A presentation of the category of stochastic matrices

A cartesian presentation of weak n-categories

We propose a notion of weak (n+k,n)-category, which we call (n+k,n)-Theta-spaces. The (n+k,n)-Theta-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category Theta_n. This notion is a generalization of that of complete Segal spaces (which are precisely the (infty,1)-Theta-spaces). Our main result is that the above model category is cartesian.

拟范畴的重写与表示

Rewriting and presentations of quasicategories

该研究将重写理论方法应用于拟范畴层面，通过模型结构转换得到弱(∞,1)-范畴的Amick-Groves-Squier型表示，并给出拟范畴理论相关应用。

Coherent Presentations of Monoidal Categories

Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations where the objects are considered modulo an equivalence relation, which is described by equational generators. When those form a convergent (abstract) rewriting system on objects, there are three very natural constructions that can be used to define the category which is described by the presentation: one consists in turning equational generators into identi…

Cylinder topological quantum field theory: A categorical presentation of classical field theory and its symmetries

We use geometric ideas coming from certain classic algebraic constructions to associate, to every classical field theory, a symmetric monoidal double functor from the double category of cobordisms with corners to a certain symmetric monoidal double category. We call symmetric monoidal double functors so constructed cylinder topological quantum field theories. Our initial formulation of cylinder topological quantum field theory is presented in a purely topological context. We present appropriate refinements in order to accommodate information relevant to classical field theory.

范畴的同伦呈现

Homotopical presentation of categories

本文给出小范畴间函子生成 Dugger 万有模型范畴小呈现的判定准则。

Joyal's Suspension Functor on $Θ$ and Kan's Combinatorial Spectra

In [Joyal] where the category $Θ$ is first defined it is noted that the dimensional shift on $Θ$ suggests an elegant presentation of the unreduced suspension on cellular sets. In this note we prove that the reduced suspension associated to that presentation is left Quillen with respect to the Cisinski model category structure presenting the $\left(\infty,1\right)$-category of pointed spaces and enjoys the correct universal property. More, we go on to describe how, in forthcoming work, inspired by the combinatorial spectra described in [Kan], this suspension functor entails a description of spe…

Decreasing diagrams and coherent presentations

We show how decreasing diagrams introduced in the theory of rewriting systems can be used to prove coherence type theorems in category theory. We apply this method to describe a coherent presentation of the $0$-Hecke monoid $\mathcal{H}(Σ_n)$ of the symmetric group $Σ_n$, i.e. a presentation by generators, relations, and relations between relations.

The Quotient of a Category by the Action of a Monoidal Category

We introduce the notion of the quotient of a category $C$ by the action $A : M \longrightarrow C \times C$ of a unital symmetric monoidal category $M$. The quotient $C/M$ is a 2-category. We prove its existence and uniqueness by first showing that every small 2-category has a presentation in terms of generators and relations and then describing the generators and relations needed for the quotient $C/M$.

范畴的子范畴的充实单子与簇的图示表示

Diagrammatic presentations of enriched monads and varieties for a subcategory of arities

本文针对子范畴的范畴，提出了图示 $J$-表示和 $J$-元簇的形式体系，用于直接描述充实代数结构，并证明 $J$-元簇与 $J$-元 $V$-单子对偶等价，进而推广了预理论-单子伴随及Birkhoff的Galois联络。

Theory Presentation Combinators

We motivate and give semantics to theory presentation combinators as the foundational building blocks for a scalable library of theories. The key observation is that the category of contexts and fibered categories are the ideal theoretical tools for this purpose.

