definition of trifunctor: by constructing the data needed for the new trifunctor from the data of the original trifunctor and the biequivalences, and then proving that the axioms hold using diagram manipulation techniques.

The low-dimensional structures formed by tricategories

2009-01-28

Richard Garner , Nick Gurski

We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called … We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.

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Alternative notion to intercategories: part I. A tricategory of double categories

2020-01-01

Bojana Femić

This is a first of a series of two papers. Our motive is to tackle the question raised in Böhm's "The Gray Monoidal Product of Double Categories" from Applied Categorical … This is a first of a series of two papers. Our motive is to tackle the question raised in Böhm's "The Gray Monoidal Product of Double Categories" from Applied Categorical Structures: which would be an alternative notion to intercategories of Grandis and Paré, so that monoids in Böhm's monoidal category $Dbl$ of strict double categories and double pseudo functors be an example of it? Before addressing this question we observe that although bicategories embed into pseudo double categories, this embedding is not monoidal, with the usual notion of a monoidal pseudo double category. We then prove that monoidal bicategories embed into the mentioned monoids of Böhm. In order to fit Böhm's monoid into an intercategory-type object, we start by upgrading the category $Dbl$ to a 2-category and end up rather with a tricategory $\DblPs$. We propose an alternative definition of intercategories as internal categories in this tricategory, enabling $Dbl$ to be an example of this gadget. The formal definition of a category internal to the (type of a) tricategory (of) $\DblPs$, as well as another important example of these in the literature, we leave for a subsequent paper.

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The low-dimensional structures that tricategories form

2007-11-12

Richard Garner , Nick Gurski

We form tricategories and the homomorphisms between them into a bicategory. We then enrich this bicategory into an example of a three-dimensional structure called a locally double bicategory, this being … We form tricategories and the homomorphisms between them into a bicategory. We then enrich this bicategory into an example of a three-dimensional structure called a locally double bicategory, this being a bicategory enriched in the monoidal 2-category of weak double categories. Finally, we show that every sufficiently well-behaved locally double bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.

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Tricategorical Universal Properties Via Enriched Homotopy Theory

2024-09-03

Adrian Miranda

We develop the theory of tricategorical limits and colimits, and show that they can be modelled up to biequivalence via certain homotopically well-behaved limits and colimits enriched over the monoidal … We develop the theory of tricategorical limits and colimits, and show that they can be modelled up to biequivalence via certain homotopically well-behaved limits and colimits enriched over the monoidal model category $\mathbf{Gray}$ of $2$-categories and $2$-functors. This categorifies the relationship that bicategorical limits and colimits have with the so called `flexible' enriched limits in $2$-category theory. As examples, we establish the tricategorical universal properties of Kleisli constructions for pseudomonads, Eilenberg-Moore and Kleisli constructions for (op)monoidal pseudomonads, centre constructions for $\mathbf{Gray}$-monoids, and strictifications of bicategories and pseudo-double categories.

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Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories

2021-01-01

Bojana Femić

This paper emerged as a result of tackling the following three issues. Firstly, we would like the well known embedding of bicategories into pseudo double categories to be monoidal, which … This paper emerged as a result of tackling the following three issues. Firstly, we would like the well known embedding of bicategories into pseudo double categories to be monoidal, which it is not if one uses the usual notion of a monoidal pseudo double category. Secondly, in \cite{Gabi} the question was raised: which would be an alternative notion to intercategories of Grandis and Par\'e, so that monoids in B\"ohm's monoidal category $(Dbl,\ot)$ of strict double categories and strict double functors with a Gray type monoidal product be an example of it? We obtain and prove that precisely the monoidal structure of $(Dbl,\ot)$ resolves the first question. On the other hand, resolving the second question, we upgrade the category $Dbl$ to a tricategory $\DblPs$ and propose %an alternative definition of intercategories as to consider internal categories in this tricategory. %, enabling monoids in $(Dbl,\ot)$ to be examples of this gadget. Apart from monoids in $(Dbl,\ot)$ - more importanlty, weak pseudomonoids in a tricategory containing $(Dbl,\ot)$ as a sub 1-category - most of the examples of intercategories are also examples of this gadget, the ones that escape are those that rely on laxness of the product on the pullback, as duoidal categories. For the latter purpose we define categories internal to tricategories (of the type of $\DblPs$), which simultaneously serves our third motive. Namely, inspired by the tricategory and $(1\times 2)$-category of tensor categories, we prove under mild conditions that categories enriched over certain type of tricategories may be made into categories internal in them. We illustrate this occurrence for tensor categories with respect to the ambient tricategory $2\x\Cat_{wk}$ of 2-categories, pseudofunctors, pseudonatural transformations and modifications.

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Internalization and enrichment via spans and matrices in a tricategory

2022-01-01

Bojana Femić , Enrico Ghiorzi

We introduce categories $\M$ and $\S$ internal in the tricategory $\Bicat_3$ of bicategories, pseudofunctors, pseudonatural transformations and modifications, for matrices and spans in a 1-strict tricategory $V$. Their horizontal tricategories … We introduce categories $\M$ and $\S$ internal in the tricategory $\Bicat_3$ of bicategories, pseudofunctors, pseudonatural transformations and modifications, for matrices and spans in a 1-strict tricategory $V$. Their horizontal tricategories are the tricategories of matrices and spans in $V$. Both the internal and the enriched constructions are tricategorifications of the corresponding constructions in 1-categories. Following \cite{FGK} we introduce monads and their vertical morphisms in categories internal in tricategories. We prove an equivalent condition for when the internal categories for matrices $\M$ and spans $\S$ in a 1-strict tricategory $V$ are equivalent, and deduce that in that case their corresponding categories of (strict) monads and vertical monad morphisms are equivalent, too. We prove that the latter categories are isomorphic to those of categories enriched and discretely internal in $V$, respectively. As a byproduct of our tricategorical constructions we recover some results from \cite{Fem}. Truncating to 1-categories we recover results from \cite{CFP} and \cite{Ehr} on the equivalence of enriched and discretely internal 1-categories.

