Evidence is given for the correctness of the Joyal-Riehl-Verity construction of the homotopy bicategory of the $(\infty, 2)$-category of $(\infty, 1)$-categories; in particular, it is shown that the analogous construction …
Evidence is given for the correctness of the Joyal-Riehl-Verity construction of the homotopy bicategory of the $(\infty, 2)$-category of $(\infty, 1)$-categories; in particular, it is shown that the analogous construction using complete Segal spaces instead of quasicategories yields a bicategorically equivalent 2-category.
Nous definissons la notion de 2-categorie 2-filtrante et donnons une construction explicite de la bicolimite d'un 2-foncteur a valeurs dans les categories. Une categorie consideree comme etant une 2-categorie triviale …
Nous definissons la notion de 2-categorie 2-filtrante et donnons une construction explicite de la bicolimite d'un 2-foncteur a valeurs dans les categories. Une categorie consideree comme etant une 2-categorie triviale est 2-filtrante si et seulement si c'est une categorie filtrante, et notre construction conduit a une categorie equivalente a la categorie qui s'obtient par la construction usuelle des colimites filtrantes de categories. Pour cette construction des axiomes plus faibles suffisent, et nous appelons la notion correspondante 2-categorie pre 2-filtrante. L'ensemble complet des axiomes est necessaire pour montrer que les bicolimites 2-filtrantes ont les proprietes correspondantes aux proprietes essentielles des colimites.
We define the notion of 2-filtered 2-category and give an explicit construction of the bicolimit of a category valued 2-functor. A category considered as a trivial 2-category is 2-filtered if …
We define the notion of 2-filtered 2-category and give an explicit construction of the bicolimit of a category valued 2-functor. A category considered as a trivial 2-category is 2-filtered if and only if it is a filtered category, and our construction yields a category equivalent to the category resulting from the usual construction of filtered colimits of categories. Weaker axioms suffice for this construction, and we call the corresponding notion pre 2-filtered 2-category. The full set of axioms is necessary to prove that 2-filtered bicolimits have the properties corresponding to the essential properties of filtered bicolimits. Kennison already considered filterness conditions on a 2-category under the name of bifiltered 2-category. It is easy to check that a bifiltered 2-category is 2-filtered, so our results apply to bifiltered 2-categories. Actually Kennison's notion is equivalent to ours, but the other direction of this equivalence is not entirely trivial.
We define the notion of 2-filtered 2-category and give an explicit construction of the bicolimit of a category valued 2-functor. A category considered as a trivial 2-category is 2-filtered if …
We define the notion of 2-filtered 2-category and give an explicit construction of the bicolimit of a category valued 2-functor. A category considered as a trivial 2-category is 2-filtered if and only if it is a filtered category, and our construction yields a category equivalent to the category resulting from the usual construction of filtered colimits of categories. Weaker axioms suffice for this construction, and we call the corresponding notion pre 2-filtered 2-category. The full set of axioms is necessary to prove that 2-filtered bicolimits have the properties corresponding to the essential properties of filtered bicolimits. Kennison already considered filterness conditions on a 2-category under the name of bifiltered 2-category. It is easy to check that a bifiltered 2-category is 2-filtered, so our results apply to bifiltered 2-categories. Actually Kennison's notion is equivalent to ours, but the other direction of this equivalence is not entirely trivial.
We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to …
We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to mean that some external information is taken into account. For the particular use of the authors, such an open functor is described by two components: a presheaf representing the possible external influences for each input, and a classical functor from the category of elements of this presheaf to the category of results. Considering the appropriate notion of composition then leads to a bicategory. This report describes this bicategory with as little auxiliary constructions as possible and gives all the details of all the proofs needed to establish the bicategory, as explicitly as possible. Subsequent reports will give other presentations of this bicategory and compare it to other existing constructions, e.g. spans, fibrations, pseudoadjunctions, Kleisli bicategories of pseudo-monads, and profunctors (or distributors).
