On the existence of a category with a given matrix

Internalizing decorated bicategories: The globularily generated condition

This is the first part of a series of papers studying the problem of existence of double categories for which horizontal bicategory and object category are given. We refer to this problem as the problem of existence of internalizations for decorated bicategories. We establish a formal framework within which the problem of existence of internalizations can be correctly formulated. Further, we introduce the condition of a double category being globularily generated. We prove that the problem of existence of internalizations for a decorated bicategory admits a solution if and only if it admits a …

三角范畴的再分解与三角矩阵范畴的奇点范畴

Recollements of Triangulated Categories and the Singularity Category of a Triangular Matrix Category

利用再分解框架，在特定同调条件下证明三角矩阵范畴的奇点范畴等价于其右下子范畴的奇点范畴，推广了Pin Liu和Ming Lu的结果。

函子范畴、Recollements 与三角矩阵范畴

Functor categories, Recollements, Triangular Matrix category

本文通过阿贝尔范畴的 recollements 理论研究三角矩阵范畴，构造了两个典范 recollements，并证明在相关函子正合时中间项范畴仍为三角矩阵范畴，推广了 Liping Li 的结果；同时给出模范畴具有非平凡挠对时原范畴等价于三角矩阵范畴的判定。

A$_{\infty}$-范畴范畴中余等化子不存在的一个例子

An example of the non-existence of coequalizers in the category of the A$_{\infty}$-categories

本文证明严格单位与非单位的A∞-代数及A∞-范畴范畴在相应态射下均无余等化子，给出反例。

On the cocartesian image of preorders and equivalence relations in regular categories

In a regular category $\mathbb E$, the direct image along a regular epimorphism $f$ of a preorder is not a preorder in general. In $Set$, its best preorder approximation is then its cocartesian image above $f$. In a regular category, the existence of such a cocartesian image above $f$ of a preorder $S$ is actually equivalent to the existence of the supremum $R[f]\vee S$ among the preorders. We investigate here some conditions ensuring the existence of these cocartesian images or equivalently of these suprema. They applied to two very dissimilar contexts: any topos $\mathbb E$ with suprema of c…

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.

来自n重矩阵因子化的三角范畴

Triangulated categories arising from n-fold matrix factorizations

本文研究了n重矩阵因子化所诱导的三角范畴结构，证明了在特定条件下该范畴具有三角结构。

Matrix factorizations for nonaffine LG-models

We propose a natural definition of a category of matrix factorizations for nonaffine Landau-Ginzburg models. For any LG-model we construct a fully faithful functor from the category of matrix factorizations defined in this way to the triangulated category of singularities of the corresponding fiber. We also show that this functor is an equivalence if the total space of the LG-model is smooth.

$\infty$-cosheafification

Cosheaves are a dual notion of sheaves. In this paper, we prove existence of a dual of sheafifications, called \textit{cosheafifications}, in the $\infty$-category theory. We also prove that the $\infty$-category of $\infty$-cosheaves is presentable and equivalent to an $\infty$-category of left adjoint functors.

Complete intersections and equivalences with categories of matrix factorizations

We prove that one can realize certain triangulated subcategories of the singularity category of a complete intersection as homotopy categories of matrix factorizations. Moreover, we prove that for any commutative ring and non-zerodivisor, the homotopy category of matrix factorizations embeds into the homotopy category of totally acyclic complexes of finitely generated projective modules over the factor ring.

Triangular Matrix Categories II: Recollements and functorially finite subcategories

In this paper we continue the study of triangular matrix categories $\mathbfΛ=\left[ \begin{smallmatrix} \mathcal{T} & 0 \\ M & \mathcal{U} \end{smallmatrix}\right]$ initiated in [21]. First, given an additive category $\mathcal{C}$ and an ideal $\mathcal{I}_{\mathcal{B}}$ in $\mathcal{C}$, we prove a well known result that there is a canonical recollement $\xymatrix{\mathrm{Mod}(\mathcal{C}/\mathcal{I}_{\mathcal{B}})\ar[r]_{} & \mathrm{Mod}(\mathcal{C})\ar[r]_{}\ar@<-1ex>[l]_{}\ar@<1ex>[l]_{} & \mathrm{Mod}(\mathcal{B})\ar@<-1ex>[l]_{}\ar@<1ex>[l]_{}}$. We show that given a recollement betwee…

Middle-Four Maps and Net Categories

We briefly relate the existence of a middle-four interchange map in a category with two monoidal structures, to the standard Cockett and Seely notion of a weakly distributive category.

