A cohomology theory for commutative monoids.
type: Item
instance of: scholarly article
title: A cohomology theory for commutative monoids.
zbMATH Open document ID: 1337.20071
author: M. Calvo-Cervera
author: Antonio M. Cegarra
published in: Mathematics
publication date: 2016-01-12T00:00:00Z
review text: The authors present a new approach for a cohomology theory of commutative monoids. Let \(M\) be a commutative monoid and \(\mathcal A\colon\mathbb HM\to\mathbf{Ab}\) a group valued functor, where \(\mathbb HM\) is the Leech category with object set \(M\) and morphism set \(M\times M\), such that \((a,b)\colon a\to ab\) and the composition is given by \((ab,c)(a,b)=(a,bc)\). The cohomology groups of \(M\) with coefficients in \(\mathcal A\), denoted \(H_c^n(M,\mathcal A)\), are defined by \(H_c^n(M,\mathcal A)=H^{n+1}(\overline W^2M,\mathcal A)\), \(n\geq 1\), where \(\overline W^2M\colon\Delta^{op}\to\mathbf{Set}\) is a commutative simplicial monoid. This allows to interprete the elements of \(H_c^3(M,\mathcal A)\) in terms of equivalence classes of monoidal abelian groupoids \((\mathcal M,\otimes, c)\) endowed with coherent and natural isomorphisms (the braidings) \(c_{x,y}\colon x\otimes y\cong y\otimes x\). Then for each \(n\leq 3\), there is a natural isomorphism \(H_c^n(M,\mathcal A)\cong H^n(C_c^*(M,\mathcal A))\), where \(C_c^*(M,\mathcal A)\) is a certain complex of normalized commutative cochains on \(M\) with values in \(\mathcal A\). Denoting Grillet's symmetric cohomology groups by \(H_s^n(M,\mathcal A)\), it is proved that \(H_s^n(M,\mathcal A)\cong H_c^n(M,\mathcal A)\) for \(n=1,2\), and that \(H_s^3(M,\mathcal A)\hookrightarrow H_c^3(M,\mathcal A)\) is a natural monomorphism. The article deals also with a cohomological classification of braided monoidal abelian groupoids. It is proved (among other results) that for any braided abelian \(\otimes\)-groupoid \(\mathcal M\), there exist a commutative monoid \(M\), a functor \(\mathcal A\colon\mathbb HM\to\mathbf{Ab}\), a commutative three-cocycle \((h,\mu)\in Z_c^3(M,\mathcal A)\) and a braided \(\otimes\)-equivalence \(\mathcal A\rtimes_{h,\mu}M\simeq\mathcal M\).
Mathematics Subject Classification ID: 20M50
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 18B40
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18G60
zbMATH DE Number: 6528859
zbMATH Keywords: commutative monoids
zbMATH Keywords: cohomology groups
zbMATH Keywords: simplicial sets
zbMATH Keywords: braided monoidal categories
zbMATH Keywords: Leech categories
zbMATH Keywords: monoidal Abelian groupoids
reviewed by: Peeter Normak
MaRDI profile type: Publication
full work available at URL: math3041001
OpenAlex ID: W1833781378
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DOI: 10.3390/MATH3041001
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description: scientific article; zbMATH DE number 6528859