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Enrichment and internalization in tricategories, the case of tensor categories and alternative notion to intercategories

2021-01-05

Bojana Femić

This paper emerged as a result of tackling the following three issues. Firstly, we would like the well known embedding of bicategories into pseudo double categories to be monoidal, which … This paper emerged as a result of tackling the following three issues. Firstly, we would like the well known embedding of bicategories into pseudo double categories to be monoidal, which it is not if one uses the usual notion of a monoidal pseudo double category. Secondly, in \cite{Gabi} the question was raised: which would be an alternative notion to intercategories of Grandis and Pare, so that monoids in Bohm's monoidal category $(Dbl,\ot)$ of strict double categories and strict double functors with a Gray type monoidal product be an example of it? We obtain and prove that precisely the monoidal structure of $(Dbl,\ot)$ resolves the first question. On the other hand, resolving the second question, we upgrade the category $Dbl$ to a tricategory $\DblPs$ and propose %an alternative definition of intercategories as to consider internal categories in this tricategory. %, enabling monoids in $(Dbl,\ot)$ to be examples of this gadget. Apart from monoids in $(Dbl,\ot)$ - more importanlty, weak pseudomonoids in a tricategory containing $(Dbl,\ot)$ as a sub 1-category - most of the examples of intercategories are also examples of this gadget, the ones that escape are those that rely on laxness of the product on the pullback, as duoidal categories. For the latter purpose we define categories internal to tricategories (of the type of $\DblPs$), which simultaneously serves our third motive. Namely, inspired by the tricategory and $(1\times 2)$-category of tensor categories, we prove under mild conditions that categories enriched over certain type of tricategories may be made into categories internal in them. We illustrate this occurrence for tensor categories with respect to the ambient tricategory $2\x\Cat_{wk}$ of 2-categories, pseudofunctors, pseudonatural transformations and modifications.

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Weak vertical composition

2022-01-01

Eugenia Cheng , Alexander S. Corner

We study semi-strict tricategories in which the only weakness is in vertical composition. We construct these as categories enriched in the category of bicategories with strict functors, with respect to … We study semi-strict tricategories in which the only weakness is in vertical composition. We construct these as categories enriched in the category of bicategories with strict functors, with respect to the cartesian monoidal structure. As these are a form of tricategory it follows that doubly-degenerate ones are braided monoidal categories. We show that this form of semi-strict tricategory is weak enough to produce all braided monoidal categories. That is, given any braided monoidal category $B$ there is a doubly-degenerate ``vertically weak'' semi-strict tricategory whose associated braided monoidal category is braided monoidal equivalent to $B$.

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Weak vertical composition II: totalities

2023-01-01

Eugenia Cheng , Alexander S. Corner

We continue our study of semi-strict tricategories in which the only weakness is in vertical composition. We assemble the doubly-degenerate such tricategories into a 2-category, defining weak functors and transformations. … We continue our study of semi-strict tricategories in which the only weakness is in vertical composition. We assemble the doubly-degenerate such tricategories into a 2-category, defining weak functors and transformations. We exhibit a biadjoint biequivalence between this 2-category and the 2-category of braided (weakly) monoidal categories, braided (weakly) monoidal functors, and monoidal transformations.

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Internal bicategories

2012-01-01

Christopher L. Douglas , André Henriques

We define bicategories internal to 2-categories. When the ambient 2-category is symmetric monoidal categories, this provides a convenient framework for encoding the structures of a symmetric monoidal 3-category. This framework … We define bicategories internal to 2-categories. When the ambient 2-category is symmetric monoidal categories, this provides a convenient framework for encoding the structures of a symmetric monoidal 3-category. This framework is well suited to examples arising in geometry and algebra, such as the 3-category of bordisms or the 3-category of conformal nets.

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Turning monoidal categories into strict ones.

2001-01-01

Peter Schauenburg

It is well-known that every monoidal category is equivalent to a strict one. We show that for categories of sets with additional structure (which we define quite formally below) it … It is well-known that every monoidal category is equivalent to a strict one. We show that for categories of sets with additional structure (which we define quite formally below) it is not even necessary to change the category: The same category has a different (but isomorphic) tensor product, with which it is a strict monoidal category. The result applies to ordinary (bi)modules, where it shows that one can choose a realization of the tensor product for each pair of modules in such a way that tensor products are strictly associative. Perhaps more surprisingly, the result also applies to such nontrivially nonstrict categories as the category of modules over a quasibialgebra.

Iterated icons

2013-01-01

Eugenia Cheng , Nick Gurski

We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is … We study the totality of categories weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of "2-tuply monoidal categories" missing from our earlier studies of the Periodic Table. We propose a generalisation to all k-tuply monoidal n-categories.

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Internalization and enrichment via spans and matrices in a tricategory

2023-01-04

Bojana Femić , Enrico Ghiorzi

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Constructing symmetric monoidal bicategories functorially

2019-01-01

Linde Wester Hansen , Michael Shulman

We present a method of constructing monoidal, braided monoidal, and symmetric monoidal bicategories from corresponding types of monoidal double categories that satisfy a lifting condition. Many important monoidal bicategories arise … We present a method of constructing monoidal, braided monoidal, and symmetric monoidal bicategories from corresponding types of monoidal double categories that satisfy a lifting condition. Many important monoidal bicategories arise naturally in this way, and applying our general method is much easier than explicitly verifying the coherence laws of a monoidal bicategory for each example. Abstracting from earlier work in this direction, we express the construction as a functor between locally cubical bicategories that preserves monoid objects; this ensures that it also preserves monoidal functors, transformations, adjunctions, and so on. Examples include the monoidal bicategories of algebras and bimodules, categories and profunctors, sets and spans, open Markov processes, parametrized spectra, and various functors relating them.

