A pretorsion theory for the category of all categories

从挠理论构建预挠理论

Building pretorsion theories from torsion theories

本文展示了预挠理论在阿贝尔范畴中的自然出现，提出了两种从挠理论构造预挠理论的方法，并给出从预挠理论获取挠理论的通用方法，应用于模范畴、内部群胚、回忆和表示理论。

一类广泛的预挠理论例子及相关注记

A wide class of examples of pretorsion theories and related remarks

该数学研究构造了一类广泛的预挠理论例子，证明乘积范畴中特定结构为预挠理论的充要条件，并指出结果可推广至任意预挠理论的乘积。

Pretorsion theories in general categories

We present a setting for the study of torsion theories in general categories. The idea is to associate, with any pair ($\mathcal T$, $\mathcal F$) of full replete subcategories in a category $\mathcal C$, the corresponding full subcategory $\mathcal Z = \mathcal T \cap \mathcal F$ of \emph{trivial objects} in $\mathcal C$. The morphisms which factor through $\mathcal Z$ are called $\mathcal Z$-trivial, and these form an ideal of morphisms, with respect to which one can define $\mathcal Z$-prekernels, $\mathcal Z$-precokernels, and short $\mathcal Z$-preexact sequences. This naturally leads to …

原子幺半群的范畴代数：可表示性、正则性和预挠理论

Categorical Algebra of Atomic Monoids: Presentability, Regularity, and Pretorsion Theories

研究原子幺半群范畴$\mathsf{AtoMon}$，证明其局部有限可表示，指出其允许特定分解但非正则范畴，建立单位群伴随关系并构造余自由原子幺半群，还展示了将$\mathsf{Grp}$挠理论提升到$\mathsf{AtoMon}$预挠理论的方法。

预序群的稳定范畴

The stable category of preordered groups

本文构造了与Z-预挠理论相关的预序群稳定范畴，研究了其函子性质，描述了预序群范畴中的Z-核与Z-余核，并证明了该稳定范畴的泛性质。

The stable category of preorders in a pretopos II: the universal property

We prove that the stable category associated with the category $\mathsf{PreOrd}(\mathbb C)$ of internal preorders in a pretopos $\mathbb C$ satisfies a universal property. The canonical functor from $\mathsf{PreOrd}(\mathbb C)$ to the stable category $\mathsf{Stab}(\mathbb C)$ universally transforms a pretorsion theory in $\mathsf{PreOrd}(\mathbb C)$ into a classical torsion theory in the pointed category $\mathsf{Stab}(\mathbb C)$. This also gives a categorical insight into the construction of the stable category first considered by Facchini and Finocchiaro in the special case when $\mathbb C…

Derived categories of NDG categories

We study $N$-differential graded ($NDG$) categories and their the derived categories. First, we introduce $N$-differential modules over an $NDG$ category $\mathcal{A}$. Then we show that the category $\mathsf{C}_{Ndg}(\mathcal{A})$ of $N$-differential $\mathcal{A}$-modules is a Frobenius category, and that its homotopy category $\mathsf{K}_{Ndg}(\mathcal{a})$ is a triangulated category. Second, we study the properties of the derived category $\mathsf{D}_{Ndg}(\mathcal{A})$ and give triangle equivalences of Morita type between derived categories of $NDG$ categories. Finally, we show $\mathsf{D}…

An adjunction between a category of complete restriction monoids and a category of étale categories

We prove that there is an adjunction between what we call étale topological categories and restriction quantal frames that leads to an adjunction with a category of complete restriction monoids. This generalizes the adjunction between étale groupoids and pseudogroups.

The category of categories with pullbacks is cartesian closed

We show that the category of categories with pullbacks and pullback preserving functors is cartesian closed.

2 Category of FRBSU Monoidal Categories and Crossed Modules

In that paper, we prove that the collection of all FRBSU monoidal categories and the collection of all crossed modules form a 2 category.

A Quillen model category structure on some categories of comonoids

We prove that for certain monoidal (Quillen) model categories, the category of comonoids therein also admits a model structure.

置换范畴的对称幺半2-范畴

The symmetric monoidal 2-category of permutative categories

本文为置换范畴定义张量积，证明其使置换范畴的2-范畴成为闭对称幺半双范畴，奠定了相关结构的基础。

Tangent Categories from the Coalgebras of Differential Categories

Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result appli…

The (2,1)-category of small coherent categories

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.