范畴中方程的图示表示

The diagrammatic presentation of equations in categories

本文定义了精确表示“同解”的范畴图等价，并证明图范畴沿该等价的局部化与切片范畴沿初始函子的局部化同构，且推广至局部可展示 2-范畴。

Left-induced model structures and diagram categories

We prove existence results a la Jeff Smith for left-induced model category structures, of which the injective model structure on a diagram category is an important example. We further develop the notions of fibrant generation and Postnikov presentation from Hess, which are dual to a weak form of cofibrant generation and cellular presentation. As examples, for k a field and H a differential graded Hopf algebra over k, we produce a left-induced model structure on augmented H-comodule algebras and show that the category of bounded below chain complexes of finite-dimensional k-vector spaces has a …

将有限 Reedy 范畴呈现为有限直范畴的局部化

Presentation of finite Reedy categories as localizations of finite direct categories

本文通过构造直范畴 Down(C) 及其到 Reedy 范畴 C 的函子，将有限 Reedy 范畴呈现为有限直范畴的 (∞,1)-范畴局部化，从而为同伦类型论的有限截断单纯类型元理论提供工具。

Coherent presentations of Artin monoids

We compute coherent presentations of Artin monoids, that is presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squier's and Knuth-Bendix's completions into a homotopical completion-reduction, applied to Artin's and Garside's presentations. The main result of the paper states that the so-called Tits-Zamolodchikov 3-cells extend Artin's presentation into a coherent presentation. As a byproduct, we give a new constructive proof of a theorem of Deligne on the actions of an Artin monoid on a category.

Grothendieck拓扑是筛子的外延表示

Grothendieck Topologies Are Extensional Presentations of the Form of Sieves

本文通过引入分配性形式，将Grothendieck拓扑刻画为筛子形式的外延表示，并应用于多种拓扑结构。

Sur l'existence d'une catégorie ayant une matrice strictement positive donnée

The Leinster matrix corresponding to a finite category has entries counting the number of morphisms between objects. A first question is to know which positive integer matrices come from at least one finite category. Here, that question will be answered when the entries are strictly positive.

Linear polygraphs applied to categorification

We introduce two applications of polygraphs to categorification problems. We compute first, from a coherent presentation of an $n$-category, a coherent presentation of its Karoubi envelope. For this, we extend the construction of Karoubi envelope to $n$-polygraphs and linear $(n,n-1)$-polygraphs. The second problem treated in this paper is the construction of Grothendieck decategorifications for $(n,n-1)$-polygraphs. This construction yields a rewriting system presenting for example algebras categorified by a linear monoidal category. We finally link quasi-convergence of such rewriting systems…

On the existence of a category with a given matrix

We classify the matrices M which correspond to finite categories

Coherent confluence modulo relations and double groupoids

A coherent presentation of an n-category is a presentation by generators, relations and relations among relations. Confluent and terminating rewriting systems generate coherent presentations, whose relations among relations are defined by confluence diagrams of critical branchings. This article introduces a procedure to compute coherent presentations when the rewrite relations are defined modulo a set of axioms. Our coherence results are formulated using the structure of n-categories enriched in double groupoids, whose horizontal cells represent rewriting paths, vertical cells represent the co…

Simple free star-autonomous categories and full coherence

This paper gives a simple presentation of the free star-autonomous category over a category, based on Eilenberg-Kelly-MacLane graphs and Trimble rewiring, yielding a full coherence theorem: the commutativity of diagrams of canonical maps is decidable.

范畴化路径演算

Categorified Path Calculus

本文将路径演算范畴化，提出基于双幺半范畴上矩阵双范畴的曲面图演算，并展示双积、对偶和dagger等结构如何丰富该演算，最终给出隐形传态协议的新图形化证明。

自由生成的n-范畴、上插入器与低维范畴的表示

Freely generated $n$-categories, coinserters and presentations of low dimensional categories

研究自由生成n-范畴等相关问题，核心方法是通过上插入器和上均衡器给出计算表示的统一描述，主要贡献为从计算体恢复自由严格n-范畴，满足双范畴泛性质，给出群胚亏格等相关结论及同调下界等。

Rewriting in higher dimensional linear categories and application to the affine oriented Brauer category

In this paper, we introduce a rewriting theory of linear monoidal categories. Those categories are a particular case of what we will define as linear (n, p)-categories. We will also define linear (n, p)-polygraphs, a linear adapation of n-polygraphs, to present linear (n -- 1, p)-categories. We focus then on linear (3, 2)-polygraphs to give presentations of linear monoidal categories. We finally give an application of this theory in linear (3, 2)-polygraphs to prove a basis theorem on the category AOB with a new method using a rewriting property defined by van Ostroom: decreasingness.

