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 prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples.
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…
The Ehresmann-Schein-Nambooripad (ESN) Theorem asserts an equivalence between the category of inverse semigroups and the category of inductive groupoids. In this paper, we consider the category of inverse categories and functors -- a natural generalization of inverse semigroups -- and extend the ESN theorem to an equivalence between this category and the category of top-heavy locally inductive groupoids and locally inductive functors. From the proof of this extension, we also generalize the ESN Theorem to an equivalence between the category of inverse semicategories and the category of locally…
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.
We introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopenka's principle, the same happens in the regular case. The category of complete partial orders is the natural example of a nearly locally finitely presentable category which is not locally presentable.
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 exhibit an equivalence between the model-theoretic framework of universal classes and the category-theoretic framework of locally multipresentable categories. We similarly give an equivalence between abstract elementary classes (AECs) admitting intersections and locally polypresentable categories. We use these results to shed light on Shelah's presentation theorem for AECs.
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）
It is proved that the projective model structure of the category of topologically enriched diagrams of topological spaces over a topologically enriched locally contractible small category is Quillen equivalent to the standard Quillen model structure of topological spaces. We give a geometric interpretation of this fact in directed homotopy.
For a locally finitely presented Grothendieck category $\mathcal{A}$, we consider a certain subcategory of the homotopy category of FP-injective objects in $\mathcal{A}$ which we show is compactly generated. In the case where $\mathcal{A}$ is locally coherent, we identify this subcategory with the derived category of FP-injective objects in $\mathcal{A}$. Our results are, in a sense, dual to the ones obtained by Neeman on the homotopy category of flat modules. Our proof is based on extending a characterization of the pure acyclic complexes which is due to Emmanouil.
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）
The main objective of this paper is to construct a symmetric monoidal closed model category of coherently commutative monoidal quasi-categories. We construct another model category structure whose fibrant objects are (essentially) those coCartesian fibrations which represent objects that are known as symmetric monoidal quasi-categories in the literature. We go on to establish a Quillen equivalence between the two model categories.
We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type $\text{FP}_n$ and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type $\text{FP}_n$ categories as a generalization of locally finitely generated and locally finitely presented categories. We also define and study the injective objects that are Ext-orthogonal to the class of objects of type $\text{FP}_n$, called $\text{FP}_n$-injective objects, which will be the right half of a complete cotorsion pair. As a generalizatio…
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…
We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for project…
After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed infinity-categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every infinity-topos has a hierarchy of "universal" univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying (n-2)-truncated maps. We show that univalent families are preserved (and detected) by right adjoints t…
In this paper, we investigate the property (P) that finite products commute with arbitrary coequalizers in pointed categories. Examples of such categories include any regular unital or (pointed) majority category with coequalizers, as well as any pointed factor permutable category with coequalizers. We establish a Mal'tsev term condition characterizing pointed varieties of universal algebras satisfying (P). We then consider categories satisfying (P) locally, i.e., those categories for which every fibre $Pt_{\mathbb{C}}(X)$ of the fibration of points $π: Pt_{\mathbb{C}} \rightarrow \mathbb{C}$ …
We show that, under particular conditions, if a t-structure in the unbounded derived category of a locally coherent Grothendieck category restricts to the bounded derived category of its category of finitely presented objects, then its heart is itself a locally coherent Grothendieck category. Those particular conditions are always satisfied when the Grothendieck category is arbitrary and one considers the t-structure associated to a torsion pair in the category of finitely presented objects. They are also satisfied when one takes any compactly generated t-structure in the derived category of a…
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…
