Differential calculus in categories. I

切触范畴中的嘉当演算

Cartan calculus in tangent categories

研究切触范畴中构建嘉当演算所需结构，核心是由交换环对象\(R\)进行标量乘法，使切丛成\(R -\)模。在此基础上向量场李代数满足相关法则，各对象有李 - 莱因哈特形式的嘉当演算，还给出多种示例范畴。

The Maximality of the Typed Lambda Calculus and of Cartesian Closed Categories

From the analogue of Boehm's Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here of these results, which were obtained previously by Richard Statman and Alex K. Simpson.

From triangulated categories to module categories via localisation II: Calculus of fractions

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor Hom(T, -), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equivalent to the category of finite dimensional modules over the endomorphism algebra of T in C.

String Diagrams For Double Categories and Equipments

A popular graphical calculus for monoidal categories makes computations tactile and intuitive. Complicated diagram chases can be expressed in a few pictures and discovered by playing with a shoelace. Joyal and Street's proof of the soundness of this calculus says that any deformation of a diagram, any bending of the strings, describes the same morphism. In this paper, we extend the graphical calculus to double categories and proarrow equipments in order to bring the string diagrammatic method to formal category theory. Our main theorem proves this calculus sound with the help of Dawson and Par…

Module Eilenberg-Watts calculus

The categorical formulation of the Eilenberg-Watts calculus relates, for any pair of finite categories M and N, the finite categories Fun^{le}(N,M) and Fun^{re}(N,M) of linear left or right exact functors and the Deligne product \bar N \boxtimes M by adjoint equivalences. We establish a variant of this calculus for the case that the finite categories M and N are module categories over a finite tensor category. This provides in particular canonical and explicitly computable equivalences between categories of left or right exact module functors (or, more generally, balanced functors) and certain…

无偏幺半范畴中图式演算的无穷模型

An Infinitary Model of Diagrammatic Calculus in Unbiased Monoidal Categories

本文通过引入本质上抵抗同痕形变的结构，将无穷张量积和超限复合相干地引入图式形式体系，从而扩展了幺半范畴的弦图理论，构建了无穷幺半范畴模型。

The Sequent Calculus of Skew Monoidal Categories

Szlachányi's skew monoidal categories are a well-motivated variation of monoidal categories in which the unitors and associator are not required to be natural isomorphisms, but merely natural transformations in a particular direction. We present a sequent calculus for skew monoidal categories, building on the recent formulation by one of the authors of a sequent calculus for the Tamari order (skew semigroup categories). In this calculus, antecedents consist of a stoup (an optional formula) followed by a context, and the connectives behave like in the standard monoidal sequent calculus except t…

A calculus on lax functors

We verify that Kelly's constructions of the internal Hom for enriched categories extends naturally to lax functors taking their values in a symmetric monoidal category. Our motivation is to set up a `calculus on lax functors' that will host the theory of weakly enriched categories that are defined by lax diagrams.

Cartesian Difference Categories: Extended Report

Cartesian differential categories are categories equipped with a differential combinator which axiomatizes the directional derivative. Important models of Cartesian differential categories include classical differential calculus of smooth functions and categorical models of the differential lambda-calculus. However, Cartesian differential categories cannot account for other interesting notions of differentiation such as the calculus of finite differences or the Boolean differential calculus. On the other hand, change action models have been shown to capture these examples as well as more "exot…

Kan extensions and the calculus of modules for $\infty$-categories

Various models of $(\infty,1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\infty$-cosmos. In a generic $\infty$-cosmos, whose objects we call $\infty$-categories, we introduce modules (also called profunctors or correspondences) between $\infty$-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from $A$ to $B$ is an $\infty$-category equipped with a left action of $A$ and a right action of $B$, in a suitable se…

$2$-Categories from a Gray Perspective

In this paper we present $2$-category theory from the perspective of Gray-categories using the graphical calculus of separated surface diagrams. As an extended example we consider cones and limits of $2$-functors. Then we use the canonical adjunction between $2$-computads and $2$-categories to interpret the comparison structure of lax functors and extend the surface diagram calculus with compositor sheets in order to represent and reason about them.