重新审视 $\mathbf{Cat}$ 中的极限和同伦范畴

Revisiting colimits in $\mathbf{Cat}$ and homotopy category

本文通过明确的方法证明了 $\mathbf{Cat}$ 中 (co) 限的存在，通过同伦范畴 functor 与特定加权极限的关系，以及利用 simplicial sets 和 nerve functor 的性质，证明了 $\mathbf{Cat}$ 的 (co) 完备性。

论$n$-正合范畴 I：极大$n$-正合结构的存在性与唯一性

On $n$-exact categories I: The existence and uniqueness of maximal $n$-exact structures

本文作为$n$-正合范畴系列研究的首篇，遵循Enomoto的思路构造范畴上$n$-正合结构与函子范畴特定子范畴的双射，证明了幂等完备加法范畴上唯一极大$n$-正合结构的存在性。

Thom-Sebastiani & Duality for Matrix Factorizations

The derived category of a hypersurface has an action by "cohomology operations" k[t], deg t=-2, underlying the 2-periodic structure on its category of singularities (as matrix factorizations). We prove a Thom-Sebastiani type Theorem, identifying the k[t]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving k[t]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support conditi…

Normalizers in the non-pointed context: a weak case of extremal decomposition

The aim of this work is to point out a strong structural phenomenon hidden behind the existence of normalizers through the investigation of this property in the non-pointed context: given any category E, a certain property of the fibration of points: Pt(E) --> E guarentees the existence of normalizers. This property becomes a characterization of this existence when E is quasi-pointed and protomodular. This property is also showed to be equivalent to a property of the category GrdE of internal groupoids in E which is a kind of opposite, for the monomorphic internal functors, of the comprehensiv…

Limits and Colimits in a Category of Lenses

Lenses are an important tool in applied category theory. While individual lenses have been widely used in applications, many of the mathematical properties of the corresponding categories of lenses have remained unknown. In this paper, we study the category of small categories and asymmetric delta lenses, and prove that it has several good exactness properties. These properties include the existence of certain limits and colimits, as well as so-called imported limits, such as imported products and imported pullbacks, which have arisen previously in applications. The category is also shown to b…

关于意义、对象、名称和存在的范畴论独立性

On the Category-Theoretic Independence of Meaning, Object, Name and Existence

该研究通过范畴论方法证明意义、对象、名称和存在四个概念在结构上的独立性，并展示了在拓扑斯中内部存在与全局命名的分离，为信息网络的基础框架提供了理论支持。

An almost full embedding of the category of graphs into the category of abelian groups

We construct an embedding G of the category of graphs into the category of abelian groups such that for graphs X and Y we have Hom(GX,GY)=Z[Hom(X,Y)], the free abelian group whose basis is the set Hom(X,Y). The isomorphism is functorial in X and Y. The existence of such an embedding implies that, contrary to a common belief, the category of abelian groups is as complex and comprehensive as any other concrete category. We use this embedding to settle an old problem of Isbell whether every full subcategory of the category of abelian groups, which is closed under limits, is reflective. A positive…

Coherent analogues of matrix factorizations and relative singularity categories

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its full subcategory. The latter category is identified with the kernel o…

范畴矩阵分解与单射范畴

Categorical matrix factorizations and monomorphism categories

本文在纯范畴背景下推广了Eisenbud和Yoshino关于矩阵分解与超曲面环上最大Cohen-Macaulay模的对应关系，通过引入超曲面范畴和Frobenius精确子范畴，证明分解稳定范畴与单射链范畴之间存在三角等价。

Sheaves that fail to represent matrix rings

There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is naturally isomorphic to the identity, such that F restricts to the Zariski spectrum functor Spec on the category of commutative rings (in a compatible way with the natural isomorphism). Both of these no-go re…