刻画（余）自由dagger范畴

Characterizing (Co)Free Dagger Categories

本文通过内部dagger范畴结构，利用矩形带和之字形概念，分别给出了余自由与自由dagger范畴的充要刻画，并联系到伴随诱导的（余）单子（余）代数。

Stacks and sheaves of categories as fibrant objects

We show that the category of categories fibred over a site is a generalized Quillen model category in which the weak equivalences are the local equivalences and the fibrant objects are the stacks, as they were defined by J. Giraud. The generalized model category restricts to one on the full subcategory whose objects are the categories fibred in groupoids. We show that the category of sheaves of categories is a model category that is Quillen equivalent to the generalized model category for stacks and to the model category for strong stacks due to A. Joyal and M. Tierney.

Cocompletion of restriction categories

Restriction categories were introduced as a way of generalising the notion of partial map categories. In this paper, we define cocomplete restriction category, and give the free cocompletion of a small restriction category as a suitably defined category of restriction presheaves. We also consider the case where our restriction category is locally small.

Definable Categories

We introduce the notion of a definable category--a category equivalent to a full subcategory of a locally finitely presentable category that is closed under products, directed colimits and pure subobjects. Definable subcategories are precisely the finite-injectivity classes. We prove a $2$-duality between the $2$-category of small exact categories and the $2$-category of definable categories, and provide a new proof of its additive version. We further introduce a third vertex of the $2$-category of regular toposes and show that the diagram of $2$-(anti-)equivalences between three $2$-categorie…

Simple omega-categories and chain complexes

The category of strict omega-categories has an important full subcategory whose objects are the simple omega-categories freely generated by planar trees or by globular cardinals. We give a simple description of this subcategory in terms of chain complexes, and we give a similar description of the opposite category, the category of finite discs, in terms of cochain complexes. Berger has shown that the category of simple omega-categories has a filtration by iterated wreath products of the simplex category. We generalise his result by considering wreath products of categories of chain complexes o…

Majority categories

We introduce the notion of a majority category --- the categorical counterpart of varieties of universal algebras admitting a majority term. This notion can be thought to capture properties of the category of lattices, in a way that parallels how Mal'tsev categories capture properties of the category of groups. Among algebraic majority categories are the categories of lattices, Boolean algebras, and Heyting algebras. Many geometric categories such as the category of topological spaces, metric spaces, ordered sets, any topos, ect., are comajority categories (i.e.~their duals are majority catego…

Abelian categories versus abelian 2-categories

Recently Dupont proved that the categories of discrete and codiscrete (or connected) objects in an abelian 2-category are equivalent abelian categories. He posses also a question whether any abelian category comes in this way. We will give a rather trivial solution of this problem in the case when a given abelian category has enough projective or injective objects.

Reflection on the enrichment of double categories

Enrichment and internal categories are two different way to generalize the notion of category. As such, enriching double categories (which are categories internal to Cat) is not a clear concepts. One can look at the internal categories of the category of C-enriched categories or the internal C-category in Cat. This concepts, unfortunately, are not powerful enough, even thin (posetal) double categories can't be seen this way. However, an orthogonal investigation, that of the enrichment by a double category, will help relaxing the internal C-category to a incorporate the internal categories of t…

Hom weak $ω$-categories of a weak $ω$-category

Classical definitions of weak higher-dimensional categories are given inductively; for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak $n$-categories for all natural numbers $n$, or of weak $ω$-categories, take more sophisticated approaches, and the nature of the "hom" is often not immediate from the definitions. In this paper, we focus on Leinster's definition of weak $ω$-category based on an earlier definition by Batanin, and construct for each weak $ω$-category $\mathcal{A}$,…

关系的 dagger 范畴：延迟 dagger 范畴与满正则独立范畴的等价性

Dagger categories of relations: The equivalence of dilatory dagger categories and epi-regular independence categories

针对若干看似关系范畴却不符合正则范畴关系理论的反常例子，本文在满正则独立范畴中建立平行关系理论，证明其与延迟 dagger 范畴的等价性，为解释这些例子提供新框架。

A general method to construct cube-like categories and applications to homotopy theory

In this paper, we introduce a method to construct new categories which look like "cubes", and discuss model structures on the presheaf categories over them. First, we introduce a notion of thin-powered structure on small categories, which provides a generalized notion of "power-sets" on categories. Next, we see that if a small category $\mathcal{R}$ admits a good thin-powered structure, we can construct a new category $\square(\mathcal{R})$ called the cubicalization of the category. We also see that $\square(\mathcal{R})$ is equipped with enough structures so that many arguments made for the c…

Restriction categories III: colimits, partial limits, and extensivity

A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about partial maps as they have a purely algebraic formulation. In this paper we consider colimits and limits in restriction categories. As the notion of restriction category is not self-dual, we should not expect …

作为斜充实范畴的笛卡尔微分范畴

Cartesian differential categories as skew enriched categories

本文将笛卡尔微分范畴刻画为基于斜幺半范畴的充实范畴，并证明每个小笛卡尔微分范畴可完全保持结构地嵌入由微分模态诱导的范畴，从而解决了该领域的一个公开问题。

The Relation between Ann-Categories and Ring Categories

There are different categorizations of the definition of a {\it ring} such as {\it Ann-category} (see N. T. Quang [6]), {\it ring category} (see M. Kapranov and V.Voevodsky [2]),... The main result of this paper is to prove that every axiom of the definition of a {\it ring category}, without the axiom $x_{0}=y_{0},$ can be deduced from the axiomatics of an {\it Ann-category}.