Deductive Systems and Coherence for Skew Prounital Closed Categories

In this paper, we develop the proof theory of skew prounital closed categories. These are variants of the skew closed categories of Street where the unit is not represented. Skew closed categories in turn are a weakening of the closed categories of Eilenberg and Kelly where no structural law is required to be invertible. The presence of a monoidal structure in these categories is not required. We construct several equivalent presentations of the free skew prounital closed category on a given set of generating objects: a categorical calculus (Hilbert-style system), a cut-free sequent calculus a…

笛卡尔差分范畴

Cartesian Difference Categories

本文提出笛卡尔差分范畴，作为笛卡尔微分范畴与变化作用模型的桥梁，统一建模光滑函数微分与有限差分，并证明其具有切丛单子结构。

Eilenberg-Watts calculus for finite categories and a bimodule Radford $S^4$ theorem

We obtain Morita invariant versions of Eilenberg-Watts type theorems, relating Deligne products of finite linear categories to categories of left exact as well as of right exact functors. This makes it possible to switch between different functor categories as well as Deligne products, which is often very convenient. For instance, we can show that applying the equivalence from left exact to right exact functors to the identity functor, regarded as a left exact functor, gives a Nakayama functor. The equivalences of categories we exhibit are compatible with the structure of module categories ove…

A graphical calculus for semi-groupal categories

Around the year 1988, Joyal and Street established a graphical calculus for monoidal categories, which provides a firm foundation for many explorations of graphical notations in mathematics and physics. For a deeper understanding of their work, we consider a similar graphical calculus for semi-groupal categories. We introduce two frameworks to formalize this graphical calculus, a topological one based on the notion of a processive plane graph and a combinatorial one based on the notion of a planarly ordered processive graph, which serves as a combinatorial counterpart of a deformation class of…

On the 3-arrow calculus for homotopy categories

We develop a localisation theory for certain categories, yielding a 3-arrow calculus: Every morphism in the localisation is represented by a diagram of length 3, and two such diagrams represent the same morphism if and only if they can be embedded in a 3-by-3 diagram in an appropriate way. The methods to construct this localisation are similar to the Ore localisation for a 2-arrow calculus; in particular, we do not have to use zigzags of arbitrary length. Applications include the localisation of an arbitrary model category with respect to its weak equivalences as well as the localisation of it…

切 $\infty$-范畴与 Goodwillie 计算

Tangent $\infty$-categories and Goodwillie calculus

本文通过引入一个更高范畴框架，将同调论中的 Goodwillie 函数计算与光滑流形的微分计算之间的类比精确化，展示了切 $\infty$-范畴的概念及其在 Goodwillie 计算中的应用。

Lie Calculus

We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher order theory naturally involves higher algebra (n-fold groupoids).(conceptual, topological) differential calculus, groupoids, higher algebra($n$-fold groupoids), Lie group, Lie groupoid, tangent groupoid, cubes of rings

笛卡尔微分范畴的泰勒级数收敛的超度量

An Ultrametric for Cartesian Differential Categories for Taylor Series Convergence

本文在任意笛卡尔微分范畴中建立了无需极限或无穷和的泰勒级数形式理论，通过泰勒多项式诱导超度量，证明了泰勒范畴中泰勒级数收敛于原映射，统一了微分学与微分λ-演算的展开。

Cartesian Differential Comonads and New Models of Cartesian Differential Categories

Cartesian differential categories come equipped with a differential combinator that formalizes the derivative from multi-variable differential calculus, and also provide the categorical semantics of the differential $λ$-calculus. An important source of examples of Cartesian differential categories are the coKleisli categories of the comonads of differential categories, where the latter concept provides the categorical semantics of differential linear logic. In this paper, we generalize this construction by introducing Cartesian differential comonads, which are precisely the comonads whose coKl…