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Iterated icons

2013-08-29

Eugenia Cheng , Nick Gurski

We study the totality of weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric … We study the totality of weakly enriched in a monoidal bicategory using a notion of enriched icon as 2-cells. We show that when the monoidal bicategory in question is symmetric then this process can be iterated. We show that starting from the symmetric monoidal bicategory Cat and performing the construction twice yields a convenient symmetric monoidal bicategory of partially strict tricategories. We show that restricting to the doubly degenerate ones immediately gives the correct bicategory of 2-tuply monoidal categories missing from our earlier studies of the Periodic Table. We propose a generalisation to all k-tuply monoidal n-categories.

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fc-multicategories

1999-02-28

Tom Leinster

fc-multicategories are a very general kind of two-dimensional structure, encompassing bicategories, monoidal categories, double categories and ordinary multicategories. We define them and explain how they provide a natural setting for … fc-multicategories are a very general kind of two-dimensional structure, encompassing bicategories, monoidal categories, double categories and ordinary multicategories. We define them and explain how they provide a natural setting for two familiar categorical ideas. The first is the bimodules construction, traditionally carried out on suitably cocomplete bicategories but perhaps more naturally carried out on fc-multicategories. The second is enrichment: there is a theory of categories enriched in an fc-multicategory, extending the usual theory of enrichment in a monoidal category. We finish by indicating how this work is just the simplest case of a much larger phenomenon.

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fc-multicategories

1999-01-01

Tom Leinster

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Equivalences in Bicategories

2018-01-01

Omar Abbad

Unbiased multicategory theory

2024-09-16

Caterina Pisani

We present an unbiased theory of symmetric multicategories, where sequences are replaced by families. To be effective, this approach requires an explicit consideration of indexing and reindexing of objects and … We present an unbiased theory of symmetric multicategories, where sequences are replaced by families. To be effective, this approach requires an explicit consideration of indexing and reindexing of objects and arrows, handled by the double category $\dPb$ of pullback squares in finite sets: a symmetric multicategory is a sum preserving discrete fibration of double categories $M: \dM\to \dPb$. If the \"loose" part of $M$ is an opfibration we get unbiased symmetric monoidal categories. The definition can be usefully generalized by replacing $\dPb$ with another double prop $\dP$, as an indexing base, giving $\dP$-multicategories. For instance, we can remove the finiteness condition to obtain infinitary symmetric multicategories, or enhance $\dPb$ by totally ordering the fibers of its loose arrows to obtain plain multicategories. We show how several concepts and properties find a natural setting in this framework. We also consider cartesian multicategories as algebras for a monad $(-)^\cart$ on $\sMlt$, where the loose arrows of $\dM^\cart$ are \"spans" of a tight and a loose arrow in $\dM$.

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A 3‐categorical perspective on G$G$‐crossed braided categories

2022-11-03

Corey Jones , David Penneys , David Reutter

A braided monoidal category may be considered a 3-category with one object and one 1-morphism. In this paper, we show that, more generally, 3-categories with one object and 1-morphisms given … A braided monoidal category may be considered a 3-category with one object and one 1-morphism. In this paper, we show that, more generally, 3-categories with one object and 1-morphisms given by elements of a group G $G$ correspond to G $G$ -crossed braided categories, certain mathematical structures which have emerged as important invariants of low-dimensional quantum field theories. More precisely, we show that the 4-category of 3-categories C $\mathcal {C}$ equipped with a 3-functor B G → C $\mathrm{B}G \rightarrow \mathcal {C}$ which is essentially surjective on objects and 1-morphisms is equivalent to the 2-category of G $G$ -crossed braided categories. This provides a uniform approach to various constructions of G $G$ -crossed braided categories.

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Norms in motivic homotopy theory

2017-11-08

Tom Bachmann , Marc Hoyois

If $f:S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal functor $f_\otimes: \mathcal H_*(S') \to\mathcal H_*(S)$, where $\mathcal H_*(S)$ is the pointed unstable … If $f:S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal functor $f_\otimes: \mathcal H_*(S') \to\mathcal H_*(S)$, where $\mathcal H_*(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite etale, we show that it stabilizes to a functor $f_\otimes: \mathcal{SH}(S') \to \mathcal{SH}(S)$, where $\mathcal{SH}(S)$ is the $\mathbb P^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb Z$, the homotopy K-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb Z$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.

Morita Bicategories of Algebras and Duality Involutions

2019-02-13

Jonathan Lorand , Alessandro Valentino

The notion of a weak duality involution on a bicategory was recently introduced by Shulman in [arXiv:1606.05058]. We construct a weak duality involution on the fully dualisable part of $\text{Alg}$, … The notion of a weak duality involution on a bicategory was recently introduced by Shulman in [arXiv:1606.05058]. We construct a weak duality involution on the fully dualisable part of $\text{Alg}$, the Morita bicategory of finite-dimensional k-algebras. The 2-category $\text{KV}$ of Kapranov-Voevodsky k-vector spaces may be equipped with a canonical strict duality involution. We show that the pseudofunctor $\text{Rep}: \text{Alg}^{fd} \to \text{KV}$ sending an algebra to its category of finite-dimensional modules may be canonically equipped with the structure of a duality pseudofunctor. Thus $\text{Rep}$ is a strictification in the sense of Shulman's strictification theorem for bicategories with a weak duality involution. Finally, we present a general setting for duality involutions on the Morita bicategory of algebras in a semisimple symmetric finite tensor category.

