We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond …
We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond to trihomomorphisms from a bicategory into Bicat. We describe the Grothendieck construction for each kind of fibration and present a few examples of each. Fibrations in our sense, between bicategories, are closed under composition and are stable under equiv-comma. The free such fibration on a homomorphism is obtained by taking an oplax comma along an identity.
We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond …
We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond to trihomomorphisms from a bicategory into Bicat. We describe the Grothendieck construction for each kind of fibration and present a few examples of each. Fibrations in our sense, between bicategories, are closed under composition and are stable under equiv-comma. The free such fibration on a homomorphism is obtained by taking an oplax comma along an identity.
In this paper,given a 2-category S,we construct two 2-categories Rand E,and show that the pullback in S,the final object in Rand the product inE are the same.Thus,we get some equivalent …
In this paper,given a 2-category S,we construct two 2-categories Rand E,and show that the pullback in S,the final object in Rand the product inE are the same.Thus,we get some equivalent definitions of pullback in 2-category.
In this paper, we discuss the Bass class and the Auslander class with respect to a semidualizing module over an associative ring. Let _SC_R be a semidualizing module, we proved …
In this paper, we discuss the Bass class and the Auslander class with respect to a semidualizing module over an associative ring. Let _SC_R be a semidualizing module, we proved that the Bass class B_C (R) is a right orthogonal subcategory of some right R-module; and that the Auslander class A_C (S) is a left orthogonal subcategory of the character module of some left S-module. As an application, we introduce the notion of the minimal semidualizing module, and get a one to one correspondence between the isomorphism classes of minimal semidualizing R-modules and maximal classes among coresolving preenvelope classes of Mod R with the same Ext-projective generators in gen^* R.
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.
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.
In this paper we introduce sigma limits (which we write $\sigma$-limits), a concept that interpolates between lax and pseudolimits: for a fixed family $\Sigma$ of arrows of a 2-category $\mathcal{A}$, …
In this paper we introduce sigma limits (which we write $\sigma$-limits), a concept that interpolates between lax and pseudolimits: for a fixed family $\Sigma$ of arrows of a 2-category $\mathcal{A}$, a $\sigma$-cone for a $2$-functor $\mathcal{A} \stackrel{F}{\rightarrow} \mathcal{B}$ is a lax cone such that the structural 2-cells corresponding to the arrows of $\Sigma$ are invertible. The conical $\sigma$-limit of $F$ is the universal $\sigma$-cone. Similary we define $\sigma$-natural transformations and weighted $\sigma$-limits. We consider also the case of bilimits. We develop the theory of $\sigma$-limits and $\sigma$-bilimits, whose importance relies on the following key fact: any weighted $\sigma$-limit (or $\sigma$-bilimit) can be expressed as a conical one. From this we obtain, in particular, a canonical expression of an arbitrary $\mathcal{C}at$-valued 2-functor as a conical $\sigma$-bicolimit of representable 2-functors, for a suitable choice of $\Sigma$, which is equivalent to the well known bicoend formula. As an application, we establish the 2-dimensional theory of flat pseudofunctors. We define a $\mathcal{C}at$-valued pseudofunctor to be flat when its left bi-Kan extension along the Yoneda 2-functor preserves finite weighted bilimits. We introduce a notion of 2-filteredness of a 2-category with respect to a class $\Sigma$, which we call $\sigma$-filtered. Our main result is: A pseudofunctor $\mathcal{A} \rightarrow \mathcal{C}at$ is flat if and only if it is a $\sigma$-filtered $\sigma$-bicolimit of representable 2-functors. In particular the reader will notice the relevance of this result for the development of a theory of 2-topoi.
We associate to a given 2-category K a new 2-category Bimon(K), whose 0-cells are the bimonads in K. We show that this construction denes an endofunctor of the category 2-CAT …
We associate to a given 2-category K a new 2-category Bimon(K), whose 0-cells are the bimonads in K. We show that this construction denes an endofunctor of the category 2-CAT of all 2-categories, which is represented by a certain 2-category Bimon.
