In this article we introduce four variance flavours of cartesian 2-fibrations of $\infty$-bicategories with $\infty$-bicategorical fibres, in the framework of scaled simplicial sets. Given a map $p\colon \mathcal{E} \rightarrow\mathcal{B}$ of …
In this article we introduce four variance flavours of cartesian 2-fibrations of $\infty$-bicategories with $\infty$-bicategorical fibres, in the framework of scaled simplicial sets. Given a map $p\colon \mathcal{E} \rightarrow\mathcal{B}$ of $\infty$-bicategories, we define $p$-(co)cartesian arrows and inner/outer triangles by means of lifting properties against $p$. Inner/outer (co)cartesian 2-fibrations are then defined to be maps with enough (co)cartesian lifts for arrows and enough inner/outer lifts for triangles, together with a compatibility property with respect to whiskerings in the outer case. By doing so, we also recover in particular the case of $\infty$-bicategories fibred in $\infty$-categories studied in previous work. We also prove that equivalences of such 2-fibrations can be tested fiberwise. As a motivating example, we show that the domain projection $\mathrm{d}\colon\mathrm{RMap}(\Delta^1,\mathcal{C})\rightarrow \mathcal{C}$ is a prototypical example of an outer cartesian 2-fibration, where $\mathrm{RMap}(X,Y)$ denotes the $\infty$-bicategory of functors, lax natural transformations and modifications. We then define inner/outer (co)cartesian 2-fibrations of categories enriched in $\infty$-categories, and we show that a fibration $p\colon \mathcal{E} \rightarrow \mathcal{B}$ of such categories is a (co)cartesian inner/outer 2-fibration if and only if the corresponding scaled nerve $\mathrm{N}^{\mathrm{sc}}(p)\colon \mathrm{N}^{\mathrm{sc}}\mathcal{E} \rightarrow \mathrm{N}^{\mathrm{sc}}\mathcal{B}$ is a fibration of this type between $\infty$-bicategories.
We introduce unary operadic 2-categories as a framework for operadic Grothendieck construction for categorical $\mathbb{O}$-operads, $\mathbb{O}$ being a unary operadic category. The construction is a fully faithful functor $\int_\mathbb{O}$ which …
We introduce unary operadic 2-categories as a framework for operadic Grothendieck construction for categorical $\mathbb{O}$-operads, $\mathbb{O}$ being a unary operadic category. The construction is a fully faithful functor $\int_\mathbb{O}$ which takes categorical $\mathbb{O}$-operads to operadic functors over $\mathbb{O}$, and we characterise its essential image by certain lifting properties. Such operadic functors are called operadic fibration. Our theory is an extension of the discrete (unary) operadic case and, in some sense, of the classical Grothendieck construction of a categorical presheaf. For the terminal unary operadic category $\odot$, a categorical $\odot$-operad is a strict monoidal category $\mathscr{V}$ and its Grothendieck construction $\int_\odot \mathscr{V}$ is connected to the `Para' construction appearing in machine learning. The 2-categorical setting provides a characterisation of $\mathbb{O}$-operads valued in $\mathscr{V}$ as operadic functors $\mathbb{O} \to \int_\odot \mathscr{V}$. Last, we describe a left adjoint to $\int_\odot$.
Abstract We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on …
Abstract We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using these results, we describe the homotopical behaviour not only of conical limits but also of weighted limits. Finally, pseudo-limits are related to homotopy limits.
Fibrations over a category B, introduced to category theory by Grothendieck, determine pseudo-functors B op ! Cat. A two-sided discrete variation models functors B op A! Set. By work of …
Fibrations over a category B, introduced to category theory by Grothendieck, determine pseudo-functors B op ! Cat. A two-sided discrete variation models functors B op A! Set. By work of Street, both notions can be dened internally to an arbitrary 2-category or bicategory. While the two- sided discrete brations model profunctors internally to Cat, unexpectedly, the dual two-sided codiscrete cobrations are necessary to model V-profunctors internally toV-Cat. There are many categorical prerequisites, particularly in the later sections, but we believe they are strictly easier than the topics below that take advantage of them. These notes were written to accompany a talk given in the Algebraic Topology and Category Theory Proseminar in the fall of 2010 at the University of Chicago.
