Cohomology of small categories
type: Item
instance of: scholarly article
title: Cohomology of small categories
zbMATH Open document ID: 0587.18006
DOI: 10.1016/0022-4049(85)90008-8
author: Hans-Joachim Baues
author: Günther J. Wirsching
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: In this interesting paper, a new cohomology theory for categories is defined, and its properties are explored. The coefficients are called natural systems. Explicitly, a natural system is a functor \(D: F\mathcal C\to\mathcal{A}b\) where \(F\mathcal C\) is the twisted morphism category of Dwyer and Kan (objects of \(F\mathcal C\) are the morphisms of \(\mathcal C\), and arrows of \(F\mathcal C\) from \(f\) to \(g\) are pairs \((\alpha,\beta)\) of morphisms of \(\mathcal C\) satisfying \(g=\alpha f(\beta)\). Derivations \(\mathcal C\to D\) and inner derivations are defined, and \(H^1(\mathcal C,D)\) is derivations modulo inner derivations. Similarly, \(H^2(\mathcal C,D)\) is interpreted in terms of equivalence classes of linear extensions of \(\mathcal C\) by \(D\); algebraic and topological examples are given in section 3. Computations are facilitated by using ``covers'' of \(\mathcal C\) by subcategories, as well as by an isomorphism \(H^n(\mathcal C,D)=\mathrm{Ext}^n_{F\mathcal C}(Z,D)\). For appropriate choices of \(\mathcal C\) and \(D\), one retrieves the earlier theories of Hochschild, Mitchell, Roos, Quillen, Grothendieck, and Watts, as well as ordinary group cohomology (whose techniques are used throughout the paper). Understanding of this theory should lead to a deeper understanding of various algebraic and topological problems.
reviewed by: Don H. Van Osdol
Mathematics Subject Classification ID: 18G50
Mathematics Subject Classification ID: 55U99
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 55N35
zbMATH DE Number: 3941767
zbMATH Keywords: extension
zbMATH Keywords: cohomology theory for categories
zbMATH Keywords: natural systems
zbMATH Keywords: linear extensions
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90008-8
OpenAlex ID: W2080181577
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description: scientific article; zbMATH DE number 3941767