Abstract In this chapter we shall study the remarkable 2-categorical structure of the 2-category BTop/S of bounded S-toposes, where S is a fixed topos. We shall restrict ourselves entirely to …
Abstract In this chapter we shall study the remarkable 2-categorical structure of the 2-category BTop/S of bounded S-toposes, where S is a fixed topos. We shall restrict ourselves entirely to bounded geometric morphisms throughout the chapter: although some of our results can be extended to unbounded morphisms, the structure of Top/S is much more delicate, and the gain in generality does not seem worth the extra trouble that would be involved in formulating all our results so as to apply to unbounded morphisms. In this first section, our aim is to construct finite weighted limits (in the sense of 1.1.5) in BTop/S.
We study the interaction between the notions of filteredness, fractions and fibrations in the theory of bicategories, generalizing classical results for categories. We give an explicit formula for filtered pseudo-colimits …
We study the interaction between the notions of filteredness, fractions and fibrations in the theory of bicategories, generalizing classical results for categories. We give an explicit formula for filtered pseudo-colimits of categories indexed by a bicategory, and we use it to compute the hom-categories of a bicategory of fractions. As a consequence, we show that the canonical pseudo-functor into a bicategory of fractions is exact.
Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk's classification of regular Lie groupoids in foliation theory, …
Actions of bicategories arise as categorification of actions of categories. They appear in a variety of different contexts in mathematics, from Moerdijk's classification of regular Lie groupoids in foliation theory, to Waldmann's work on deformation quantization. For any such action we introduce an action bicategory, together with a canonical projection (strict) 2-functor to the bicategory which acts. When the bicategory is a bigroupoid, we can impose the additional condition that action is principal in bicategorical sense, giving rise to a bigroupoid 2-torsor. In that case, the Duskin nerve of the canonical projection is precisely the Duskin-Glenn simplicial 2-torsor.
On développe une théorie de l'homotopie des 2-catégories analogue à la théorie de l'homotopie des catégories développée par Grothendieck dans "À la poursuite des champs". Il s'agit de la thèse …
On développe une théorie de l'homotopie des 2-catégories analogue à la théorie de l'homotopie des catégories développée par Grothendieck dans "À la poursuite des champs". Il s'agit de la thèse de doctorat de l'auteur. We develop a homotopy theory of 2-categories analogous to Grothendieck's homotopy theory of categories developed in "Pursuing Stacks." This is the author's PhD thesis.
We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In …
We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we investigate the relation between this `global' monoidal structure where the total category is monoidal and the fibration strictly preserves the structure, and a `fibrewise' one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, lax monoidal structures on the functor to the 2-category of categories correspond to lifts of the functor to the 2-category of monoidal categories. Finally, we give examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.
Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to …
Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between these phenomena. In this rst part of a two-part series on this subject, we show that the assignment to each symmetric monoidal closed category V its associated V -enriched category V extends to a 2-functor valued in an op-2-bred 2-category of symmetric monoidal closed categories enriched over various bases. For a xed V , we show that this induces a 2-functorial passage from symmetric monoidal closed categories over V (i.e., equipped with a morphism to V ) to symmetric monoidal closed V -categories over V. As a consequence, we nd that the enriched adjunction determined a symmetric monoidal closed adjunction can be obtained by applying a 2-functor and, consequently, is an adjunction in the 2-category of symmetric monoidal closed V -categories.
Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce to encode these ways of combining networks. …
Networks can be combined in various ways, such as overlaying one on top of another or setting two side by side. We introduce to encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to $\mathbf{Cat}$, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.
We describe a non-extensional variant of Martin-L\"of type theory which we call two-dimensional type theory, and equip it with a sound and complete semantics valued in 2-categories.
We describe a non-extensional variant of Martin-L\"of type theory which we call two-dimensional type theory, and equip it with a sound and complete semantics valued in 2-categories.
Hofmann and Streicher showed that there is a model of the intensional form of Martin-Lof’s type theory obtained by interpreting closed types as groupoids. We show that there is also …
Hofmann and Streicher showed that there is a model of the intensional form of Martin-Lof’s type theory obtained by interpreting closed types as groupoids. We show that there is also a model when closed types are interpreted as strict ω-groupoids. The nonderivability of various truncation and uniqueness principles in intensional type theory is then an immediate consequence. In the process of constructing the interpretation we develop some ω-categorical machinery including a version of the Grothendieck construction for strict ω-categories.