Given 2-categories $\mathcal{C}$ and $\mathcal{D}$, let $\textrm{Lax}(\mathcal{C},\mathcal{D})$ denote the 2-category of lax functors, lax natural transformations and modifications, and $[\mathcal{C},\mathcal{D}]_\mathrm{lnt}$ its full sub-2-category of (strict) 2-functors. We give two isomorphic …
Given 2-categories $\mathcal{C}$ and $\mathcal{D}$, let $\textrm{Lax}(\mathcal{C},\mathcal{D})$ denote the 2-category of lax functors, lax natural transformations and modifications, and $[\mathcal{C},\mathcal{D}]_\mathrm{lnt}$ its full sub-2-category of (strict) 2-functors. We give two isomorphic constructions of a 2-category $\mathcal{C}\boxtimes\mathcal{D}$ satisfying $\textrm{Lax}(\mathcal{C},\textrm{Lax}(\mathcal{D},\mathcal{E})) \cong [\mathcal{C}\boxtimes \mathcal{D},\mathcal{E}]_\mathrm{lnt}$, hence generalising the case of the free distributive law $1\boxtimes 1$. We also discuss dual constructions.
Given 2-categories $\mathcal{C}$ and $\mathcal{D}$, let $\textrm{Lax}(\mathcal{C},\mathcal{D})$ denote the 2-category of lax functors, lax natural transformations and modifications, and $[\mathcal{C},\mathcal{D}]_\mathrm{lnt}$ its full sub-2-category of (strict) 2-functors. We give two isomorphic …
Given 2-categories $\mathcal{C}$ and $\mathcal{D}$, let $\textrm{Lax}(\mathcal{C},\mathcal{D})$ denote the 2-category of lax functors, lax natural transformations and modifications, and $[\mathcal{C},\mathcal{D}]_\mathrm{lnt}$ its full sub-2-category of (strict) 2-functors. We give two isomorphic constructions of a 2-category $\mathcal{C}\boxtimes\mathcal{D}$ satisfying $\textrm{Lax}(\mathcal{C},\textrm{Lax}(\mathcal{D},\mathcal{E})) \cong [\mathcal{C}\boxtimes \mathcal{D},\mathcal{E}]_\mathrm{lnt}$, hence generalising the case of the free distributive law $1\boxtimes 1$. We also discuss dual constructions.
Given 2-categories $\mathcal{C}$ and $\mathcal{D}$, let $\textrm{Lax}(\mathcal{C},\mathcal{D})$ denote the 2-category of lax functors, lax natural transformations and modifications, and $[\mathcal{C},\mathcal{D}]_\mathrm{lnt}$ its full sub-2-category of (strict) 2-functors. We give two isomorphic …
Given 2-categories $\mathcal{C}$ and $\mathcal{D}$, let $\textrm{Lax}(\mathcal{C},\mathcal{D})$ denote the 2-category of lax functors, lax natural transformations and modifications, and $[\mathcal{C},\mathcal{D}]_\mathrm{lnt}$ its full sub-2-category of (strict) 2-functors. We give two isomorphic constructions of a 2-category $\mathcal{C}\boxtimes\mathcal{D}$ satisfying $\textrm{Lax}(\mathcal{C},\textrm{Lax}(\mathcal{D},\mathcal{E})) \cong [\mathcal{C}\boxtimes \mathcal{D},\mathcal{E}]_\mathrm{lnt}$, hence generalising the case of the free distributive law $1\boxtimes 1$. We also discuss dual constructions.
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.