The tricategory of formal composites and its strictification

2019-01-01

Peter Guthmann

The results of this thesis allows one to replace calculations in tricategories with equivalent calculations in Gray categories (aka semistrict tricategories). In particular the rewriting calculus for Gray categories as … The results of this thesis allows one to replace calculations in tricategories with equivalent calculations in Gray categories (aka semistrict tricategories). In particular the rewriting calculus for Gray categories as used for example by the online proof assistant globular (arXiv:1612.01093), or equivalently the Gray-diagrams of arXiv:1211.0529 can then be used also in the case of a fully weak tricategory.

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Cartesian closed bicategories: type theory and coherence

2020-07-15

Philip Saville

In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a … In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for cartesian closed bicategories. Cartesian closed bicategories---2-categories `up to isomorphism' equipped with similarly weak products and exponentials---arise in logic, categorical algebra, and game semantics. I show that there is at most one 2-cell between any parallel pair of 1-cells in the free cartesian closed bicategory on a set and hence---in terms of the difficulty of calculating---bring the data of cartesian closed bicategories down to the familiar level of cartesian closed categories. In fact, I prove this result in two ways. The first argument is closely related to Power's coherence theorem for bicategories with flexible bilimits. For the second, which is the central preoccupation of this thesis, the proof strategy has two parts: the construction of a type theory, and the proof that it satisfies a form of normalisation I call local coherence. I synthesise the type theory from algebraic principles using a novel generalisation of the (multisorted) abstract clones of universal algebra, called biclones. Using a bicategorical treatment of M. Fiore's categorical analysis of normalisation-by-evaluation, I then prove a normalisation result which entails the coherence theorem for cartesian closed bicategories. In contrast to previous coherence results for bicategories, the argument does not rely on the theory of rewriting or strictify using the Yoneda embedding. Along the way I prove bicategorical generalisations of a series of well-established category-theoretic results, present a notion of glueing of bicategories, and bicategorify the folklore result providing sufficient conditions for a glueing category to be cartesian closed.

Contravariance through enrichment

2016-01-01

Michael Shulman

We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict … We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution. For this purpose we introduce "2-categories with contravariance", a sort of enhanced 2-category with a basic notion of "contravariant morphism", which can be regarded either as generalized multicategories or as enriched categories. This enables a universal characterization of duality involutions using absolute weighted colimits, leading to a conceptual proof of the coherence theorem.

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The categorified Grothendieck--Riemann--Roch theorem

2018-04-03

Marc Hoyois , Pavel Safronov , Sarah Scherotzke +1 more

In this paper we prove a categorification of the Grothendieck-Riemann-Roch theorem. Our result implies in particular a Grothendieck-Riemann-Roch theorem for Toen and Vezzosi's secondary Chern character. As a main application, … In this paper we prove a categorification of the Grothendieck-Riemann-Roch theorem. Our result implies in particular a Grothendieck-Riemann-Roch theorem for Toen and Vezzosi's secondary Chern character. As a main application, we establish a comparison between the Toen-Vezzosi Chern character and the classical Chern character, and show that the categorified Chern character recovers the classical de Rham realization.

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Infinite loop spaces, and coherence for symmetric monoidal bicategories

2013-07-25

Nick Gurski , Angélica M. Osorno

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K-theory for 2-categories

2017-11-13

Nick Gurski , Niles Johnson , Angélica M. Osorno

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List of Main Facts

2021-01-31

Extract Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Extract Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12

Dedication

2021-01-31

Extract The first author dedicates this book to his wife, Nemili. The second author dedicates this book to Eun Soo and Jacqueline. Extract The first author dedicates this book to his wife, Nemili. The second author dedicates this book to Eun Soo and Jacqueline.

Copyright Page

2021-01-31

Extract Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, … Extract Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Niles Johnson and Donald Yau 2021 The moral rights of the authors have been asserted First Edition published in 2021 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address...

Preface

2021-01-31

Extract 2-Dimensional Categories The theory of 2-dimensional categories, which includes 2-categories and bicategories, is a fundamental part of modern category theory with a wide range of applications not only in … Extract 2-Dimensional Categories The theory of 2-dimensional categories, which includes 2-categories and bicategories, is a fundamental part of modern category theory with a wide range of applications not only in mathematics but also in physics [BN96, KV94a, KV94b, KTZ20, Par18, SP∞], computer science [PL07], and linguistics [Lam04, Lam11]. The basic definitions and properties of 2-categories and bicategories were introduced by Bénabou in [Bén65] and [Bén67], respectively. The one-object case is illustrative: a monoid, which is a set with a unital and associative multiplication, is a one-object category. A monoidal category, which is a category with a product that is associative and unital up to coherent isomorphisms, is a one-object bicategory. The definition of a bicategory is obtained from that of a category by replacing the hom sets with hom categories, the composition and identities with functors,...