We present two Dialectica-like constructions for models of intensional Martin-Löf type theory based on Gödel's original Dialectica interpretation and the Diller-Nahm variant, bringing dependent types to categorical proof theory. We …
We present two Dialectica-like constructions for models of intensional Martin-Löf type theory based on Gödel's original Dialectica interpretation and the Diller-Nahm variant, bringing dependent types to categorical proof theory. We set both constructions within a logical predicates style theory for display map categories where we show that 'quasifibred' versions of dependent products and universes suffice to construct their standard counterparts. To support the logic required for dependent products in the first construction, we propose a new semantic notion of finite sum for dependent types, generalizing finitely-complete extensive categories. The second avoids extensivity assumptions using biproducts in a Kleisli category for a fibred additive monad.
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ …
An abstract is not available for this content so a preview has been provided. As you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
Abstract We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin–Löf type theory, two-level type theory, and cubical type theory. We …
Abstract We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin–Löf type theory, two-level type theory, and cubical type theory. We establish basic results in the semantics of type theory: every type theory has a bi-initial model; every model of a type theory has its internal language; the category of theories over a type theory is bi-equivalent to a full sub-2-category of the 2-category of models of the type theory.
The contributions of this paper are twofold. Within the framework of Grothendieck's fibrational category theory, we present a web of fundamental 2-adjunctions surrounding the formation of the category of all …
The contributions of this paper are twofold. Within the framework of Grothendieck's fibrational category theory, we present a web of fundamental 2-adjunctions surrounding the formation of the category of all small diagrams in a given category and the formation of the Grothendieck category of a functor into the category of small categories. We demonstrate the utility of these adjunctions, in part by deriving three formulae for (co-)limits: a `twisted' generalization of the well-known Fubini formula, as first established by Chacholski and Scherer; a new `general colimit decomposition formula'; and a special case of the general formula, which actually initiated this work, and which was proved independently by Batanin and Berger. We give three proofs for this colimit decomposition formula, using methods that provide quite distinct insights.
The `base' of our web of 2-adjunctions extends earlier work of the Ehresmann school and Guitart and promises to be of independent interest. It involves forming the diagram category of an arbitrary functor, seen as an object of the arrow category of the category of locally small categories, rather than that of a mere category. The left adjoint of the emerging generalized Guitart 2-adjunction factors through the 2-equivalence of split Grothendieck (co-)fibrations and strictly (co-)indexed categories, which we present here most generally by allowing 2-dimensional variation in the base categories.
Categories of lenses/optics and Dialectica categories are both comprised of bidirectional morphisms of basically the same form. In this work we show how they can be considered a special case …
Categories of lenses/optics and Dialectica categories are both comprised of bidirectional morphisms of basically the same form. In this work we show how they can be considered a special case of an overarching fibrational construction, generalizing Hofstra's construction of Dialectica fibrations and Spivak's construction of generalized lenses. This construction turns a tower of Grothendieck fibrations into another tower of fibrations by iteratively twisting each of the components, using the opposite fibration construction. Comment: v3: 18 pp. Project results from the American Mathematical Society's Math Research Community on Applied Category Theory 2022. Final version for proceedings of MFPS 2024. Updated author affiliation
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> .
One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor $L …
One goal of applied category theory is to understand open systems. We compare two ways of describing open systems as cospans equipped with extra data. First, given a functor $L \colon \mathsf{A} \to \mathsf{X}$, a "structured cospan" is a diagram in $\mathsf{X}$ of the form $L(a) \rightarrow x \leftarrow L(b)$. If $\mathsf{A}$ and $\mathsf{X}$ have finite colimits and $L$ preserves them, it is known that there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are structured cospans. Second, given a pseudofunctor $F \colon \mathsf{A} \to \mathbf{Cat}$, a "decorated cospan" is a diagram in $\mathsf{A}$ of the form $a \rightarrow m \leftarrow b$ together with an object of $F(m)$. Generalizing the work of Fong, we show that if $\mathsf{A}$ has finite colimits and $F \colon (\mathsf{A},+) \to (\mathsf{Cat},\times)$ is symmetric lax monoidal, there is a symmetric monoidal double category whose objects are those of $\mathsf{A}$ and whose horizontal 1-cells are decorated cospans. We prove that under certain conditions, these two constructions become isomorphic when we take $\mathsf{X} = \int F$ to be the Grothendieck category of $F$. We illustrate these ideas with applications to electrical circuits, Petri nets, dynamical systems and epidemiological modeling.