External triangulation of the homotopy category of exact quasi-category

Extriangulated categories axiomatize extension-closed subcategories of triangulated categories. We show that the homotopy category of an exact quasi-category can be equipped with a natural extriangulated structure.

Constructing model categories with prescribed fibrant objects

We present a weak form of a recognition principle for Quillen model categories due to J.H. Smith. We use it to put a model category structure on the category of small categories enriched over a suitable monoidal simplicial model category. The proof uses a part of the model structure on small simplicial categories due to J. Bergner. We give an application of the weak form of Smith's result to left Bousfield localizations of categories of monoids in a suitable monoidal model category.

Factorization systems in $\infty$-categories

We extend the notion of a factorization system in a category to the realm of $\infty$-categories. To this end, we provide a description of the category of $\infty$-categories with factorization systems as the category of presheaves of spaces on a certain category that satisfy a version of the Segal condition. Additionally, we study the notion of distributive laws between monads and, more generally, lax functors in the context of categories with factorization systems. In particular, we also characterize categories with factorization systems as distributive laws in the bicategory of spans.

幺正、内积与Dagger范畴

Unitary, Inner product, and Dagger categories

本文通过内积范畴重新定义了Dagger范畴，指出内积范畴等同于弱Dagger范畴，通过内积结构可将弱Dagger范畴严格化为Dagger范畴。

Minimal accessible categories

We give a purely category-theoretic proof of the result of Makkai and Paré saying that the category $\bf Lin$ of linearly ordered sets and order preserving injective mappings is a minimal finitely accessible category. We also discuss the existence of a minimal $\aleph_1$-accessible category.

Introduction to Ann-categories

In this paper, we present new concepts of Ann-categories, Ann-functors, and a transmission of the structure of categories based on Ann-equivalences. We build Ann-category of Pic-funtors and prove that each Ann-category can be faithfully embedded into an almost strictly Ann-category.

Strict $ω$-categories are monadic over polygraphs

We give a direct proof that the category of strict $ω$-categories is monadic over the category of polygraphs.

实数充实范畴导引笔记

Introductory notes on real-enriched categories

本文介绍了由单位区间和连续 t-范数构成的实数充实范畴，重点探讨了该范畴中的预层单子及其子单子。

Compact closed categories and $Γ$-categories (with an appendix by André Joyal)

In this paper we study compact closed categories within the context of homotopical algebra. We construct two new model category structures by localizing two (Quillen equivalent) model categories of symmetric monoidal categories with the objective of establishing the free compact closed category on one generator as a fibrant replacement of the free symmetric monoidal category on one generator, in our localized model categories. We go on to show that the fibrant objects in our model categories are compact closed categories.

Right Delocalization of Model Categories

Model categories have long been a useful tool in homotopy theory, allowing many generalizations of results in topological spaces to other categories. Giving a localization of a model category provides an additional model category structure on the same base category, which alters what objects are being considered equivalent by increasing the class of weak equivalences. In some situations, a model category where the class of weak equivalences is restricted from the original one could be more desirable. In this situation we need the notion of a delocalization. In this paper, right Bousfield deloc…

Cohomological classification of braided $Ann$-categories

A braided $Ann$-category $\mathcal A$ is an $Ann$-category $\mathcal A$ together with a braiding $c$ such that $(\mathcal A, \otimes, a, c, (1,l,r))$ is a braided tensor category, moreover $c$ is compatible with the distributivity constraints. According to the structure transport theorem, the paper shows that each braided $Ann$-category is equivalent to a braided $Ann$-category of the type $(R,M)$, hence the proof of the classification theorem for braided $Ann$-categories by the cohomology of commutative rings is presented.

Categories of operators and actions of group operads

We propose a new model for multicategories with symmetries with respect to Zhang's group operads. The fully faithful embedding of the category of group operads into that of crossed interval groups is made use of, and it is shown that every multicategory gives rise to a fibration, in a sense, over a quotient of the total category of group operads. The symmetric structures can be presented as structures of internal presheaves over a category internal to the category of small categories, in other words a double category.

