A cohomology theory for commutative monoids. (Q901455)

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 \(\mathcal A\rtimes_{h,\mu}M\simeq\mathcal M\).