We investigate fibrancy conditions in the Thomason model structure on the category of small categories.In particular, we show that the category of weak equivalences of a partial model category is …
We investigate fibrancy conditions in the Thomason model structure on the category of small categories.In particular, we show that the category of weak equivalences of a partial model category is fibrant.Furthermore, we describe connections to calculi of fractions.
We investigate fibrancy conditions in the Thomason model structure on the category of small categories. In particular, we show that the category of weak equivalences of a partial model category …
We investigate fibrancy conditions in the Thomason model structure on the category of small categories. In particular, we show that the category of weak equivalences of a partial model category is fibrant. Furthermore, we describe connections to calculi of fractions.
We investigate fibrancy conditions in the Thomason model structure on the category of small categories. In particular, we show that the category of weak equivalences of a partial model category …
We investigate fibrancy conditions in the Thomason model structure on the category of small categories. In particular, we show that the category of weak equivalences of a partial model category is fibrant. Furthermore, we describe connections to calculi of fractions.
We construct a flagged ∞-category Corr of ∞-categories and bimodules among them. We prove that Corr classifies exponentiable fibrations. This representability of exponentiable fibrations extends that established by Lurie of …
We construct a flagged ∞-category Corr of ∞-categories and bimodules among them. We prove that Corr classifies exponentiable fibrations. This representability of exponentiable fibrations extends that established by Lurie of both coCartesian fibrations and Cartesian fibrations, as they are classified by the ∞-category of ∞-categories and its opposite, respectively. We introduce the flagged ∞-subcategories LCorr and RCorr of Corr , whose morphisms are those bimodules which are left-final and right-initial , respectively. We identify the notions of fibrations these flagged ∞-subcategories classify, and show that these ∞-categories carry universal left/right fibrations.
Waldhausen's $S_\bullet$-construction gives a way to define the algebraic $K$-theory space of a category with cofibrations. Specifically, the $K$-theory space of a category with cofibrations $\mathcal{C}$ can be defined as …
Waldhausen's $S_\bullet$-construction gives a way to define the algebraic $K$-theory space of a category with cofibrations. Specifically, the $K$-theory space of a category with cofibrations $\mathcal{C}$ can be defined as the loop space of the realization of the simplicial topological space $|iS_\bullet \mathcal{C} |$. Dyckerhoff and Kapranov observed that if $\mathcal{C}$ is chosen to be a proto-exact category, then this simplicial topological space is 2-Segal. A natural question is then what variants of this $S_\bullet$-construction give 2-Segal spaces. We find that for $|iS_\bullet \mathcal{C}|$, $S_\bullet\mathcal{C}$, $wS_\bullet\mathcal{C}$, and the simplicial set whose $n$th level is the set of isomorphism classes of $S_\bullet\mathcal{C}$, there are certain $2$-Segal maps which are always equivalences. However for all of these simplicial objects, none of the rest of the $2$-Segal maps have to be equivalences. We also reduce the question of whether $|wS_\bullet \mathcal{C}|$ is $2$-Segal in nice cases to the question of whether a simpler simplicial space is $2$-Segal. Additionally, we give a sufficient condition for $S_\bullet \mathcal{C}$ to be $2$-Segal. Along the way we introduce the notion of a generated category with cofibrations and provide an example where the levelwise realization of a simplicial category which is not $2$-Segal is $2$-Segal.
We construct a flagged $\infty$-category ${\sf Corr}$ of $\infty$-categories and bimodules among them. We prove that ${\sf Corr}$ classifies exponentiable fibrations. This representability of exponentiable fibrations extends that established by …
We construct a flagged $\infty$-category ${\sf Corr}$ of $\infty$-categories and bimodules among them. We prove that ${\sf Corr}$ classifies exponentiable fibrations. This representability of exponentiable fibrations extends that established by Lurie of both coCartesian fibrations and Cartesian fibrations, as they are classified by the $\infty$-category of $\infty$-categories and its opposite, respectively. We introduce the flagged $\infty$-subcategories ${\sf LCorr}$ and ${\sf RCorr}$ of ${\sf Corr}$, whose morphisms are those bimodules which are \emph{left final} and \emph{right initial}, respectively. We identify the notions of fibrations these flagged $\infty$-subcategories classify, and show that these $\infty$-categories carry universal left/right fibrations.