We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to …
We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to mean that some external information is taken into account. For the particular use of the authors, such an open functor is described by two components: a presheaf representing the possible external influences for each input, and a classical functor from the category of elements of this presheaf to the category of results. Considering the appropriate notion of composition then leads to a bicategory. This report describes this bicategory with as little auxiliary constructions as possible and gives all the details of all the proofs needed to establish the bicategory, as explicitly as possible. Subsequent reports will give other presentations of this bicategory and compare it to other existing constructions, e.g. spans, fibrations, pseudoadjunctions, Kleisli bicategories of pseudo-monads, and profunctors (or distributors).
We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a Hom-finite hereditary abelian category with enough projectives, we prove that the Auslander-Reiten-Smal{\o}-Ringel formula of …
We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a Hom-finite hereditary abelian category with enough projectives, we prove that the Auslander-Reiten-Smal{\o}-Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations of strongly locally finite quivers.
In the category of locally convex (topological vectorial) Hausdorff spaces, by means of the Banach spaces of bounded functions that are defined on a set, sufficiently many injective objects are …
In the category of locally convex (topological vectorial) Hausdorff spaces, by means of the Banach spaces of bounded functions that are defined on a set, sufficiently many injective objects are built. In this category by means of injective spaces a proper class of bicategory structures with sufficient injective objects and with the classes of injections right complete is constructed. 1. LOCALLY CONVEX INJECTIVE SPACES Vector spaces over the field K of real or complex numbers are examined. The results are formulated for the category of locally convex spaces Hausdorff, but they also can be formulated for the category of locally convex groups Haus- dorff. Similar results hold also if the condition that the spaces are Hausdorff is removed. In the category C2V of locally convex spaces Hausdorff we use the following notation: Eu - the class of universal epimorphisms, that coincides with the class of surjective applications; Mp - the class of precise morphisms; Mp = E ⌊. This class coincides with the class of topological inclusions (ho- momorphisms). In this way (Eu, Mp) is a bicategory structure in the category C2V; Mf - the class of strict monomorphisms that coincides with the class of topological inclusions with the closed image.
This paper adresses two issues in dealing with bicategories of fractions. The first is to introduce a set of conditions on a class of arrows in a bicategory which is …
This paper adresses two issues in dealing with bicategories of fractions. The first is to introduce a set of conditions on a class of arrows in a bicategory which is weaker than the one given in Pronk, Etendues and stacks as bicategories of fractions, but still allows a bicalculus of fractions. These conditions allow us to invert a smaller collection of arrows so that in some cases we may obtain a bicategory of fractions with small hom-categories. We adapt the construction of the bicategory of fractions to work with the weaker conditions. The second issue is the difficulty in dealing with 2-cells, which are defined by equivalence classes. We discuss conditions under which there are canonical representatives for 2-cells, and how pasting of 2-cells can be simplified in the presence of certain pseudo pullbacks. We also discuss how both of these improvements apply in the category of orbispaces.
This paper adresses two issues in dealing with bicategories of fractions. The first is to introduce a set of conditions on a class of arrows in a bicategory which is …
This paper adresses two issues in dealing with bicategories of fractions. The first is to introduce a set of conditions on a class of arrows in a bicategory which is weaker than the one given in Pronk, Etendues and stacks as bicategories of fractions, but still allows a bicalculus of fractions. These conditions allow us to invert a smaller collection of arrows so that in some cases we may obtain a bicategory of fractions with small hom-categories. We adapt the construction of the bicategory of fractions to work with the weaker conditions. The second issue is the difficulty in dealing with 2-cells, which are defined by equivalence classes. We discuss conditions under which there are canonical representatives for 2-cells, and how pasting of 2-cells can be simplified in the presence of certain pseudo pullbacks. We also discuss how both of these improvements apply in the category of orbispaces.