In this paper we start by pointing out that Yoneda's notion of a regular span $S \colon \mathcal{X} \to \mathcal{A} \times \mathcal{B}$ can be interpreted as a special kind of …
In this paper we start by pointing out that Yoneda's notion of a regular span $S \colon \mathcal{X} \to \mathcal{A} \times \mathcal{B}$ can be interpreted as a special kind of morphism, that we call fiberwise opfibration, in the 2-category $\mathsf{Fib}(\mathcal{A})$. We study the relationship between these notions and those of internal opfibration and two-sided fibration. This fibrational point of view makes it possible to interpret Yoneda's Classification Theorem given in his 1960 paper as the result of a canonical factorization, and to extend it to a non-symmetric situation, where the fibration given by the product projection $Pr_0 \colon \mathcal{A} \times \mathcal{B} \to \mathcal{A}$ is replaced by any split fibration over $\mathcal{A}$. This new setting allows us to transfer Yoneda's theory of extensions to the non-additive analog given by crossed extensions for the cases of groups and other algebraic structures.
The straightening–unstraightening correspondence of Grothendieck–Lurie provides an equivalence between cocartesian fibrations between $(\infty, 1)$ -categories and diagrams of $(\infty, 1)$ -categories. We provide an alternative proof of this correspondence, as …
The straightening–unstraightening correspondence of Grothendieck–Lurie provides an equivalence between cocartesian fibrations between $(\infty, 1)$ -categories and diagrams of $(\infty, 1)$ -categories. We provide an alternative proof of this correspondence, as well as an extension of straightening–unstraightening to all higher categorical dimensions. This is based on an explicit combinatorial result relating two types of fibrations between double categories, which can be applied inductively to construct the straightening of a cocartesian fibration between higher categories.
We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as relK, spnK, parK ,a nd proK for a suitable category …
We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as relK, spnK, parK ,a nd proK for a suitable category K, along with related constructs such as the V-pro arising from a suitable monoidal category V. We further exhibit the equipments as the objects of a 2-category, in such a way that arbitrary functors F : L ✲ K induce equipment arrows relF : relL ✲ relK, spnF : spnL ✲ spnK, and so on, and similarly for arbitrary monoidal functors V ✲ W. The article I with the title above dealt with those equipments M having each M(A, B) only an ordered set, and contained a detailed analysis of the case M = relK; in the present article we allow the M(A, B) to be general categories, and illustrate our results by a detailed study of the case M = spnK. We show in particular that spn is a locally-fully-faithful 2-functor to the 2-category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2-category of equipments, we are able to give a simple characterization of those spnG which arise from a geometric morphism G.
A factorization system (E;M) on a category A gives rise to the covariant category-valued pseudofunctor P of A sending each object to its slice category over M. This article characterizes …
A factorization system (E;M) on a category A gives rise to the covariant category-valued pseudofunctor P of A sending each object to its slice category over M. This article characterizes the P so obtained as follows: their object images have terminal objects, and they admit bicategorically cartesian liftings, up to equivalence, of slice-category projections. It is clear that, and how, (E;M) can be recovered from such a P. The correspondence thus described is actually the second of three similar ones between certain (E;M) and certain P that the article presents. In the rst one the characterization of the P has all ultimately bicategorical ingredients replaced with their categorical analogues. A category A with such a P is precisely what the author has called a \slicing site. In the article's terms the associated (E;M) are again factor- ization systems | but the concept conveyed extends the standard one by not obliging isomorphisms to belong to either factor class |, namely those that are \right semire- plete (isomorphisms do belong toM) and \left semistrict (morphisms inM are monic relative toE). The third correspondence subsumes the other two; here the (E;M) are all right-semireplete factorization systems.