Matrix factorizations for self-orthogonal categories of modules

For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in \mathsf{C}$, we show there is a natural embedding of the homotopy category of $\mathsf{C}$-factorizations of $f$ into a corresponding homotopy category of totally acyclic complexes. Moreover, we prove this is an equivalence if $\mathsf{C}$ is the category of projective or flat-cotorsion $S$-modules. Dually, using divisibility in place of regularity, we observ…

代数代数的若干方面

Aspects of algebraic Algebras

本文研究Kock-Zöberlein单子下Kleisli与Eilenberg-Moore范畴之间的中间范畴，刻画自由代数、研究加权极限存在性，并应用于滤子单子的幂等完备化。

Extension of Matrix Pencil Reduction to Abelian Categories

Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition Theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part made of nilpotent matrices, and two other dual parts, which we call the observation and control part. The goal of this paper is to show that large portions of that decomposition are still valid for pairs of morphisms of modules or abelian groups, and more generally in any abelian category. % This gives a new perspective even in the vector space case, as we hav…

Compact generators in categories of matrix factorizations

We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field. We deduce a quasi-equivalence between the category of matrix factorizations and the dg derived category of an explicitly computable dg algebra. Building on this result, we employ a variant of Toen's derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain …

Matrix taxonomy and Bourn localization

In a recent paper, an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices -- the so called `matrix properties'. In this paper we extend this algorithm to include matrix properties involving pointedness of a category, such as the properties of a category to be unital, strongly unital or subtractive, for example. Moreover, this extended algorithm can also be used to determine whether a given matrix property is the Bourn localization of another, thus leading to new characterizations of Mal'tsev, majority and ar…

The existence and utility of Giry algebras in probability theory

Giry algebras are barycenters maps, which are coequalizers of contractible coequalizer pairs (like any algebras), and their existence, in general, requires the measurable space be coseparated by the discrete two point space, and the hypothesis that no measurable cardinals exist. Under that hypothesis, every measurable space which is coseparated has an algebra, and the category of Giry algebras provides a convenient setting for probability theory because it is a symmetric monoidal closed category with all limits and colimits, as well as having a seperator and coseperator. This is in stark contr…

推动-拉span的更高范畴，II：矩阵因子化

Higher categories of push-pull spans, II: Matrix factorizations

本文基于前文构建的对称单向$(\infty,3)$-范畴$\mathscr{CRW}$，扩展了Brunner等人关于仿射Rozansky-Witten模型的工作，建立矩阵因子化与同伦2-范畴之间的函子，并计算相应的TFT。

命题演绎系统的范畴

The category of propositional deductive systems

研究命题演绎系统范畴相关问题，通过定义范畴$\QM$等，证明余积等的存在性，定义非满子范畴$\DS_0$并证明其与特定命题逻辑等价，还探讨了$\DS_0$中的融合，明确了其中余积的性质。

The first obstructions to enhancing a triangulated category

In this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement. We show that these obstructions do not vanish in an example of triangulated category without models. We also obtain cohomological characterizations of pre-triangulated DG, A-infinity, and spectral categories.

On the representation categories of matrix quantum groups of type $A$

A quantum groups of type $A$ is defined in terms of a Hecke symmetry. We show in this paper that the representation category of such a quantum group is uniquely determined as an abelian braided monoidal category by the bi-rank of the Hecke symmetry.

Categories with dependent arrows

We present an abstract, categorical formulation of dependent functions in a fundamental manner and independently from the Sigma-construction. For that, we define first the notion of a category with family-arrows, or a $\f$-category. A $(\f, Σ)$-category is a $\f$-category with Sigma-objects, where a $(\f, Σ)$-category with a terminal object is exactly a type-category of Pitts, or a category with attributes of Cartmell. We introduce categories with dependent arrows, or $\di$-categories, and we show that every $(\f, Σ)$-category is a $\di$-category in a canonical way. The notion of a Sigma-objec…

A Geometric Approach to Orlov's Theorem

A famous theorem of D. Orlov describes the derived bounded category of coherent sheaves on projective hypersurfaces in terms of an algebraic construction called graded matrix factorizations. In this article, I implement a proposal of E. Segal to prove Orlov's theorem in the Calabi-Yau setting using a globalization of the category of graded matrix factorizations (graded D-branes). Let X be a projective hypersurface. Already, Segal has established an equivalence between Orlov's category of graded matrix factorizations and the category of graded D-branes on the canonical bundle K of the ambient p…