List of Notations

2021-01-31

Fell bundles over quasi-lattice ordered groups and $\mathrm{C}^*$-algebras of compactly aligned product systems

2020-01-01

Camila F. Sehnem

We define notions of semi-saturatedness and orthogonality for a Fell bundle over a quasi-lattice ordered group. We show that a compactly aligned product system of Hilbert bimodules can be naturally … We define notions of semi-saturatedness and orthogonality for a Fell bundle over a quasi-lattice ordered group. We show that a compactly aligned product system of Hilbert bimodules can be naturally extended to a semi-saturated and orthogonal Fell bundle whenever it is simplifiable. Conversely, a semi-saturated and orthogonal Fell bundle is completely determined by the positive fibres and its cross sectional $\mathrm{C}^*$-algebra is isomorphic to a relative Cuntz--Pimsner algebra of a simplifiable product system of Hilbert bimodules. We show that this correspondence is part of an equivalence between bicategories and use this to generalise several results of Meyer and the author in the context of single correspondences. We apply functoriality for relative Cuntz--Pimsner algebras to study Morita equivalence between $\mathrm{C}^*$-algebras attached to compactly aligned product systems over Morita equivalent $\mathrm{C}^*$-algebras.

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Stable homotopy hypothesis in the Tamsamani model

2022-04-09

Lyne Moser , Viktoriya Ozornova , Simona Paoli +2 more

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Bicategories and 2-Categories

2021-01-31

Niles Johnson , Donald Yau

Abstract In this chapter, 2-categories and bicategories are defined, along with basic examples. Several useful unity properties in bicategories, generalizing those in monoidal categories and underlying many fundamental results in … Abstract In this chapter, 2-categories and bicategories are defined, along with basic examples. Several useful unity properties in bicategories, generalizing those in monoidal categories and underlying many fundamental results in bicategory theory, are discussed. In addition to well-known examples, the 2-categories of multicategories and of polycategories are constructed. This chapter ends with a discussion of duality of bicategories.

A 3-categorical perspective on G-crossed braided categories

2020-09-01

Corey Jones , David Penneys , David Reutter

A braided monoidal category may be considered a $3$-category with one object and one $1$-morphism. In this paper, we show that, more generally, $3$-categories with one object and $1$-morphisms given … A braided monoidal category may be considered a $3$-category with one object and one $1$-morphism. In this paper, we show that, more generally, $3$-categories with one object and $1$-morphisms given by elements of a group $G$ correspond to $G$-crossed braided categories, certain mathematical structures which have emerged as important invariants of low-dimensional quantum field theories. More precisely, we show that the 4-category of $3$-categories $\mathcal{C}$ equipped with a 3-functor $\mathrm{B}G \to \mathcal{C}$ which is essentially surjective on objects and $1$-morphisms is equivalent to the $2$-category of $G$-crossed braided categories. This provides a uniform approach to various constructions of $G$-crossed braided categories.

The 2-dimensional stable homotopy hypothesis

2019-01-23

Nick Gurski , Niles Johnson , Angélica M. Osorno

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Data structures for quasistrict higher categories

2017-06-01

Krzysztof Bar , Jamie Vicary

We present new data structures for quasistrict higher categories, in which associativity and unit laws hold strictly. Our approach has low axiomatic complexity compared to traditional algebraic approaches, and gives … We present new data structures for quasistrict higher categories, in which associativity and unit laws hold strictly. Our approach has low axiomatic complexity compared to traditional algebraic approaches, and gives a practical method for performing calculations in quasistrict 4-categories. It is amenable to computer implementation, and we exploit this to give a machine-verified algebraic proof that every adjunction of 1-cells in a quasistrict 4-category can be promoted to a coherent adjunction satisfying the butterfly equations.

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What is an equivalence in a higher category?

2023-11-14

Viktoriya Ozornova , Martina Rovelli

Abstract The purpose of this survey is to present in a uniform way the notion of equivalence between strict ‐categories or ‐categories, and inside a strict ‐category or ‐category. Abstract The purpose of this survey is to present in a uniform way the notion of equivalence between strict ‐categories or ‐categories, and inside a strict ‐category or ‐category.

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A model structure for weakly horizontally invariant double categories

2023-06-14

Lyne Moser , Maru Sarazola , Paula Verdugo

We construct a model structure on the category DblCat of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal … We construct a model structure on the category DblCat of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and vertical morphisms, and fully faithful on squares; and whose fibrant objects are the weakly horizontally invariant double categories.We show that the functor H ≃ : 2Cat → DblCat, a more homotopical version of the usual horizontal embedding H, is right Quillen and homotopically fully faithful when considering Lack's model structure on 2Cat.In particular, H ≃ exhibits a levelwise fibrant replacement of H.Moreover, Lack's model structure on 2Cat is right-induced along H ≃ from the model structure for weakly horizontally invariant double categories.We also show that this model structure is monoidal with respect to Böhm's Gray tensor product.Finally, we prove a Whitehead Theorem characterizing the weak equivalences with fibrant source as the double functors which admit a pseudo inverse up to horizontal pseudo natural equivalence.

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The categorified Grothendieck–Riemann–Roch theorem

2021-01-01

Marc Hoyois , Pavel Safronov , Sarah Scherotzke +1 more

In this paper we prove a categorification of the Grothendieck–Riemann–Roch theorem. Our result implies in particular a Grothendieck–Riemann–Roch theorem for Toën and Vezzosi's secondary Chern character. As a main application, … In this paper we prove a categorification of the Grothendieck–Riemann–Roch theorem. Our result implies in particular a Grothendieck–Riemann–Roch theorem for Toën and Vezzosi's secondary Chern character. As a main application, we establish a comparison between the Toën–Vezzosi Chern character and the classical Chern character, and show that the categorified Chern character recovers the classical de Rham realization.

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Coherence in Three-Dimensional Category Theory

2013-03-21

Nick Gurski

Dimension three is an important test-bed for hypotheses in higher category theory and occupies something of a unique position in the categorical landscape. At the heart of matters is the … Dimension three is an important test-bed for hypotheses in higher category theory and occupies something of a unique position in the categorical landscape. At the heart of matters is the coherence theorem, of which this book provides a definitive treatment, as well as covering related results. Along the way the author treats such material as the Gray tensor product and gives a construction of the fundamental 3-groupoid of a space. The book serves as a comprehensive introduction, covering essential material for any student of coherence and assuming only a basic understanding of higher category theory. It is also a reference point for many key concepts in the field and therefore a vital resource for researchers wishing to apply higher categories or coherence results in fields such as algebraic topology or theoretical computer science.