This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a dual …
This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions. First, we present a dual to the sound induction rule for inductive types that we developed previously. That is, we present a sound coinduction rule for any data type arising as the carrier of the final coalgebra of a functor, thus relaxing Hermida and Jacobs' restriction to polynomial functors. To achieve this we introduce the notion of a quotient category with equality (QCE) that i) abstracts the standard notion of a fibration of relations constructed from a given fibration; and ii) plays a role in the theory of coinduction dual to that played by a comprehension category with unit (CCU) in the theory of induction. Secondly, we show that inductive and coinductive indexed types also admit sound induction and coinduction rules. Indexed data types often arise as carriers of initial algebras and final coalgebras of functors on slice categories, so we give sufficient conditions under which we can construct, from a CCU (QCE) U:E \rightarrow B, a fibration with base B/I that models indexing by I and is also a CCU (resp., QCE). We finish the paper by considering the more general case of sound induction and coinduction rules for indexed data types when the indexing is itself given by a fibration.
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.
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that …
In this paper we show that the strict and lax pullbacks of a 2-categorical opfibration along an arbitrary 2-functor are homotopy equivalent. We give two applications. First, we show that the strict fibers of an opfibration model the homotopy fibers. This is a version of Quillen's Theorem B amenable to applications. Second, we compute the $E^2$ page of a homology spectral sequence associated to an opfibration and apply this machinery to a 2-categorical construction of $S^{-1}S$. We show that if $S$ is a symmetric monoidal 2-groupoid with faithful translations then $S^{-1}S$ models the group completion of $S$.
Author(s): Moeller, Joseph Patrick | Advisor(s): Baez, John C | Abstract: In this thesis, we present a flexible framework for specifying and constructing operads which are suited to reasoning about …
Author(s): Moeller, Joseph Patrick | Advisor(s): Baez, John C | Abstract: In this thesis, we present a flexible framework for specifying and constructing operads which are suited to reasoning about network construction. The data used to present these operads is called a network model, a monoidal variant of Joyal's combinatorial species. The construction of the operad required that we develop a monoidal lift of the Grothendieck construction. We then demonstrate how concepts like priority and dependency can be represented in this framework. For the former, we generalize Green's graph products of groups to the context of universal algebra. For the latter, we examine the emergence of monoidal fibrations from the presence of catalysts in Petri nets.
This paper is concerned with developing a 2-dimensional analogue of the notion of an ordinary discrete fibration. A definition is proposed, and it is shown that such discrete 2-fibrations correspond …
This paper is concerned with developing a 2-dimensional analogue of the notion of an ordinary discrete fibration. A definition is proposed, and it is shown that such discrete 2-fibrations correspond via a 2-equivalence to certain category-valued 2-functors. The ultimate goal of the paper is to show that discrete 2-fibrations are 2-monadic over a slice of the 2-category of categories.
Abstract Via the adjunction $$ - *\mathbbm {1} \dashv \mathcal V(\mathbbm {1},-) :\textsf {Span}({\mathcal {V}}) \rightarrow {\mathcal {V}} \text {-} \textsf {Mat} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mn>1</mml:mn> …
Abstract Via the adjunction $$ - *\mathbbm {1} \dashv \mathcal V(\mathbbm {1},-) :\textsf {Span}({\mathcal {V}}) \rightarrow {\mathcal {V}} \text {-} \textsf {Mat} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mn>1</mml:mn> <mml:mo>⊣</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mo>)</mml:mo> <mml:mo>:</mml:mo> <mml:mi>Span</mml:mi> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> <mml:mo>→</mml:mo> <mml:mi>V</mml:mi> <mml:mtext>-</mml:mtext> <mml:mi>Mat</mml:mi> </mml:mrow> </mml:math> and a cartesian monad T on an extensive category $$ {\mathcal {V}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> with finite limits, we construct an adjunction $$ - *\mathbbm {1} \dashv {\mathcal {V}}(\mathbbm {1},-) :\textsf {Cat}(T,{\mathcal {V}}) \rightarrow ({\overline{T}}, \mathcal V)\text{- }\textsf{Cat} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mn>1</mml:mn> <mml:mo>⊣</mml:mo> <mml:mi>V</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>-</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi>Cat</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>→</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mover> <mml:mi>T</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mtext>-</mml:mtext> <mml:mspace/> <mml:mi>Cat</mml:mi> </mml:mrow> </mml:math> between categories of generalized enriched multicategories and generalized internal multicategories, provided the monad T satisfies a suitable property, which holds for several examples. We verify, moreover, that the left adjoint is fully faithful, and preserves pullbacks, provided that the copower functor $$ - *\mathbbm {1} :\textsf {Set} \rightarrow {\mathcal {V}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mn>1</mml:mn> <mml:mo>:</mml:mo> <mml:mi>Set</mml:mi> <mml:mo>→</mml:mo> <mml:mi>V</mml:mi> </mml:mrow> </mml:math> is fully faithful. We also apply this result to study descent theory of generalized enriched multicategorical structures. These results are built upon the study of base-change for generalized multicategories, which, in turn, was carried out in the context of categories of horizontal lax algebras arising out of a monad in a suitable 2-category of pseudodouble categories.