We construct a flagged $\infty$-category ${\sf Corr}$ of $\infty$-categories and bimodules among them. We prove that ${\sf Corr}$ classifies exponentiable fibrations. This representability of exponentiable fibrations extends that established by …
We construct a flagged $\infty$-category ${\sf Corr}$ of $\infty$-categories and bimodules among them. We prove that ${\sf Corr}$ classifies exponentiable fibrations. This representability of exponentiable fibrations extends that established by Lurie of both coCartesian fibrations and Cartesian fibrations, as they are classified by the $\infty$-category of $\infty$-categories and its opposite, respectively. We introduce the flagged $\infty$-subcategories ${\sf LCorr}$ and ${\sf RCorr}$ of ${\sf Corr}$, whose morphisms are those bimodules which are \emph{left final} and \emph{right initial}, respectively. We identify the notions of fibrations these flagged $\infty$-subcategories classify, and show that these $\infty$-categories carry universal left/right fibrations.
Suppose an extension map $U\colon \mathbb{T}_1 \to \mathbb{T}_0$ in the 2-category $\mathfrak{Con}$ of contexts for arithmetic universes satisfies a Chevalley criterion for being an (op)fibration in $\mathfrak{Con}$. If $M$ is …
Suppose an extension map $U\colon \mathbb{T}_1 \to \mathbb{T}_0$ in the 2-category $\mathfrak{Con}$ of contexts for arithmetic universes satisfies a Chevalley criterion for being an (op)fibration in $\mathfrak{Con}$. If $M$ is a model of $\mathbb{T}_0$ in an elementary topos $\mathcal{S}$ with nno, then the classifier $p\colon\mathcal{S}[\mathbb{T}_1/M]\to\mathcal{S}$ satisfies Johnstone's criterion for being an (op)fibration in the 2-category $\mathcal{E}\mathfrak{Top}$ of elementary toposes (with nno) and geometric morphisms. Along the way, we provide a convenient reformulation of Johnstone's criterion.
The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed categoryE, is generalised forE just having pullbacks. The 2-categorical analogue of the theory of polynomials and …
The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed categoryE, is generalised forE just having pullbacks. The 2-categorical analogue of the theory of polynomials and polynomial functors is given, and its rela- tionship with Street's theory of brations within 2-categories is explored. Johnstone's notion of \bagdomain data is adapted to the present framework to make it easier to completely exhibit examples of polynomial monads.
Let $\mathcal{C}$ be a representable 2-category, and $\mathfrak{T}_\bullet$ a 2-endofunctor of the arrow 2-category $\mathcal{C}^\downarrow$ such that (i) $\mathsf{cod} \mathfrak{T}_\bullet = \mathsf{cod}$ and (ii) $\mathfrak{T}_\bullet$ preserves proneness of morphisms in …
Let $\mathcal{C}$ be a representable 2-category, and $\mathfrak{T}_\bullet$ a 2-endofunctor of the arrow 2-category $\mathcal{C}^\downarrow$ such that (i) $\mathsf{cod} \mathfrak{T}_\bullet = \mathsf{cod}$ and (ii) $\mathfrak{T}_\bullet$ preserves proneness of morphisms in $\mathcal{C}^\downarrow$. Then $\mathfrak{T}_\bullet$ preserves fibrations and opfibrations in $\mathcal{C}$. The proof takes Street's characterization of (e.g.) opfibrations as pseudoalgebras for 2-monads $\mathfrak{L}_B$ on slice categories $\mathcal{C}/B$ and develops it by defining a 2-monad $\mathfrak{L}_\bullet$ on $\mathcal{C}^\downarrow$ that takes change of base into account, and uses known results on the lifting of 2-functors to pseudoalgebras.