Realization of rigid C⁎-bicategories as bimodules over type II1 von Neumann algebras

2023-02-01

Luca Giorgetti , Wei Yuan

We prove that every rigid C⁎-bicategory with finite-dimensional centers (finitely decomposable horizontal units) can be realized as Connes' bimodules over finite direct sums of II1 factors. In particular, we realize … We prove that every rigid C⁎-bicategory with finite-dimensional centers (finitely decomposable horizontal units) can be realized as Connes' bimodules over finite direct sums of II1 factors. In particular, we realize every multitensor C⁎-category as bimodules over a finite direct sum of II1 factors.

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Identity in the Presence of Adjunction

2024-07-15

Mateusz Stroiński

Abstract We develop a theory of adjunctions in semigroup categories, that is, monoidal categories without a unit object. We show that a rigid semigroup category is promonoidal, and thus one … Abstract We develop a theory of adjunctions in semigroup categories, that is, monoidal categories without a unit object. We show that a rigid semigroup category is promonoidal, and thus one can naturally adjoin a unit object to it. This extends the previous results of Houston in the symmetric case, and addresses a question of his. It also extends the results in the non-symmetric case with additional finiteness assumptions, obtained by Benson–Etingof–Ostrik, Coulembier, and Ko–Mazorchuk–Zhang. We give an interpretation of these results using comonad cohomology, and, in the absence of finiteness conditions, using enriched traces of monoidal categories. As an application of our results, we give a characterization of finite tensor categories in terms of the finitary $2$-representation theory of Mazorchuk–Miemietz.

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Norms in motivic homotopy theory

2021-09-01

Tom Bachmann , Marc Hoyois

If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed … If $f : S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes : \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes : \mathcal{S}\mathcal{H}(S') \to \mathcal{S}\mathcal{H}(S)$, where $\mathcal{S}\mathcal{H}(S)$ is the $\mathbb{P}^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendieck's Galois theory, with Betti realization, and with Voevodsky's slice filtration; we prove that the norm functors categorify Rost's multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $H\mathbb{Z}$, the homotopy $K$-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $H\mathbb{Z}$ is a common refinement of Fulton and MacPherson's mutliplicative transfers on Chow groups and of Voevodsky's power operations in motivic cohomology.

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2-Dimensional Categories

2021-01-31

Niles Johnson , Donald Yau

Abstract 2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic … Abstract 2-Dimensional Categories provides an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories; pasting diagrams; lax functors; 2-/bilimits; the Duskin nerve; the 2-nerve; internal adjunctions; monads in bicategories; 2-monads; biequivalences; the Bicategorical Yoneda Lemma; and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.

Constructing Higher Inductive Types as Groupoid Quotients

2021-04-22

Niccolò Veltri , Niels van der Weide

In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. … In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. We define an internal notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. Then we construct a biinitial object in the bicategory of algebras in groupoids, which gives the desired algebra. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. We present several examples of HITs which are definable using our notion of signature. In particular, we show that each signature gives rise to a HIT corresponding to the freely generated algebraic structure over it. We also start the development of universal algebra in 1-types. We show that the bicategory of algebras has PIE limits, i.e. products, inserters and equifiers, and we prove a version of the first isomorphism theorem for 1-types. Finally, we give an alternative characterization of the foundamental groups of some HITs, exploiting our construction of HITs via the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.

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A category of kernels for equivariant factorizations, II: further implications

2013-10-09

Matthew R. Ballard , David Favero , Ludmil Katzarkov

We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories … We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories of coherent sheaves on varieties. In particular, we provide a conjectural geometric framework to further understand M. Kontsevich's Homological Mirror Symmetry conjecture. We obtain new cases of a conjecture of Orlov concerning the Rouquier dimension of the bounded derived category of coherent sheaves on a smooth variety. Further, we introduce actions of $A$-graded commutative rings on triangulated categories and their associated Noether-Lefschetz spectra as a new invariant of triangulated categories. They are intended to encode information about algebraic classes in the cohomology of an algebraic variety. We provide some examples to motivate the connection.

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A category of kernels for graded matrix factorizations and its implications for Hodge theory

2011-05-16

Matthew R. Ballard , David Favero , Ludmil Katzarkov

We provide a matrix factorization model for the derived internal Hom (continuous), in the homotopy category of k-linear dg-categories, between categories of graded matrix factorizations. This description is used to … We provide a matrix factorization model for the derived internal Hom (continuous), in the homotopy category of k-linear dg-categories, between categories of graded matrix factorizations. This description is used to calculate the derived natural transformations between twists functors on categories of graded matrix factorizations. Furthermore, we combine our model with a theorem of Orlov to establish a geometric picture related to Kontsevich's Homological Mirror Symmetry Conjecture. As applications, we obtain new cases of a conjecture of Orlov concerning the Rouquier dimension of the bounded derived category of coherent sheaves on a smooth variety and a proof of the Hodge conjecture for n-fold products of a K3 surface closely related to the Fermat cubic fourfold. We also introduce Noether-Lefschetz spectra as a new Morita invariant of dg-categories. They are intended to encode information about algebraic classes in the cohomology on an algebraic variety.