作为边界极限的维数商

Dimension quotients as boundary limits

本文定义函子的边界极限，并证明群的第四维数商可自然表示为某个简单函子的边界极限。

Duoidal R-矩阵

Duoidal R-Matrices

本文在双重张量积的余幺半群幺半群设定下定义了双代数R-矩阵的类似物，证明其与Eilenberg--Moore范畴上的duoidal结构双射对应，并探讨了其余交换版本对正规duoidal范畴线性分配结构的提升作用。

从矩阵到态射 II：MATLAB 与 Octave 中的计算范畴示例

From Matrices to Morphisms II: Examples of Computational Categories in MATLAB and Octave

本文提出计算范畴概念，以 MATLAB、Octave 数据结构表示多种数学结构，构建统一框架并实现算法化，为范畴论与矩阵类科学计算提供双向视角。

对称幺半范畴的Operad

Operads for Symmetric Monoidal Categories

本文显式构造了以对称幺半范畴为代数的范畴operad，并将其嵌入四个operad的序列，从而得到四种逐级更严格的对称幺半范畴概念。

Coherence for Categorified Operadic Theories

Given an algebraic theory which can be described by a (possibly symmetric) operad $P$, we propose a definition of the \emph{weakening} (or \emph{categorification}) of the theory, in which equations that hold strictly for $P$-algebras hold only up to coherent isomorphism. This generalizes the theories of monoidal categories and symmetric monoidal categories, and several related notions defined in the literature. Using this definition, we generalize the result that every monoidal category is monoidally equivalent to a strict monoidal category, and show that the "strictification" functor has an i…

薄平曲面范畴的一种范式

A normal form for the Thin Flat Surfaces category

该研究针对Khovanov等人引入的薄平曲面范畴TFS，证明了连通可视tf-配边的范式定理与关系充分性定理，给出了TFS的完整生成元-关系表示。

Categories of partial equivalence relations as localizations

We construct a category of fibrant objects $\mathbb{C}\langle P\rangle$ in the sense of K. Brown from any indexed frame (a kind of indexed poset generalizing triposes) $P$, and show that its homotopy category is the Barr-exact category $\mathbb{C}[P]$ of partial equivalence relations and compatible functional relations. In particular this gives a presentation of realizability toposes as homotopy categories. We give criteria for the existence of left and right derived functors to functors $\mathbb{C}\langleΦ\rangle : \mathbb{C}\langle P\rangle\to \mathbb{C}\langle Q\rangle$ induced by finite-me…

A Graphical Calculus for Lagrangian Relations

Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precis…

Identities among relations for higher-dimensional rewriting systems

We generalize the notion of identities among relations, well known for presentations of groups, to presentations of n-categories by polygraphs. To each polygraph, we associate a track n-category, generalizing the notion of crossed module for groups, in order to define the natural system of identities among relations. We relate the facts that this natural system is finitely generated and that the polygraph has finite derivation type.

Towards an Algebraic Characterization of 3-dimensional Cobordisms

The goal of this paper is to find a close to isomorphic presentation of 3-manifolds in terms of Hopf algebraic expressions. To this end we define and compare three different braided tensor categories that arise naturally in the study of Hopf algebras and 3-dimensional topology. The first is the category \Cob of connected surfaces with one boundary component and 3-dimensional relative cobordisms, the second is a category \Tgl of tangles with relations, and the third is a natural algebraic category \Alg freely generated by a Hopf algebra object. From previous work we know that {\Tgl} and \Cob ar…

Quantum Representation Theory and Manin matrices I: finite-dimensional case

We construct Quantum Representation Theory which describes quantum analogue of representations in frame of "non-commutative linear geometry" developed by Manin. To do it we generalise the internal hom-functor to the case of adjunction with a parameter and construct a general approach to representations of a monoid in a symmetric monoidal category with a parameter subcategory. Quantum Representation Theory is obtained by application of this approach to a monoidal category of some class of graded algebras with Manin product, where the parameter subcategory consists of connected finitely generate…