Khovanov 的 Heisenberg 分类中的半单子类

A semisimple subcategory of Khovanov's Heisenberg category

本文研究了 Khovanov 的 Heisenberg 分类中的半单子类，展示了其在非交换张量三角几何中的应用，揭示了 Balmer 谱的张量积性质与单侧厚张量理想分类的矛盾。

Combinatorial and accessible weak model categories

In a previous work, we have introduced a weakening of Quillen model categories called weak model categories. They still allow all the usual constructions of model category theory, but are easier to construct and are in some sense better behaved. In this paper we continue to develop their general theory by introducing combinatorial and accessible weak model categories. We give simple necessary and sufficient conditions under which such a weak model category can be extended into a left and/or right semi-model category. As an application, we recover Cisinski-Olschok theory and generalize it to we…

Equalizers in the category of cocomplete cocategories

We prove existence of equalizers in certain categories of cocomplete cocategories. This allows us to complete the proof of the fact that A-infinity functor categories arise as internal Hom-objects in the category of differential graded cocomplete augmented cocategories.

A folk Quillen model structure for operads

We establish, by elementary means, the existence of a cofibrantly generated monoidal model structure on the category of operads. By slicing over a suitable operad the classical Rezk model structure on the category of small categories is recovered.

右诱导模型结构存在的判据

A criterion for existence of right-induced model structures

本文提出一个判断右诱导模型结构存在性的条件，并通过多个例子展示了其应用。

Recursion categories of coalgebras

We construct recursion categories from categories of coalgebras. Let $F$ be a nontrivial endofunctor on the category of sets that weakly preserves pullbacks and such that the category $\textbf{Set}_F$ of $F$-coalgebras is complete. The category $\textbf{Set}_F$ may be embedded in the category $\mathbf{Pfn}_F$ of $F$-coalgebras and partial morphisms, which is a $P$-category that is prodominical but not dominical in general. An existence theorem of A. Heller is applied to certain subcategories of $\textbf{Pfn}_F$ to obtain examples of recursion categories of coalgebras.

Triangular Matrix Categories over path Categories and Quasi-hereditary Categories, as well as one point extensions by Projectives

In this paper, we prove that the lower triangular matrix category $Λ=\left [ \begin{smallmatrix} \mathcal{T}&0\\ M&\mathcal{U} \end{smallmatrix} \right ]$, where $\mathcal{T}$ and $\mathcal{U}$ are quasi-hereditary $\mathrm{Hom}$-finite Krull-Schmidt $K$-categories and $M$ is a $\mathcal U\otimes_K \mathcal T^{op}$-module that satisfies suitable conditions, is quasi-hereditary in the sense of \cite{LGOS1} and \cite{Martin}. Moreover, we solve the problem of finding quotients of path categories isomorphic to the lower triangular matrix category $Λ$, where $\mathcal T=K\mathcal{R/J}$ and $\mathc…

The homotopy type of the $\infty$-category associated to a simplicial complex

This paper is part of a series of papers about homotopy theory of strict $n$-categories. In the first paper of this series, we gave conditions that guarantee the existence of a Thomason model category structure on the category of strict $n$-categories. The main goal of our paper is to show one of these conditions. To do so, we associate to any simplicial complex a strict $\infty$-category generated by a computad. We conjecture that this $\infty$-category has the same homotopy type as the corresponding simplicial complex and we prove this conjecture when the simplicial complex comes from a pose…

Bondarenko 范畴的三角化结构

Triangulated structure for Bondarenko's Categories

本文针对 Bondarenko 范畴非阿贝尔但与 gentle 代数导出范畴密切相关的问题，在其商范畴上构造了三角范畴结构，为运用三角范畴技术研究偏序集表示提供了新工具。

拟细胞范畴与 Brauer 范畴的结构

Quasi-cellular categories and the structure of the Brauer category

本文提出拟细胞范畴概念，研究 Brauer 范畴的拟细胞结构，构造到矩阵 Brauer 范畴的线性函子，证明非奇异参数下该函子为同构且 Brauer 范畴与置换子范畴 Morita 等价，推广了 Brauer 代数的相关结果。

