Amir Jafari
We generalize the notion of an exact category and introduce weakly exact categories. A proof of the snake lemma in this general setting is given. Some applications are given to illustrate how one can do homological algebra in a weakly exact category.
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The well-known snake lemma is proved entirely within category theory, without the help of "points with value in..." à la Grothendieck, nor pseudo-elements as in Guglielmetti & Zaganidis. Instead, we define and use consistently semi-cartesian squares, which were promoted by C. Chevalley.
Given an abelian category, we characterize the long exact sequences of length six which can be obtained from the snake lemma. Equivalently, these are the long exact sequences which arise as the homology of a triangle in the corresponding derived bounded category.
We obtain a generalization of the Two-Square Lemma proved for abelian categories by Fay, Hardie, and Hilton in 1989 and (in a special case) for preabelian categories by Generalov in 1994. We also prove the equivalence up to sign of two definitions of a connecting morphism of the Snake Lemma.
A two-categorical Snake Lemma
本文在2-范畴中证明蛇引理，通过引入2-核、2-余核及di-正合性等概念，建立2-正合六项序列，并展示三个模型实例，推广了经典蛇引理。
We prove an injective version of Schanuel's lemma from homological algebra in the setting of exact categories.
We prove that a regular category $\mathcal C$ is a Mal'tsev category if and only if a strong form of the denormalised $3 \times 3$ Lemma holds true in $\mathcal C$. In this version of the $3 \times 3$ Lemma, the vertical exact forks are replaced by pullbacks of regular epimorphisms along arbitrary morphisms. The shape of the diagram it determines suggests to call it the Cuboid Lemma. This new characterisation of regular categories that are Mal'tsev categories (= $2$-permutable) is similar to the one previously obtained for Goursat categories (= $3$-permutable). We also analyse the "relative" v…
The Kaplansky-Walker Lemma for additive categories
本文针对带直和的幂等完全加法范畴建立Kaplansky-Walker型引理，将其应用于紧生成三角范畴，得到投射模相关定理及可数对象直和的同构细化结果。
The snail lemma and the long homology sequence
本文提出了同伦版本的蜗牛引理，并通过可序列箭头家族构建了长同调序列，适用于阿贝尔范畴中的链复形扩展。
$2$-star-permutable categories were introduced in a joint work with Z. Janelidze and A. Ursini as a common generalisation of regular Mal'tsev categories and of normal subtractive categories. In the present article we first characterise these categories in terms of what we call star-regular pushouts. We then show that the $3 \times 3$ Lemma characterising normal subtractive categories and the Cuboid Lemma characterising regular Mal'tsev categories are special instances of a more general homological lemma for star-exact sequences. We prove that $2$-star-permutability is equivalent to the validit…
We prove that Mal'tsev and Goursat categories may be characterised through stronger variations of the Shifting Lemma, that is classically expressed in terms of three congruences $R$, $S$ and $T$, and characterises congruence modular varieties. We first show that a regular category $\mathcal C$ is a Mal'tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in $\mathcal C$. Moreover, we prove that a regular category $\mathcal C$ is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation $S$ and reflexive and positive relation…
Homological lemmas for (non-abelian) group-like structures by diagram chasing in a self-dual context
本文通过引入'诺特形式'的上下文，展示了如何在非阿贝尔群状结构范畴中，利用自对偶方法处理同调引理。
The aim of this article is to better understand the correspondence between $n$-cubic extensions and $3^n$-diagrams, which may be seen as non-abelian Yoneda extensions, useful in (co)homology of non-abelian algebraic structures. We study a higher-dimensional version of the coequaliser/kernel pair adjunction, which relates $n$-fold reflexive graphs with $n$-fold arrows in any exact Mal'tsev category. We first ask ourselves how this adjunction restricts to an equivalence of categories. This leads to the concept of an effective $n$-fold equivalence relation, corresponding to the $n$-fold regular e…
Yoneda lemma and representation theorem for double categories
本文研究取值于 $\mathbb{C}\mathrm{at}$ 的正规松弛双预层，建立其米田引理、与离散双纤维化的 Grothendieck 2-等价，以及由双终对象刻画的表示定理。