Higher categorical structures in algebraic topology: classifying spaces and homotopy coherence

2016-01-01

B. A. Heredia

Burnside rings for Real $2$-representation theory: The linear theory

2019-06-26

Dmitriy Rumynin , Matthew B. Young

This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant … This paper is a fundamental study of the Real $2$-representation theory of $2$-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a $2$-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real $2$-representations as a Real variant of the Burnside ring of the fundamental group of the $2$-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to $2$-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real $2$-representation theory of $2$-groups, as studied by the second author.

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Lax familial representability and lax generic factorizations

2018-12-23

Charles Walker

A classical result due to Diers shows that a copresheaf $F\colon\mathcal{A}\to\mathbf{Set}$ on a category $\mathcal{A}$ is a coproduct of representables precisely when each connected component of $F$'s category of elements … A classical result due to Diers shows that a copresheaf $F\colon\mathcal{A}\to\mathbf{Set}$ on a category $\mathcal{A}$ is a coproduct of representables precisely when each connected component of $F$'s category of elements has an initial object. Most often, this condition is imposed on a copresheaf of the form $\mathcal{B}\left(X,T-\right)$ for a functor $T\colon\mathcal{A}\to\mathcal{B}$, in which case this property says that $T$ admits generic factorizations at $X$, or equivalently that $T$ is familial at $X$. Here we generalize these results to the two-dimensional setting, replacing $\mathcal{A}$ with an arbitrary bicategory $\mathscr{A}$, and $\mathbf{Set}$ with $\mathbf{Cat}$. In this two-dimensional setting, simply asking that a pseudofunctor $F\colon\mathscr{A}\to\mathbf{Cat}$ be a coproduct of representables is often too strong of a condition. Instead, we will only ask that $F$ be a lax conical colimit of representables. This in turn allows for the weaker notion of lax generic factorizations (and lax familial representability) for pseudofunctors of bicategories $T\colon\mathscr{A}\to\mathscr{B}$. We also compare our lax familial pseudofunctors to Weber's familial 2-functors, finding our description is more general (not requiring a terminal object in $\mathscr{A}$), though essentially equivalent when a terminal object does exist. Moreover, our description of lax generics allows for an equivalence between lax generic factorizations and lax familial representability. Finally, we characterize our lax familial pseudofunctors as right lax $\mathsf{F}$-adjoints followed by locally discrete fibrations of bicategories, which in turn yields a simple definition of parametric right adjoint pseudofunctors.

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Pasting Diagrams

2021-01-31

Niles Johnson , Donald Yau

Abstract In this chapter, pasting diagrams are defined, and pasting theorems for 2-/bicategories are proved. Pasting is an essential reasoning tool in 2-dimensional category theory. Each pasting theorem says that … Abstract In this chapter, pasting diagrams are defined, and pasting theorems for 2-/bicategories are proved. Pasting is an essential reasoning tool in 2-dimensional category theory. Each pasting theorem says that a pasting diagram, in a 2-category or a bicategory, has a unique composite. String diagrams, which provide another way to visualize and manipulate pasting diagrams, are also discussed.

Comparing geometric realizations of tricategories

2014-08-28

Antonio M. Cegarra , B. A. Heredia

This paper contains some contributions to the study of classifying spaces for tricategories, with applications to the homotopy theory of monoidal categories, bicategories, braided monoidal categories and monoidal bicategories.Any small … This paper contains some contributions to the study of classifying spaces for tricategories, with applications to the homotopy theory of monoidal categories, bicategories, braided monoidal categories and monoidal bicategories.Any small tricategory has various associated simplicial or pseudosimplicial objects and we explore the relationship between three of them: the pseudosimplicial bicategory (so-called Grothendieck nerve) of the tricategory, the simplicial bicategory termed its Segal nerve and the simplicial set called its Street geometric nerve.We prove that the geometric realizations of all of these 'nerves of the tricategory' are homotopy equivalent.By using Grothendieck nerves we state the precise form in which the process of taking classifying spaces transports tricategorical coherence to homotopy coherence.Segal nerves allow us to prove that, under natural requirements, the classifying space of a monoidal bicategory is, in a precise way, a loop space.With the use of geometric nerves, we obtain simplicial sets whose simplices have a pleasing geometrical description in terms of the cells of the tricategory and we prove that, via the classifying space construction, bicategorical groups are a convenient algebraic model for connected homotopy 3-types.

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Bicategories in univalent foundations

2021-11-01

Benedikt Ahrens , Dan Frumin , Marco Maggesi +2 more

We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples … We develop bicategory theory in univalent foundations. Guided by the notion of univalence for (1-)categories studied by Ahrens, Kapulkin, and Shulman, we define and study univalent bicategories. To construct examples of univalent bicategories in a modular fashion, we develop displayed bicategories, an analog of displayed 1-categories introduced by Ahrens and Lumsdaine. We demonstrate the applicability of this notion, and prove that several bicategories of interest are univalent. Among these are the bicategory of univalent categories with families and the bicategory of pseudofunctors between univalent bicategories. Furthermore, we show that every bicategory with univalent hom-categories is weakly equivalent to a univalent bicategory. All of our work is formalized in Coq as part of the UniMath library of univalent mathematics.

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Coinductive Invertibility in Higher Categories

2020-08-24

A. Rice

Invertibility is an important concept in category theory. In higher category theory, it becomes less obvious what the correct notion of invertibility is, as extra coherence conditions can become necessary … Invertibility is an important concept in category theory. In higher category theory, it becomes less obvious what the correct notion of invertibility is, as extra coherence conditions can become necessary for invertible structures to have desirable properties. We define some properties we expect to hold in any reasonable definition of a weak $\omega$-category. With these properties we define three notions of invertibility inspired by homotopy type theory. These are quasi-invertibility, where a two sided inverse is required, bi-invertibility, where a separate left and right inverse is given, and half-adjoint inverse, which is a quasi-inverse with an extra coherence condition. These definitions take the form of coinductive data structures. Using coinductive proofs we are able to show that these three notions are all equivalent in that given any one of these invertibility structures, the others can be obtained. The methods used to do this are generic and it is expected that the results should be applicable to any reasonable model of higher category theory. Many of the results of the paper have been formalised in Agda using coinductive records and the machinery of sized types.