We define a mapping space for Gray-enriched categories adapted to higher gauge theory. Our construction differs significantly from the canonical mapping space of enriched categories in that it is much …
We define a mapping space for Gray-enriched categories adapted to higher gauge theory. Our construction differs significantly from the canonical mapping space of enriched categories in that it is much less rigid. The two essential ingredients are a path space construction for Gray-categories and a kind of comonadic resolution of the 1-dimensional structure of a given Gray-category obtained by lifting the resolution of ordinary categories along the canonical fibration of GrayCat over Cat.
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].
We study effective descent V-functors for cartesian monoidal categories V with finite limits. This study is carried out via the properties enjoyed by the 2-functor V↦Fam(V), results about effective descent …
We study effective descent V-functors for cartesian monoidal categories V with finite limits. This study is carried out via the properties enjoyed by the 2-functor V↦Fam(V), results about effective descent of bilimits of categories, and the fact that the enrichment 2-functor preserves certain bilimits. Since these results rely on an understanding of (effective) descent morphisms in Fam(V), we carefully study these morphisms in free coproduct completions. Finally, we provide refined conditions when V is a regular category.
Categories.- 2-categories.- Bicategories.- Properties of Fun(A,B) and Pseud(A,B).- Properties of 2-comma categories.- Adjoint morphisms in 2-categories.- Quasi-adjointness.
Categories.- 2-categories.- Bicategories.- Properties of Fun(A,B) and Pseud(A,B).- Properties of 2-comma categories.- Adjoint morphisms in 2-categories.- Quasi-adjointness.
With a view to further applications, we give a self-contained account of indexed limits for 2-categories, including necessary and sufficient conditions for 2-categorical completeness. Many important 2-categories fail to be …
With a view to further applications, we give a self-contained account of indexed limits for 2-categories, including necessary and sufficient conditions for 2-categorical completeness. Many important 2-categories fail to be complete but do admit a wide class of limits. Accordingly, we introduce a variety of particular 2-categorical limits of practical importance, and show that certain of these suffice for the existence of indexed lax- and pseudo-limits. Other important 2-categories fail to admit even pseudo-limits, but do admit the weaker bilimits; we end by discussing these.
Any attempt to give “foundations”, for category theory or any domain in mathematics, could have two objectives, of course related. (0.1) Noncontradiction : Namely, to provide a formal frame rich …
Any attempt to give “foundations”, for category theory or any domain in mathematics, could have two objectives, of course related. (0.1) Noncontradiction : Namely, to provide a formal frame rich enough so that all the actual activity in the domain can be carried out within this frame, and consistent, or at least relatively consistent with a well-established and “safe” theory, e.g. Zermelo-Frankel (ZF). (0.2) Adequacy , in the following, nontechnical sense: (i) The basic notions must be simple enough to make transparent the syntactic structures involved. (ii) The translation between the formal language and the usual language must be, or very quickly become, obvious. This implies in particular that the terminology and notations in the formal system should be identical, or very similar, to the current ones. Although this may seem minor, it is in fact very important. (iii) “Foundations” can only be “foundations of a given domain at a given moment”, therefore the frame should be easily adaptable to extensions or generalizations of the domain, and, even better, in view of (i), it should suggest how to find meaningful generalizations. (iv) Sometimes (ii) and (iii) can be incompatible because the current notations are not adapted to a more general situation. A compromise is then necessary. Usually when the tradition is very strong (ii) is predominant, but this causes some incoherence for the notations in the more general case (e.g. the notation f ( x ) for the value of a function f at x obliges one, in category theory, to denote the composition of arrows ( f, g ) → g∘f , and all attempts to change this notation have, so far, failed).