We continue the study of enriched infinity categories, using a definition equivalent to that of Gepner and Haugseng. In our approach enriched infinity categories are associative monoids in an especially designed monoidal category of enriched quivers. We prove that, in case the monoidal structure in the basic category M comes from direct product, our definition is essentially equivalent to the approach via Segal objects. Furthermore, we compare our notion with the notion of category left-tensored over M, and prove a version of Yoneda lemma in this context. Version 2: An error in 2.6.2 corrected…
Yoneda lemma
本文研究富集图范畴中的名称指派性质，并将米田引理推广到富集原图传输之间的连续变换上。
Key lemma and universal localization
本文研究了在任何基变换下仍保持局部化的无穷范畴局部化方法及其实现原理。
Homological Lemmas in a Non-pointed Context
本文证明经典同调引理的非点态版本在配备适当偏序单余反射子范畴的正则原模范畴中成立，并给出酉环、布尔代数、海廷代数、MV-代数及其拓扑模型和基本拓扑斯对偶范畴等实例。
For a small category $\mathcal{D}$ we define fibrations of simplicial presheaves on the category $\mathcal{D}\timesΔ$, which we call localized $\mathcal{D}$-left fibration. We show these fibrations can be seen as fibrant objects in a model structure, the localized $\mathcal{D}$-covariant model structure, that is Quillen equivalent to a category of functors valued in simplicial presheaves on $\mathcal{D}$, where the Quillen equivalence is given via a generalization of the Grothendieck construction. We use our understanding of this construction to give a detailed characterization of fibrations a…
Several results on exact sequences in categories of modules over trusses
本文在truss上模范畴中建立了蛇引理和九引理，推进了truss上模理论的范畴论研究。
In (B-Gran, 2004), was given a categorical formulation of the Shifting Lemma which is a characterization of the Congruence Modular Varieties among all the variety of Universal Algebra, introduced in (Gumm, 1983). Starting from a characterization of this Shifting Lemma by a property of the fibers of the fibration of points $¶\EE$ (B, 2005), on the model of what happens for Mal'tsev categories, we shall investigate three directions:\\ 1) a new one: in following the golden thread of abelian split epimorphisms naturally provided by this characterization;\\ 2) a more or less expected one: in measur…
We develop some basic concepts in the theory of higher categories internal to an arbitrary $\infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal categories.
One-sided exact categories appear naturally as instances of Grothendieck pretopologies. In an additive setting they are given by considering the one-sided part of Keller's axioms defining Quillen exact categories. We study one-sided exact additive categories and a stronger version defined by adding the one-sided part of Quillen "obscure axiom". We show that some homological results, such as the Short Five Lemma and the 3 X 3 Lemma, can be proved in our context. We also note that the derived category of a one-sided exact additive category can be constructed.
We show that for a given exact category, there exists a bijection between semibricks (pairwise Hom-orthogonal set of bricks) and length wide subcategories (exact extension-closed length abelian subcategories). In particular, we show that a length exact category is abelian if and only if simple objects form a semibrick, that is, the Schur's lemma holds.
The Noether isomorphism theorems and the Zassenhaus Lemma from group theory have a non-pointed version in a suitable categorical context first considered by W. Tholen in his PhD thesis. This article leads to a unification of these results with the ones in the pointed categorical context considered by O. Wyler, by working in the framework of star-regular categories introduced by M. Gran, Z. Janelidze and A. Ursini. Some concrete examples of categories where these results hold are examined in detail.
We give a computational approach to theorem proving in homological algebra. This approach is based on computations in the free abelian category of an additive category $\mathbf{A}$. We show that the free abelian category is amenable to explicit computations whenever we can decide homotopy equations in $\mathbf{A}$. As some consequences of our investigations, we recover Dowker's explicit formula for the connecting homomorphism $\partial$ in the snake lemma, we find a universal sense in which $\partial$ is unique, and we give a refined version of the 5-lemma.