For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) …
For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric morphisms Hom(Sh(X),BB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal B-bundles. As a simple consequence, the geometric realization |NB| of the nerve NB of a locally contractible topological bicategory B is the classifying space of principal B-bundles, giving a variant of the result of Baas, Bokstedt and Kro derived in the context of bicategorical K-theory. We also define classifying topoi of a topological bicategory B using sheaves on other types of nerves of a bicategory given by Lack and Paoli, Simpson and Tamsamani by means of bisimplicial spaces, and we examine their properties.
For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) …
For any topological bicategory B, the Duskin nerve NB of B is a simplicial space. We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(NB) on the simplicial space NB. It is shown that the category of geometric morphisms Hom(Sh(X),BB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal B-bundles. As a simple consequence, the geometric realization |NB| of the nerve NB of a locally contractible topological bicategory B is the classifying space of principal B-bundles, giving a variant of the result of Baas, Bokstedt and Kro derived in the context of bicategorical K-theory. We also define classifying topoi of a topological bicategory B using sheaves on other types of nerves of a bicategory given by Lack and Paoli, Simpson and Tamsamani by means of bisimplicial spaces, and we examine their properties.
For any topological bicategory B, the Duskin nerve N B of B is a simplicial space.We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(N …
For any topological bicategory B, the Duskin nerve N B of B is a simplicial space.We introduce the classifying topos BB of B as the Deligne topos of sheaves Sh(N B) on the simplicial space N B. It is shown that the category of geometric morphisms Hom(Sh(X), BB) from the topos of sheaves Sh(X) on a topological space X to the Deligne classifying topos is naturally equivalent to the category of principal B-bundles.As a simple consequence, the geometric realization |N B| of the nerve N B of a locally contractible topological bicategory B is the classifying space of principal B-bundles, giving a variant of the result of Baas, Bökstedt and Kro derived in the context of bicategorical K-theory.We also define classifying topoi of a topological bicategory B using sheaves on other types of nerves of a bicategory given by Lack and Paoli, Simpson and Tamsamani by means of bisimplicial spaces, and we examine their properties.
The familiar construction of categories of fractions, due to Gabriel and Zisman, allows one to invert a class W of arrows in a category in a universal way. Similarly, bicategories …
The familiar construction of categories of fractions, due to Gabriel and Zisman, allows one to invert a class W of arrows in a category in a universal way. Similarly, bicategories of fractions allow one to invert a collection of arrows in a bicategory. In this case the arrows are inverted in the sense that they are made into equivalences. As with categories of fractions, bicategories of fractions suffer from the defect that they need not be locally small even when the bicategory in which W lives is locally small. Similarly, in the case where W is a class of arrows in a 2-category, the bicategory of fractions will not in general be a 2-category. In this paper we introduce two notions ---systems of fibrant objects and fibration systems--- which will allow us to associate to a bicategory B a homotopy bicategory Ho(B) in such a way that Ho(B) is the universal way to invert weak equivalences in B. This construction resolves both of the difficulties with bicategories of fractions mentioned above. We also describe a fibration system on the 2-category of prestacks on a site and prove that the resulting homotopy bicategory is the 2-category of stacks. Further examples considered include algebraic, differentiable and topological stacks.
Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins …
Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higher-dimensional work.
Elmendorf's Theorem states that the category of continuous actions of a topological group is a Grothendieck topos in the sense that it is equivalent to a category of sheaves on …
Elmendorf's Theorem states that the category of continuous actions of a topological group is a Grothendieck topos in the sense that it is equivalent to a category of sheaves on a site. This paper offers a 2-dimensional generalization by showing that a certain 2-category of continuous actions of a topological 2-group is 2-equivalent to a 2-category of 2-sheaves on a suitable 2-site.