Determinant functors on triangulated categories

We study determinant functors which are defined on a triangulated category and take values in a Picard category. The two main results are the existence of a universal determinant functor for every small triangulated category, and a comparison theorem for determinant functors on a triangulated category with a non-degenerate bounded t-structure and determinant functors on its heart. For a small triangulated category T we give a natural definition of groups K_0(T) and K_1(T) in terms of the universal determinant functor on T, and we show that K_i(T)=K_i(E) for i=0 and 1 if T has a non-degenerate …

Relative homological algebra, Waldhausen $K$-theory, and quasi-Frobenius conditions

We study the question of the existence of a Waldhausen category on any (relative) abelian category in which the contractible objects are the (relatively) projective objects. The associated $K$-theory groups are "stable algebraic $G$-theory," which in degree zero form a certain stable representation group. We prove both some existence and nonexistence results about such Waldhausen category structures, including the fact that, while it was known that the category of $R$-modules admits a model category structure if $R$ is quasi-Frobenius, that assumption is required even to get a Waldhausen categ…

Relative singular locus and Balmer spectrum of matrix factorizations

For a separated Noetherian scheme $X$ with an ample family of line bundles and a non-zero-divisor $W\inΓ(X,L)$ of a line bundle $L$ on $X$, we classify certain thick subcategories of the derived matrix factorization category ${\rm DMF}(X,L,W)$ of the Landau-Ginzburg model $(X,L,W)$. Furthermore, by using the classification result and the theory of Balmer's tensor triangular geometry, we show that the spectrum of the tensor triangulated category $({\rm DMF}(X,L,W), \otimes^{\frac{1}{2}})$ is homeomorphic to the relative singular locus ${\rm Sing}(X_0/X)$, introduced in this paper, of the zero s…

Parametrized higher category theory and higher algebra: Exposé I -- Elements of parametrized higher category theory

We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we are able at this stage. Simple operations, such as the formation of opposites and the formation of functor $\infty$-categories, become slightly more involved in the parametrized setting, but we explain precisely how to perform these constructions. All of these constructions can be performed explicitly, without resorting to such acts of desperation as straight…

微分分次三角矩阵范畴

Differential graded triangular matrix categories

本文在微分分次范畴中构造三角矩阵范畴 Λ，并证明其 dg-模范畴与相应 dg-逗号范畴等价，推广了 Artin 代数的经典结论。

Recollement for Differential Graded Algebras

A recollement of triangulated categories describes one such category as being "glued together" from two others. This paper gives a precise criterion for the existence of a recollement of the derived category of a Differential Graded Algebra in terms of two other such categories.

Category Algebras and States on Categories

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functionals defined on category algebras. We clarify that category algebras can be considered as generalized matrix algebras and that states on categories as linear functionals defined on category algebras turn out to be generalized of probability measures on sets as discrete categories. Moreover, by establishing a generalizatio…

The existence of UFO implies projectively universal morphisms

Let $\mathcal C$ be a concrete category. We prove that if $\mathcal{C}$ admits a universally free object $\mathsf F$, then there is a projectively universal morphism $u\colon \mathsf F\to \mathsf F$, i.e., a morphism $u$ such that for any $B\in \mathcal{C}$ and $τ\in {\rm Mor}(B)$ there exists an epimorphism $π\in {\rm Mor}(\mathsf F, B)$ such that $πτ= u π$. This builds upon and extends various ideas by Darji and Matheron (Proc. Am. Math. Soc. 145 (2017)) who proved such a result for the category of separable Banach spaces with contractive operators as well as certain classes of dynamical sys…

在对称序的导出和奇点范畴上构造Calabi-Yau结构

Calabi-Yau structures on derived and singularity categories of symmetric orders

本文研究了对称序的导出和奇点范畴上的Calabi-Yau结构，通过构造和证明基变换性质，展示了非交换情况下的Calabi-Yau结构存在性，并与广义簇范畴的等价性。

奇异 Hochschild 复形与 Cartan 矩阵

Singular Hochschild complex and Cartan matrix

本文研究对称代数与 Frobenius 代数上奇异 Hochschild 同调与 Cartan 矩阵对称性的关系，给出反例表明一般 Frobenius 代数不成立。