Diagonal Lemma for Presheaves on Eilenberg-Zilber Categories
本文研究了Eilenberg-Zilber范畴上的对角引理，证明了在Kan-Quillen模型结构下，双层简复形的逐层弱等价可诱导对角简复形的弱等价。
A symmetric monoidal Comparison Lemma
本文证明比较引理的对称幺半版本，建立松弛与强对称幺半超层范畴间的等价，并用于获得紧支撑上同调理论结果的松弛对称幺半推广。
A characterization of closed subfunctors through $3\times 3$-lemma property in extriangulated categories
本文在外三角范畴中引入子函子的3×3-引理性质，证明加法子函子闭当且仅当满足该性质，推广了Buan的结果，并给出饱和真类的新刻画。
In this paper, we introduce and investigate a new notion of exact sequences of semimodules over semirings relative to the canonical image factorization. Several homological results are proved using the new notion of exactness including some restricted versions of the Short Five Lemma and the Snake Lemma opening the door for introducing and investigating homology objects in such categories. Our results apply in particular to the variety of commutative monoids extending results in homological varieties.
We introduce relative homological and weakly homological categories, where ``relative'' refers to a distinguished class of normal epimorphisms. It is a generalization of homological categories, but also protomodular categories can be regarded as examples. We indicate that the relative versions of various homological lemmas can be proved in a relative homological category.
We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 x 3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact ca…
The internal Yoneda lemma for locally Cartesian closed $\infty$-categories
在有限完备、局部笛卡尔闭的∞-范畴\(\C\)中阐述并证明约当引理等的内部版本，证明仅用有限极限等，不依赖外部约当引理，并应用于初等∞-拓扑斯，恢复相关定理。
We give new characterisations of regular Mal'tsev categories with distributive lattice of equivalence relations through variations of the so-called Triangular Lemma and Trapezoid Lemma in universal algebra. We then give new characterisations of equivalence distributive Goursat categories (which extend $3$-permutable varieties) through variations of the Triangular and Trapezoid Lemmas involving reflexive and positive relations.
We prove a stronger version of the octahedral axiom in a pre-triangulated category. The proof uses a new lemma about exact sequences in pointed additive categories which is based on a weak converse of the snake lemma.
A full subcategory of a Grothendieck category is called deconstructible if it consists of all transfinite extensions of some set of objects. This concept provides a handy framework for structure theory and construction of approximations for subcategories of Grothendieck categories. It also allows to construct model structures and t-structures on categories of complexes over a Grothendieck category. In this paper we aim to establish fundamental results on deconstructible classes and outline how to apply these in the areas mentioned above. This is related to recent work of Gillespie, Enochs, Est…
Conservative functors to pointed categories
本文引入预指向范畴概念，研究指向范畴的保守函子，推广经典引理并建立普遍指向化构造。
We study ideal cotorsion pairs associated to weak proper classes of triangles in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximations theory developed by Fu, Herzog et al. for exact categories in the above mentioned context, and to provide simplified proofs for the ideal versions of some standard results as Salce's Lemma, Wakamatsu's Lemma and Christensen's Ghost Lemma. In the last part of the paper we apply the theory in order to study connections between projective classes (in particular localization or smashing subcateg…
We consider a notion of exact sequences in any -not necessarily exact- pointed category relative to a given (E;M)-factorization structure. We apply this notion to introduce and investigate a new notion of exact sequences of semimodules over semirings relative to the canonical image factorization. Several homological results are proved using the new notion of exactness including some restricted versions of the Short Five Lemma and the Snake Lemma opening the door for introducing and investigating homology objects in such categories. Our results apply in particular to the variety of commutative …
A Homological View of Categorical Algebra
该研究为范畴代数的同调与同伦方法构建基础，选取从z-正合范畴到阿贝尔范畴等多层公理框架并厘清其关系，开发含基本图引理的范畴计算工具，发现其可用于弱范畴环境。
We use the terms $\infty$-categories and $\infty$-functors to mean the objects and morphisms in an $\infty$-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of fibrant objects. Quasi-categories, Segal categories, complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, $θ_n$-spaces, and fibered versions of each of these are all $\infty$-categories in this sense. Previous work in this series shows that the basic category theory of $\infty$-categories and $\infty$-functors can be developed only in reference to the axioms …
In this paper, we study the ideal approximation theory associated to almost $n$-exact structures in the $n$-exangulated category. The notions of $n$-ideal cotorsion pairs and $n$-$\mathbb{F}$-phantom morphisms are introduced and studied. In particular, let $\mathscr{C}$ be an extriangulated category which satisfies the condition (WIC) and $\mathcal{T}$ be a nicely embedded $n$-cluster tilting subcategory of $\mathscr{C}$, we prove Salce's Lemma in $\mathcal{T}$.
