Related papers: The Snake Lemma
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…
In the first part of the paper, we establish an homotopical version of the snail lemma (which is a generalization of the classical snake lemma). In the second part, we introduce the category $\mathbf{Seq}(\mathcal A)$ of sequentiable…
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…
The Kernel Complex Lemma states that given commutative diagram with exact rows and exact columns which covers the region under a $\Gamma$-shape, then the kernel sequence on the top and the kernel sequence at the left have in each position…
Green [Geometric and Functional Analysis 15 (2005), 340--376] established a version of the Szemer\'edi Regularity Lemma for abelian groups and derived the Removal Lemma for abelian groups as its corollary. We provide another proof of his…
In this paper we propose a conjecture concerning partial sums of an arbitrary finite subset of an abelian group, that naturally arises investigating simple Heffter systems. Then, we show its connection with related open problems and we…
This paper introduces the concept of slender generalized groups, extending the classical notion of slender abelian groups to the setting of generalized groups (completely simple semigroups). We establish fundamental properties of slender…
It is well known that the classical diagram lemmas of homological algebra for abelian groups can be generalized to non-abelian group-like structures, such as groups, rings, algebras, loops, etc. In this paper we establish such a…
In this article we study domino snake problems on finitely generated groups. We provide general properties of these problems and introduce new tools for their study. The first is the use of symbolic dynamics to understand the set of all…
The well-known snake lemma is proved entirely within category theory, without the help of "points with value in..." \`a la Grothendieck, nor pseudo-elements as in Guglielmetti & Zaganidis. Instead, we define and use consistently…
Categorical aspects of the theory of modules over trusses were studied in recent years. The snake lemma and the nine lemma in categories of modules over trusses are formulated in this paper.
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…
Zalcman's Lemma makes significant applications in normal families, complex dynamics and related problems in complex analysis. In the present paper, we are devoted to generalizing the classical Zalcman's lemma to complex Lie groups by means…
In a stable abelian group, we characterize generic types of cosets of type-definable subgroups.
In the present note a generalization of Borel-Cantelli Lemma is proposed.
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…
A group is called square-like if it is universally equivalent to its direct square. It is known that the class of all square-like groups admits an explicit first order axiomatization but its theory is undecidable. We prove that the theory…
We generalize Bulitko's Lemma to equations over (or homomorphisms into) groups that have $\kappa$-acylindrical splittings.
We prove several generalisations of the ping-pong lemma for negatively curved groups.
We prove an arithmetic regularity lemma for stable subsets of finite abelian groups, generalising our previous result for high-dimensional vector spaces over finite fields of prime order. A qualitative version of this generalisation was…