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On C*-algebras associated to product systems

2018-01-01

Camila F. Sehnem

We consider a class of Fell bundles over quasi-lattice ordered groups. We show that these are completely determined by the positive fibres and that their cross sectional C*-algebras are relative … We consider a class of Fell bundles over quasi-lattice ordered groups. We show that these are completely determined by the positive fibres and that their cross sectional C*-algebras are relative Cuntz–Pimsner algebras associated to simplifiable product systems of Hilbert bimodules. Conversely, we show that such product systems can be naturally extended to Fell bundles and this correspondence is part of an equivalence between bicategories. We also relate amenability for this class of Fell bundles to amenability of quasi-lattice orders by showing that Fell bundles extended from free semigroups are amenable. A similar result is proved for Baumslag–Solitar groups. Moreover, we construct a relative Cuntz–Pimsner algebra of a compactly aligned product system as a quotient of the associated Nica–Toeplitz algebra. We show that this construction yields a reflector from a bicategory of compactly aligned product systems into its sub-bicategory of simplifiable product systems of Hilbert bimodules. We use this to study Morita equivalence between relative Cuntz–Pimsner algebras. In a second part, we let P be a unital subsemigroup of a group G. We propose an approach to C*-algebras associated to product systems over P. We call the C*-algebra of a given product system E its covariance algebra and denote it by A x_E P, where A is the coefficient C*-algebra. We prove that our construction does not depend on the embedding P->G and that a representation of A x_E P is faithful on the fixed-point algebra for the canonical coaction of G if and only if it is faithful on A. We compare this with other constructions in the setting of irreversible dynamical systems, such as Cuntz–Nica–Pimsner algebras, Fowler’s Cuntz–Pimsner algebra, semigroup C*-algebras of Xin Li and Exel’s crossed products by interaction groups.

ON PROPERTY-LIKE STRUCTURES

1997-01-01

G. M. Kelly , Stephen Lack

A may bear many monoidal structures, but (to within a isomorphism) only one of category with finite products. To capture such distinctions, we consider on a 2-category those 2-monads for … A may bear many monoidal structures, but (to within a isomorphism) only one of category with finite products. To capture such distinctions, we consider on a 2-category those 2-monads for which algebra is if it exists, giving a precise mathematical definition of essentially unique and investigating its consequences. We call such 2-monads property-like. We further consider the more restricted class of fully property-like 2-monads, consisting of those property-like 2-monads for which all 2-cells between (even lax) algebra morphisms are algebra 2-cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which structure is adjoint to unit, and which we now call lax-idempotent 2-monads: both these and their colax-idempotent duals are fully property-like. We end by showing that (at least for finitary 2-monads) the classes of property-likes, fully property-likes, and lax-idempotents are each coreflective among all 2-monads.

Monoidal Bicategories and Hopf Algebroids

1997-07-01

Brian Day , Ross Street

Braided Tensor Categories

1993-11-01

André Joyal , Ross Street

Coherence for tricategories

1995-01-01

Robert J. Gordon , John Power , Ross Street

Two-dimensional monad theory

1989-07-01

Robert Blackwell , G. M. Kelly , John Power

A Quillen model structure for Gray-categories

2010-09-24

Stephen Lack

Abstract A Quillen model structure on the category Gray-Cat of Gray -categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full … Abstract A Quillen model structure on the category Gray-Cat of Gray -categories is described, for which the weak equivalences are the triequivalences. It is shown to restrict to the full subcategory Gray-Gpd of Gray -groupoids. This is used to provide a functorial and model-theoretic proof of the unpublished theorem of Joyal and Tierney that Gray -groupoids model homotopy 3-types. The model structure on Gray-Cat is conjectured to be Quillen equivalent to a model structure on the category Tricat of tricategories and strict homomorphisms of tricategories.

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A Coherent Approach to Pseudomonads

2000-06-01

Stephen Lack

Coherence for bicategories and indexed categories

1985-01-01

Saunders MacLane , Robert Paré

Categories for the Working Mathematician

1971-01-01

Saunders Mac Lane

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Higher-Dimensional Algebra V: 2-Groups

2003-01-01

John C. Baez , Aaron D. Lauda

A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a … A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call "weak" and "coherent" 2-groups. A weak 2-group is a weak monoidal category in which every morphism has an inverse and every object x has a "weak inverse": an object y such that x tensor y and y tensor x are isomorphic to 1. A coherent 2-group is a weak 2-group in which every object x is equipped with a specified weak inverse x* and isomorphisms i_x: 1 -> x tensor x* and e_x: x* tensor x -> 1 forming an adjunction. We describe 2-categories of weak and coherent 2-groups and an "improvement" 2-functor that turns weak 2-groups into coherent ones, and prove that this 2-functor is a 2-equivalence of 2-categories. We internalize the concept of coherent 2-group, which gives a quick way to define Lie 2-groups. We give a tour of examples, including the "fundamental 2-group" of a space and various Lie 2-groups. We also explain how coherent 2-groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simply-connected compact simple Lie group G a family of 2-groups G_hbar (for integral values of hbar) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2-groups are built using Chern-Simons theory, and are closely related to the Lie 2-algebras g_hbar (for real hbar) described in a companion paper.

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