In this paper we introduce a description of ordered groupoids as a particular type of double categories. This enables us to turn Lawson's correspondence between ordered groupoids and left-cancellative categories into a biequivalence. We use this to identify which ordered functors are maps of sites in the sense that they give rise to geometric morphisms between the induced sheaf categories, and establish a Comparison Lemma for maps between Ehresmann sites.
The theory of abelian categories proved very useful, providing an axiomatic framework for homology and cohomology of modules over a ring and, in particular, of abelian groups. For many years, a similar categorical framework has been lacking for non-abelian (co)homology, the subject of which includes the categories of non-abelian groups, of rings, Lie algebras, etc. The notion of semi-abelian category due to Janelidze, Marki and Tholen allows a unified study of many important homological properties of such categories. For instance, in any semi-abelian category, the classical diagram lemmas (the…
Transfer theory for noetherian forms
本文提出一种在给定范畴上建立诺特形式存在性的转移策略，并证明广泛类别的本质代数与拓扑范畴（如小范畴、拓扑空间等）均具有诺特形式。
Everyone knows that if you have a bivariant homology theory satisfying a base change formula, you get an representation of a category of correspondences. For theories in which the covariant and contravariant transfer maps are in mutual adjunction, these data are actually equivalent. In other words, a 2-category of correspondences is the universal way to attach to a given 1-category a set of right adjoints that satisfy a base change formula. Through a bivariant version of the Yoneda paradigm, I give a definition of correspondences in higher category theory and prove an extension theorem for biv…
One-sided exact categories are obtained via a weakening of a Quillen exact category. Such one-sided exact categories are homologically similar to Quillen exact categories: a one-sided exact category $\mathcal{E}$ can be (essentially uniquely) embedded into its exact hull ${\mathcal{E}}^{\textrm{ex}}$; this embedding induces a derived equivalence $\textbf{D}^b(\mathcal{E}) \to \textbf{D}^b({\mathcal{E}}^{\textrm{ex}})$. Whereas it is well known that Quillen's obscure axioms are redundant for exact categories, some one-sided exact categories are known to not satisfy the corresponding obscure axi…
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 $π: Pt_{\mathbb{C}} \rightarrow \mathbb{C}$ …
Lambek Invariants of Commutative Squares in a Homological Category
本文在 Grandis 的半正合与同调范畴中研究了 Lambek 不变量，将 Lambek 同构定理推广至该范畴，并补充了相关引理的证明。
Spurred by the new examples found by Kornel Szlachányi of a form of lax monoidal category, the author felt the time ripe to publish a reworking of Eilenberg-Kelly's original paper on closed categories appropriate to the laxer context. The new examples are connected with bialgebroids. With Stephen Lack, we have also used the concept to give an alternative definition of quantum category and quantum groupoid. Szlachányi has called the lax notion {\em skew monoidal}. This paper defines {\em skew closed category}, proves Yoneda lemmas for categories enriched over such, and looks at closed cocomplet…
Diagram chasing is a customary proof method used in category theory and homological algebra. It involves an element-theoretic approach to show that certain properties hold for a commutative diagram. When dealing with abelian categories for the first time, one would work using a diagrammatic approach without relying on the notion of elements. However, constantly manipulating universal properties of various diagrams can be quite cumbersome. That said, we believe that it is still important to draw a contrast between both viewpoints in order to motivate the field of category theory. We focus our s…