关于将F-变换相关范畴与模糊预拓扑空间统一为Qua范畴

On unification of categories associated with F -transforms and fuzzy pretopological spaces as Qua category

本文提出新范畴Qua，将F-变换相关范畴与模糊预拓扑空间范畴统一，其对象为答案的成功度量，态射为变换的成功度量对，并揭示了多个L值模糊结构范畴与Qua之间的有趣关系。

A model category structure on the category of simplicial categories

In this paper we put a cofibrantly generated model category structure on the category of small simplicial categories. The weak equivalences are a simplicial analogue of the notion of equivalence of categories.

Reedy categories which encode the notion of category actions

We study a certain type of action of categories on categories and on operads. Using the structure of the categories Δ and Ω governing category and operad structures, respectively, we define categories which instead encode the structure of a category acting on a category, or a category acting on an operad. We prove that the former has the structure of an elegant Reedy category, whereas the latter has the structure of a generalized Reedy category. In particular, this approach gives a new way to regard group actions on categories and on operads.

A model 2-category of enriched combinatorial premodel categories

In his book on model categories, Hovey asked whether the 2-category $\mathbf{Mod}$ of model categories admits a "model 2-category structure" whose weak equivalences are the Quillen equivalences. We show that $\mathbf{Mod}$ does not have pullbacks and so cannot form a model 2-category. This lack of pullbacks can be traced to the two-out-of-three axiom on the weak equivalences of a model category. Accordingly, we define a premodel category to be a complete and cocomplete category equipped with two nested weak factorization systems. Combinatorial premodel categories form a complete and cocomplete…

Descent and Galois theory for Hopf categories

Descent theory for linear categories is developed. Given a linear category as an extension of a diagonal category, we introduce descent data, and the category of descent data is isomorphic to the category of representations of the diagonal category, if some flatness assumptions are satisfied. Then Hopf-Galois descent theory for linear Hopf categories, the Hopf algebra version of a linear category, is developed. This leads to the notion of Hopf-Galois category extension. We have a dual theory, where actions by dual linear Hopf categories on linear categories are considered. Hopf-Galois category…

Fibration Categories are Fibrant Relative Categories

A relative category is a category with a chosen class of weak equivalences. Barwick and Kan produced a model structure on the category of all relative categories, which is Quillen equivalent to the Joyal model structure on simplicial sets and the Rezk model structure on simplicial spaces. We will prove that the underlying relative category of a model category or even a fibration category is fibrant in the Barwick--Kan model structure.

Derived categories for Grothendieck categories of enriched functors

The derived category $D[C,V]$ of the Grothendieck category of enriched functors $[C,V]$, where $V$ is a closed symmetric monoidal Grothendieck category and $C$ is a small $V$-category, is studied. We prove that if the derived category $D(V)$ of $V$ is a compactly generated triangulated category with certain reasonable assumptions on compact generators or $K$-injective resolutions, then the derived category $D[C,V]$ is also compactly generated triangulated. Moreover, an explicit description of these generators is given.

Operadic categories and their skew monoidal categories of collections

I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an operad over the operadic category. In fact I describe two skew monoidal categories with this property. The first has the feature that the operadic category can be recovered from the skew monoidal category of collections; the second has the feature that the right unit constraint is invertible. In the case of the operadic category S of finite sets and functions…

An interpretation of dependent type theory in a model category of locally cartesian closed categories

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…

A model structure on the category of small categories for coverings

We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the fibrant replacement is the groupoidification.

The Grothendieck Construction and Gradings for Enriched Categories

The Grothendieck construction is a process to form a single category from a diagram of small categories. In this paper, we extend the definition of the Grothendieck construction to diagrams of small categories enriched over a symmetric monoidal category satisfying certain conditions. Symmetric monoidal categories satisfying the conditions in this paper include the category of $k$-modules over a commutative ring $k$, the category of chain complexes, the category of simplicial sets, the category of topological spaces, and the category of modern spectra. In particular, we obtain a generalization …

The zeta function of a finite category

We define the zeta function of a finite category. And we propose a conjecture which states the relationship between the Euler characteristic of finite categories and the zeta function of finite categories. This conjecture is verified when categories are finite groupoids, finite acyclic categories, categories with 2-objects and finite categories satisfying certain condition.

The Duskin nerve of 2-categories in Joyal's cell category Theta_2

We give an explicit and purely combinatorial description of the Duskin nerve of any (r+1)-point suspension 2-category, and in particular of any 2-category belonging to Joyal's cell category Theta_2.

Simplicial and categorical comma categories

We consider four categories: the category of diagrams of small categories indexed by a given small category O, the (comma) category of small categories over O, the category of diagrams of simplicial sets indexed by O, and the category of simplicial sets over the nerve of O. Fritsch and Golasinski claimed that these four categories have equivalent homotopy categories but, in fact, their proof contains an error and the homotopy categories are not equivalent with the weak equivalences they use in the comma categories. We show here that the correct weak equivalences are the ``weak fibre homotopy e…