Given a fibration E -> B and a class Sigma of arrows of B, one can construct the free fibration (on E over B such that all reindexing functors over …
Given a fibration E -> B and a class Sigma of arrows of B, one can construct the free fibration (on E over B such that all reindexing functors over elements of Sigma are equivalences.<br /> <br />In this work I give an explicit construction of this, and study its properties. For example, the construction preserves the property of being fibrewise discrete, and it commutes up to equivalence with fibrewise exact completions. I show that mathematically interesting situations are examples of this construction. In particular, subtoposes of the effective topos are treated.
This paper shows how it is possible to express many techniques of categorical domain theory in the general context of topical categories (where ‘topical’ means internal in the category Top …
This paper shows how it is possible to express many techniques of categorical domain theory in the general context of topical categories (where ‘topical’ means internal in the category Top of Grothendieck toposes with geometric morphisms). The underlying topos machinery is hidden by using a geometric form of constructive mathematics, which enables toposes as ‘generalized topological spaces’ to be treated in a transparently spatial way, and also shows the constructivity of the arguments. The theory of strongly algebraic (SFP) domains is given as a case study in which the topical category is Cartesian closed.
We characterize the tripos-to-topos construction of Hyland, Johnstone and Pitts as a biadjunction in a bicategory enriched category of equipment-like structures. These abstract concepts are necessary to handle the presence …
We characterize the tripos-to-topos construction of Hyland, Johnstone and Pitts as a biadjunction in a bicategory enriched category of equipment-like structures. These abstract concepts are necessary to handle the presence of oplax constructs --- the construction is only oplax functorial on certain classes of cartesian functors between triposes. A by-product of our analysis is the decomposition of the tripos-to-topos construction into two steps, the intermediate step being a weakened version of quasitoposes.
Abstract We focus on the transfer of some known orthogonal factorization systems from $$\mathsf {Cat}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Cat</mml:mi></mml:math> to the 2-category $${\mathsf {Fib}}(B)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Fib</mml:mi><mml:mo>(</mml:mo><mml:mi>B</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> of fibrations over a fixed base …
Abstract We focus on the transfer of some known orthogonal factorization systems from $$\mathsf {Cat}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Cat</mml:mi></mml:math> to the 2-category $${\mathsf {Fib}}(B)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Fib</mml:mi><mml:mo>(</mml:mo><mml:mi>B</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> of fibrations over a fixed base category B : the internal version of the comprehensive factorization , and the factorization systems given by (sequence of coidentifiers, discrete morphism) and (sequence of coinverters, conservative morphism) respectively. For the class of fibrewise opfibrations in $${\mathsf {Fib}}(B)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Fib</mml:mi><mml:mo>(</mml:mo><mml:mi>B</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> , the construction of the latter two simplify to a single coidentifier (respectively coinverter) followed by an internal discrete opfibration (resp. fibrewise opfibration in groupoids). We show how these results follow from their analogues in $$\mathsf {Cat}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Cat</mml:mi></mml:math> , providing suitable conditions on a 2-category $${\mathcal {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> , that allow the transfer of the construction of coinverters and coidentifiers from $${\mathcal {C}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> to $${\mathsf {Fib}}_{{\mathcal {C}}}(B)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Fib</mml:mi><mml:mi>C</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>B</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> .
Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put …
Presenting systems of differential equations in the form of diagrams has become common in certain parts of physics, especially electromagnetism and computational physics. In this work, we aim to put such use of diagrams on a firm mathematical footing, while also systematizing a broadly applicable framework to reason formally about systems of equations and their solutions. Our main mathematical tools are category-theoretic diagrams, which are well known, and morphisms between diagrams, which have been less appreciated. As an application of the diagrammatic framework, we show how complex, multiphysical systems can be modularly constructed from basic physical principles. A wealth of examples, drawn from electromagnetism, transport phenomena, fluid mechanics, and other fields, is included.