Abstract We develop a 2-dimensional version of accessibility and presentability compatible with the formalism of flat pseudofunctors. First we give prerequisites on the different notions of 2-dimensional colimits, filteredness and …
Abstract We develop a 2-dimensional version of accessibility and presentability compatible with the formalism of flat pseudofunctors. First we give prerequisites on the different notions of 2-dimensional colimits, filteredness and cofinality; in particular we show that $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> - filteredness and bifilteredness are actually equivalent in practice for our purposes. Then, we define bi-accessible and bipresentable 2-categories in terms of bicompact objects and bifiltered bicolimits. We then characterize them as categories of flat pseudofunctors . We also prove a bi-accessible right bi-adjoint functor theorem and deduce a 2-dimensional Gabriel-Ulmer duality relating small bilex 2-categories and finitely bipresentable 2-categories. Finally, we show that 2-categories of pseudo-algebras of finitary 2-monads on $$\textbf{Cat}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Cat</mml:mi> </mml:math> are finitely bipresentable, which in particular captures the case of $$\textbf{Lex}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Lex</mml:mi> </mml:math> , the 2-category of small lex categories. Invoking the technology of lex-colimits , we prove further that several 2-categories arising in categorical logic ( Reg, Ex, Coh, Ext, Adh, Pretop ) are also finitely bipresentable.
This paper is a first step in the study of symmetric cat-groups as the 2-dimensional analogue of abelian groups. We show that a morphism of symmetric catgroups can be factorized …
This paper is a first step in the study of symmetric cat-groups as the 2-dimensional analogue of abelian groups. We show that a morphism of symmetric catgroups can be factorized as an essentially surjective functor followed by a full and faithful one, as well as a full and essentially surjective functor followed by a faithful one. Both these factorizations give rise to a factorization system, in a suitable 2-categorical sense, in the 2-category of symmetric cat-groups. An application to exact sequences is given.
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.
K-Theory was originally defined by Grothendieck as a contravariant functor from a subcategory of schemes to abelian groups, known today as K0. The same kind of construction was then applied …
K-Theory was originally defined by Grothendieck as a contravariant functor from a subcategory of schemes to abelian groups, known today as K0. The same kind of construction was then applied to other fields of mathematics, like spaces and (not necessarily commutative) rings. In all these cases, consists of some process applied, not directly to the object one wants to study, but to some category related to it: the category of vector bundles a space, of finitely generated projective modules a ring, of free modules a scheme, for instance. Later, Quillen extracted axioms that all these categories satisfy and that allow the Grothendieck construction of K0. The categorical structure he discovered is called today a Quillen-exact category. It led him not only to broaden the domain of application of K-theory, but also to define a whole K-theory spectrum associated to such a category. Waldhausen next generalized Quillen's notion of an exact category by introducing categories with weak equivalences and cofibrations, which one nowadays calls Waldhausen categories. K-theory has since been studied as a functor from the category of suitably structured (Quillen-exact, Waldhausen, symmetric monoidal) small categories to some category of spectra1. This has given rise to a huge field of research, so much so that there is a whole journal devoted to the subject. In this thesis, we want to take advantage of these tools to begin studying K-theory from another perspective. Indeed, we have the impression that, in the generalization of topological and algebraic K-theory that has been started by Quillen, something important has been left aside. K-theory was initiated as a (contravariant) functor from the various categories of spaces, rings, schemes, …, not from the category of Waldhausen small categories. Of course, one obtains information about a ring by studying its Quillen-exact category of (finitely generated projective) modules, but still, the final goal is the study of the ring, and, more globally, of the category of rings. Thus, in a general theory, one should describe a way to associate not only a spectrum to a structured category, but also a structured category to an object. Moreover, this process should take the morphisms of these objects into account. This gives rise to two fundamental questions. What kind of mathematical objects should K-theory be applied to? Given such an object, what category over it should one consider and how does vary morphisms? Considering examples, we have made the following observations. Suppose C is the category that is to be investigated by means of K-theory, like the category of topological spaces or of schemes, for instance. The category associated to an object of C is a sub-category of the category of modules some monoid in a monoidal category with additional structure (topological, symmetric, abelian, model). The situation is highly fibred: not only morphisms of C induce (structured) functors between these