LaTeX 2e with xy-pic; 34 pages, 21 commutative diagrams; v.2: small additions and corrections, references updated, Remark 1.3 inserted; v.3: Section 5 expanded, small things corrected, references updated; v.4: small things corrected, three paragraphs inserted in the beginning of the proof of Proposition 2.5
A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable objects. We show that any exact structure on a small idempotent-complete additive category extends uniquely to a locally coherent exact structure on the category of ind-objects; in particular, any finitely accessible category has the unique maximal and the unique minimal locally coherent exact category structures. All locally coherent exact categories are of Grothendieck type in the sense of Stovicek. We also discuss the canonical embedding of a small exact category into the abelian category of additive sheaves in connection with the locally coherent exact structure on the ind-objects, and deduce two periodicity theorems as applications.
We define and study the notion of a locally bounded enriched category over a (locally bounded) symmetric monoidal closed category, generalizing the locally bounded ordinary categories of Freyd and Kelly. In addition to proving several general results for constructing examples of locally bounded enriched categories and locally bounded closed categories, we demonstrate that locally bounded enriched categories admit fully enriched analogues of many of the convenient results enjoyed by locally bounded ordinary categories. In particular, we prove full enrichments of Freyd and Kelly's reflectivity a…
We extend the 2-representation theory of finitary 2-categories to certain 2-categories with infinitely many objects, denoted locally finitary 2-categories, and extend the classical classification results of simple transitive 2-representations of weakly fiat 2-categories to this environment. We also consider locally finitary 2-categories and 2-representations with a grading, and prove the associated coalgebra 1-morphisms have a homogeneous structure. We use these results to classify simple transitive 2-representations of certain classes of cyclotomic 2-Kac-Moody algebras.
We prove that every locally Cartesian closed $\infty$-category with subobject classifier has a strict initial object and disjoint and universal binary coproducts.
This note explains how dependent sums and products are interpreted by adjoints of the base change functor in a locally cartesian closed category. An effort is made to unpack all the definitions so as to make the concepts more transparent to new learners.
There is a well-known correspondence between coherent theories (and their interpretations) and coherent categories (resp. functors), hence the (2,1)-category $\mathbf{Coh_{\sim}}$ (of small coherent categories, coherent functors and all natural isomorphisms) is of logical interest. We prove that this category admits all small 2-limits and 2-colimits (in the ($\infty $,1)-categorical sense), and prove a 2-categorical small object argument to provide weak factorisation systems for coherent functors.
We give an explicit description of the generator of finitely presented objects of the coslice of a locally finitely presentable category under a given object, as consisting of all pushouts of finitely presented maps under this object. Then we prove that the comma category under the direct image part of a morphism of locally finitely presentable category is still locally finitely presentable, and we give again an explicit description of its generator of finitely presented objects. We finally deduce that 2-category $\LFP$ has comma objects computed in $\Cat$.
We define locally wide finitary 2-categories by relaxing the definition of finitary 2-categories to allow infinitely many objects and isomorphism classes of 1-morphisms and infinite dimensional hom-spaces of 2-morphisms. After defining related concepts including transitive 2-representations in this setting, we provide a new method of constructing coalgebra 1-morphisms associated to transitive 2-representations of locally wide weakly fiat 2-categories, and demonstrate that any such transitive 2-representation is equivalent to a certain subcategory of the category of comodule 1-morphisms over th…
The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories
Seely的论文“Locally Carnival closed categories and type theory“包含了范畴类型论中一个著名的结果：局部Carnival closed categories等价于Martin-Löf类型论中的Pi-类型、Sigma-类型和外延恒等式。然而，Seely的证明依赖于一个有问题的假设，即类型中的替换可以通过回调来解释。在这里，我们证明了西利定理的一个修正版本：马丁-洛夫类型论的Bénabou-Hofmann解释在局部范畴闭范畴中产生2-范畴的双等价。为了促进技术的发展，我们采用了具有族的范畴来代替句法的马丁-洛夫类型论。作为第二个结果，我们证明了，如果我们删除PI型所产生的类别与家庭是双等价的左正合范畴。
Coherent and ideal actions in ideally exact categories
Università degli Studi di Palermo（巴勒莫大学）；Université catholique de Louvain（天主教鲁汶大学）；Università degli Studi di Milano Statale（米兰州立大学）；Università degli Studi di Messina（墨西拿大学）
Locally finitely presented Grothendieck categories with a flat generator
Universidad Rey Juan Carlos（雷乌斯·胡安·卡洛斯大学）；Universidad Austral de Chile（智利南方大学）；Universidad de Murcia（穆尔西亚大学）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）
Locally cartesian closed (lcc) categories are natural categorical models of extensional dependent type theory. This paper introduces the "gros" semantics in the category of lcc categories: Instead of constructing an interpretation in a given individual lcc category, we show that also the category of all lcc categories can be endowed with the structure of a model of dependent type theory. The original interpretation in an individual lcc category can then be recovered by slicing. As in the original interpretation, we face the issue of coherence: Categorical structure is usually preserved by func…
Let $\mathcal C$ be the category of finite graphs. Lovàsz shows that the semi-ring of isomorphism classes of $\mathcal C$ (with coproduct as sum, and product as multiplication) is embedded into the direct product of the semi-ring of natural numbers. Our aim is to generalize this result to other categories. For this, one crucial property is that every object decomposes to a finite coproduct of connected objects. We show that a locally-finite extensive category satisfies this condition. Conversely, a category where any object is decomposed into a finite coproduct of connected objects is shown to…