Topos approaches to quantum foundations are described in a unified way by means of spectral bundles, where the base space is a space of contexts and each fibre is its …
Topos approaches to quantum foundations are described in a unified way by means of spectral bundles, where the base space is a space of contexts and each fibre is its spectrum. Differences in variance are due to the bundle being a fibration or opfibration. Relative to this structure, the probabilistic predictions of the Born rule in finite dimensional settings are then described as a section of a bundle of valuations. The construction uses in an essential way the geometric nature of the valuation locale monad.
Abstract We show that in the category Kelley of Hausdorff k-spaces a map is exponentiable if and only if it is open and that any open surjection is an effective …
Abstract We show that in the category Kelley of Hausdorff k-spaces a map is exponentiable if and only if it is open and that any open surjection is an effective descent morphism.
The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks. The 2-categorical analogue of the theory of …
The theory developed by Gambino and Kock, of polynomials over a locally cartesian closed category E, is generalised for E just having pullbacks. The 2-categorical analogue of the theory of polynomials and polynomial functors is given, and its relationship with Street's theory of fibrations within 2-categories is explored. Johnstone's notion of bagdomain data is adapted to the present framework to make it easier to completely exhibit examples of polynomial monads.
We prove a general theorem relating pseudo-exponentiable objects of a bicategory K to those of the Kleisli bicategory of a pseudo-monad on K. This theorem is applied to obtain pseudo-exponentiable …
We prove a general theorem relating pseudo-exponentiable objects of a bicategory K to those of the Kleisli bicategory of a pseudo-monad on K. This theorem is applied to obtain pseudo-exponentiable objects of the homotopy slices Top//B of the category of topological spaces and the pseudo-slices Cat//B of the category of small categories.
We propose a general notion of model for two-dimensional type theory, in the form of comprehension bicategories. Examples of comprehension bicategories are plentiful; they include interpretations of directed type theory …
We propose a general notion of model for two-dimensional type theory, in the form of comprehension bicategories. Examples of comprehension bicategories are plentiful; they include interpretations of directed type theory previously studied in the literature.
A classical result due to Diers shows that a presheaf $F\colon\mathcal{A}\to\mathbf{Set}$ 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 presheaf $F\colon\mathcal{A}\to\mathbf{Set}$ 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 presheaf of the form $\mathcal{B}\left(X,L-\right)$ for a functor $L\colon\mathcal{A}\to\mathcal{B}$, in which case this property says that $L$ admits generic factorisations at $X$, or equivalently that $L$ has a left multiadjoint 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 bi-presheaf $F\colon\mathscr{A}\to\mathbf{Cat}$ 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 is turn allows for the weaker notion of lax generic factorisations (and lax multiadjoints) for pseudofunctors of bicategories $L\colon\mathscr{A}\to\mathscr{B}$.
We also compare our lax multiadjoints 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 factorisations and famility.
Finally, we characterize our lax multiadjoints as right lax $\mathsf{F}$-adjoints followed by locally discrete fibrations of bicategories, which in turn yields a more natural definition of parametric right adjoint pseudofunctors.
Categories, $n$-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define …
Categories, $n$-categories, double categories, and multicategories (among others) all have similar definitions as collections of cells with composition operations. We give an explicit description of the information required to define any higher category structure which arises as algebras for a familially representable monad on a presheaf category, then use this to describe various examples relating to higher category theory and cubical sets. The proof of this characterization avoids tedious naturality arguments by passing through the theory of categorical polynomials.