Diagram-chasing arguments frequently lead to "magical" relations between distant points of diagrams: exactness implications, connecting morphisms, etc.. These long connections are usually composites of short "unmagical" connections, but the latter, and the objects they join, are not visible in the proofs. I try to remedy this situation. Given a double complex in an abelian category, we consider, for each object A of the complex, the familiar horizontal and vertical homology objects at A, and two other objects, which we name the "donor" A_{\box} and and the "receptor" ^{\box}A at A. For each ar…
Pre-Hilbert $*$-categories: The Hilbert-space analogue of abelian categories
该研究提出准希尔伯特$*$-范畴作为阿贝尔范畴的希尔伯特空间类似物，基于对合反变自函子等结构推导其加性、拟阿贝尔等同调性质，并给出多个相关范畴实例。
We characterize double adjunctions in terms of presheaves and universal squares, and then apply these characterizations to free monads and Eilenberg--Moore objects in double categories. We improve upon our earlier result in "Monads in Double Categories", JPAA 215:6, pages 1174-1197, 2011, to conclude: if a double category with cofolding admits the construction of free monads in its horizontal 2-category, then it also admits the construction of free monads as a double category. We also prove that a double category admits Eilenberg--Moore objects if and only if a certain parameterized presheaf i…
In this note, we show that the long $n$-exact sequence theorem holds in any $n$-abelian category. As an application, the $n$-abelian version of Wakamatsu's Lemma is given.
We state a Yoneda-type lemma which leads to various functor categories being compact closed.
Derived mapping spaces of $\infty$-categories
本文证明了导出映射空间引理，推广了 Cisinski 关于 $\infty$-范畴分数演算的核心定理，为分析范畴局部化中的映射空间提供了统一框架，并给出了立方或单纯范畴是底层范畴同伦等价局部化的充分条件。
We introduce subclasses of exact categories in terms of admissible intersections or admissible sums or both at the same time. These categories are recently studied by Brüstle, Hassoun, Shah, Tattar and Wegner to give characterisations of quasi-abelian and abelian categories. We also generalise the Schur lemma to the context of exact categories.
Given Gray-categories $P$ and $L$, there is a Gray-category $\mathrm{Tricat}_{\mathrm{ls}}(P,L)$ of locally strict trihomomorphisms with domain $P$ and codomain $L$, tritransformations, trimodifications, and perturbations. If the domain $P$ is small and the codomain $L$ is cocomplete, we show that this Gray-category is isomorphic as a Gray-category to the Gray-category $\mathrm{Ps}$-$T$-$\mathrm{Alg}$ of pseudo algebras, pseudo functors, transformations, and modifications for a Gray-monad $T$ derived from left Kan extension. Inspired by a similar situation in two-dimensional monad theory, we a…
This is the first of a series of papers on enriched infinity categories, seeking to reduce enriched higher category theory to the higher algebra of presentable infinity categories, which is better understood and can be approached via universal properties. In this paper, we introduce enriched presheaves on an enriched infinity category. We prove analogues of most familiar properties of presheaves. For example, we compute limits and colimits of presheaves, prove that all presheaves are colimits of representable presheaves, and prove a version of the Yoneda lemma.
I used to believe that my conventions for drawing diagrams for categorical statements could be written down in one page or less, and that the only tricky part was the technique for reconstructing objects "from their names"... but then I found out that this is not so. This is an attempt to explain, with motivations and examples, all the conventions behind a certain diagram, called the "Basic Example" in the text. Once the conventions are understood that diagram becomes a "skeleton" for a certain lemma related to the Yoneda Lemma, in the sense that both the statement and the proof of that lemma …
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