sub-categories of modules, but the monoidal category in which theses modules take place might vary from one object of C to another. In important cases, the sub-categories of modules considered are full sub-categories of locally modules with respect to some (possibly weakened notion of) Grothendieck topology on C . That is, there are some specific modules that are considered sufficiently simple to be called trivial and trivial modules are those that are, a covering of the Grothendieck topology, isomorphic to these. In this thesis, we explore, with K-theory in view, a categorical framework that encodes these kind of data. We also study these structures for their own sake, and give examples in other fields. We do not mention in this abstract set-theoretical issues, but they are handled with care in the discussion. Moreover, an appendix is devoted to the subject. After recalling classical facts of Grothendieck fibrations (and their associated indexed categories), we provide new insights into the concept of a bifibration. We prove that there is a 2-equivalence between the 2-category of bifibrations a category ℬ and a 2-category of pseudo double functors from ℬ into the double category of adjunctions in CAT. We next turn our attention to composable pairs of fibrations , as they happen to be fundamental objects of the theory. We give a characterization of these objects in terms of pseudo-functors ℬop → FIBc into the 2-category of fibrations and Cartesian functors. We next turn to a short survey about Grothendieck (pre-)topologies. We start with the basic notion of covering function, that associate to each object of a category a family of coverings of the object. We study separately the saturation of a covering function with respect to sieves and to refinements. The Grothendieck topology generated by a pretopology is shown to be the result of these two steps. We define then, inspired by Street [89], the notion of (locally) trivial objects in a fibred category P : ℰ → ℬ equipped with some notion of covering of objects of the base ℬ. The trivial objects are objects chosen in some fibres. An object E in the fibre B ∈ ℬ is trivial if there exists a covering {fi : Bi → B}i ∈ I such the inverse image of E along fi is isomorphic to a trivial object. Among examples are torsors, principal bundles, vector bundles, schemes, constant sheaves, quasi-coherent and free sheaves of modules, finitely generated projective modules commutative rings, topological manifolds, … We give conditions under which trivial objects form a subfibration of P and describe the relationship between trivial objects with respect to subordinated covering functions. We then go into the algebraic part of the theory. We give a definition of monoidal fibred categories and show a 2-equivalence with monoidal indexed categories. We develop algebra (monoids and modules) in these two settings. Modules and monoids in a monoidal fibred category ℰ → ℬ happen to form a pair of fibrations . We end this thesis by explaining how to apply this categorical framework to K-theory and by proposing some prospects of research. ______________________________ 1 Works of Lurie, Toen and Vezzosi have shown that K-theory really depends on the (∞, 1)-category associated to a Waldhausen category [94]. Moreover, topological K-theory of spaces and Banach algebras takes the fact that the Waldhausen category is topological in account [62, 70].
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.
We introduce judgemental theories and their calculi to frame type theory and proof theory in the same mathematical framework. Our analysis sheds light on both the topics, providing a new …
We introduce judgemental theories and their calculi to frame type theory and proof theory in the same mathematical framework. Our analysis sheds light on both the topics, providing a new point of view. In the case of type theory, we provide an abstract definition of type constructor featuring the usual formation, introduction, elimination and computation rules. In proof theory we offer a deep analysis of structural rules, demystifying some of their properties, and putting them into context. We finish the paper discussing the internal logic of a topos, a predicative topos, an elementary 2-topos et similia, and show how these can be organized in judgemental theories.
An internal full subcategory of a cartesian closed category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is …
An internal full subcategory of a cartesian closed category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is shown to give rise to a structure on the 2-category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C a t left-parenthesis script upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>a</mml:mi> <mml:mi>t</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Cat(\mathcal {A})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of categories in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which introduces the notion of size into the analysis of categories in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and allows proofs by transcendental arguments. The relationship to the currently popular study of locally internal categories is examined. Internal full subcategories of locally presentable categories (in the sense of Gabriel-Ulmer) are studied in detail. An algorithm is developed for their construction and this is applied to the categories of double categories, triple categories, and so on.