We focus on two factorization systems for opfibrations in the 2-category Fib(B) of fibrations over a fixed base category B. The first one is the internal version of the so …
We focus on two factorization systems for opfibrations in the 2-category Fib(B) of fibrations over a fixed base category B. The first one is the internal version of the so called comprehensive factorization, where the right orthogonal class is given by internal discrete opfibrations. The second one has as its right orthogonal class internal opfibrations in groupoids, i.e. with groupoidal fibres. These factorizations can be obtained by means of a single step 2-colimit. Namely, their left orthogonal parts are nothing but suitable coidentifiers and coinverters respectively. We will show how these results follow from their analogues in Cat. To this end, we first provide suitable conditions on a 2-category C, allowing the transfer of the construction of coinverters and coidentifiers from C to Fib(B).
This paper shows how internal models for polymorphic lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting the famous theorem of Peter …
This paper shows how internal models for polymorphic lambda calculi arise in any 2-category with a notion of discreteness. We generalise to a 2-categorical setting the famous theorem of Peter Freyd saying that there are no sufficiently (co)complete non-degenerate categories. As a simple corollary, we obtain a variant of Freyd theorem for categories internal to any tensored category. Also, with help of introduced concept of an associated category, we prove a representation theorem relating our internal models with well-studied fibrational models for polymorphism.
Abstract We develop semantics and syntax for bicategorical type theory. Bicategorical type theory features contexts, types, terms, and directed reductions between terms. This type theory is naturally interpreted in a …
Abstract We develop semantics and syntax for bicategorical type theory. Bicategorical type theory features contexts, types, terms, and directed reductions between terms. This type theory is naturally interpreted in a class of structured bicategories. We start by developing the semantics, in the form of comprehension bicategories . Examples of comprehension bicategories are plentiful; we study both specific examples as well as classes of examples constructed from other data. From the notion of comprehension bicategory, we extract the syntax of bicategorical type theory, that is, judgment forms and structural inference rules. We prove soundness of the rules by giving an interpretation in any comprehension bicategory. The semantic aspects of our work are fully checked in the Coq proof assistant, based on the UniMath library.
The diagrams for symmetric monoidal closed categories proved commutative by Mac Lane and the author in [19] were diagrams of (generalized) natural transformations. In order to understand the connexion between …
The diagrams for symmetric monoidal closed categories proved commutative by Mac Lane and the author in [19] were diagrams of (generalized) natural transformations. In order to understand the connexion between these results and free models for the structure, the author introduced in [13] and [14] the notion of club, which was further developed in [15] and applied later to other coherence problems in [16] and elsewhere.The club idea seemed to apply to several diverse kinds of structure on a category, but still to only a restricted number of kinds. In an attempt to understand its natural limits, the author worked out a general notion of “club”, as a monad with certain properties, not necessarily on Cat now, but on any category with finite limits. A brief account of this was included in the 1978 Seminar Report [17], but was never published; the author doubted that there were enough examples to make it of general interest.During 1990 and 1991, however, we were fortunate to have with our research team at Sydney Robin Cockett, who was engaged in applying category theory to computer science. In lectures to our seminar he called attention to certain kinds of monads involved with data types, which have special properties : he was calling them shape monads, but in fact they are precisely examples of clubs in the abstract sense above.
In a recent paper, Steven Vickers introduced a 'generalized powerdomain' construction, which he called the (lower) bagdomain, for algebraic posets, and argued that it provides a more realistic model than …
In a recent paper, Steven Vickers introduced a 'generalized powerdomain' construction, which he called the (lower) bagdomain, for algebraic posets, and argued that it provides a more realistic model than the powerdomain for the theory of databases (cf. Gunter). The basic idea is that our 'partial information' about a possible database should be specified not by a set of partial records of individuals, but by an indexed family (or, in Vickers' terminology, a bag) of such records: we do not want to be forced to identify two individuals in our database merely because the information that we have about them so far happens to be identical (even though we may, at some later stage, obtain the information that they are in fact the same individual).