关系演算的丰富特性与2-置换性

Enriched aspects of calculus of relations and $2$-permutability

本文研究了Ord-范畴中关系演算的丰富特性与2-置换性，提出了Ord-Mal'tsev范畴的概念并给出了相关性质和示例。

克隆、闭范畴与组合逻辑

Clones, closed categories, and combinatory logic

本文用多元结构阐释简单类型λ演算的语义，引入外延SK-克隆并证明其可靠完备，进而证明不含产品的简单类型λ演算是SK-范畴的内部语言。

A functorial approach to differential calculus

We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets). To prepare this approach, we define a new, symmetric, presentation of differential calculus, whose main feature is the central r{ô}le played by the anchor map, which we study in detail. Our aim for developing this theory is twofold: (1) define a setting for calculus over any commutative ring, including finite rings; (2) define a setting that can be generali…

Markov categories, causal theories, and the do-calculus

We give a category-theoretic treatment of causal models that formalizes the syntax for causal reasoning over a directed acyclic graph (DAG) by associating a free Markov category with the DAG in a canonical way. This framework enables us to define and study important concepts in causal reasoning from an abstract and "purely causal" point of view, such as causal independence/separation, causal conditionals, and decomposition of intervention effects. Our results regarding these concepts abstract away from the details of the commonly adopted causal models such as (recursive) structural equation mo…

Grothendieck-Verdier范畴中Frobenius代数的曲面图与Frobenius-Schur示性数

Surface Diagrams for Frobenius Algebras and Frobenius-Schur Indicators in Grothendieck-Verdier Categories

该研究为Grothendieck-Verdier范畴建立三维曲面图演算，推广刚性幺半范畴的Frobenius代数性质，引入高阶Frobenius-Schur示性数并证明其等价不变性，配套提供图形证明助手辅助文件。

流形微分算术的背景

A Context for Manifold Calculus

本文在∞-范畴中发展了Weiss的流形微分算术，探讨多项式函子、Kan扩展与Weiss sheaves之间的联系，并分类同质函子，推广Weiss和Boavida de Brito的定理。

2-范畴中的张量演算类型化（I）

Typing Tensor Calculus in 2-Categories (I)

本研究为开发适配函数式编程与并行计算的线性代数形式化框架，将矩阵、张量建模为半加性2-范畴中的态射，构建无指标带类型的线性代数框架，并扩展至幺半半加性2-范畴完成运算验证。

无穷操作算子基础下的嵌入计算

Infinity-operadic foundations for embedding calculus

本文基于无穷操作算子构建嵌入计算框架，扩展了Goodwillie-Weiss的嵌入计算到 bordism 类别，并提出新的嵌入计算版本，如基于BTop(d)的拓扑嵌入计算和基于BAut(E_d)的配置类别计算。

Tangent bundles in differential lambda-categories

Differential lambda-categories were introduced by Bucciarelli et al. as models for the simply typed version of the differential lambda-calculus of Ehrhard and Regnier. A differential lambda-category is a cartesian closed differential category of Blute et al. in which the differential operator is compatible with the closed structure. We prove that any differential lambda-category is equipped with a canonical strong commutative monad whose construction resembles that of the tangent bundle in the category of smooth manifolds. Most of the results of this note remain valid in an arbitrary cartesian…

Convenient Antiderivatives For Differential Linear Categories

Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a natural transformation $\mathsf{K}$, which all differential categories have, is a natural isomorphism. Differential categories with antiderivatives come equipped with a canonical integration operator such that generalizations of the Fundamental Theorems of Calculus hold. In this paper, we show that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives. To help prove…

Cocycles in categories of fibrant objects

We establish that a category of fibrant objects (in the sense of Brown) admits a Dwyer-Kan homotopical calculus of right fractions. This is done using a homotopical calculus of cocycles, which is an auxiliary structure that can be defined on every category of fibrant objects. As an application, we deduce some non-abelian versions of the Verdier hypercovering theorem.

Lax 幺半伴随、二变量纤维化与配偶演算

Lax monoidal adjunctions, two-variable fibrations and the calculus of mates

本文建立面向 ∞-范畴函子的配偶演算，推广 unstraightening 等价，并通过二变量纤维化刻画 lax/oplax 幺半伴随及 Gray 张量积函子。