Variations and tensor products of \(M\)-acts (Q2461627)

For a monoid \(M=(M,\cdot ,e)\) an \(M\)-act is a pair \((A,\lambda )\) where \( A\) is a set and \(\lambda :M\times A\to M\) is a mapping such that \(\lambda (e,a)=a\) and \(\lambda (st,a)=\lambda (s,\lambda (t,a))\) for all \(a\in A\) and \( s,t\in M\) (we shall write only \(sa\) instead of \(\lambda (s,a)\)). Let \({\mathcal M}act\) be the category of all \(M\)-acts and their homomorphisms and \(| -| :{\mathcal M}act\to {\mathcal S}et\) be the underlying functor. If \(\alpha\) is an endomorphism of \(M\) then a mapping \(f:A\to B\) between \(M\)-acts is \(\alpha\)-variant if \(f(sa)=\alpha (s)f(a)\) for all \(s\in M\) and \(a\in A\). Let \(H_{\alpha} (A,B)\) denote the set of all \(\alpha\)-variant mappings from \(A\) into \(B\). Clearly, \(H_{\alpha}:{\mathcal M}act^{*}\times {\mathcal M}act\to{\mathcal S}et\) is a bifunctor. The characterization of \(\alpha\)-variant mappings is presented. A functor \(K:{\mathcal M}act^{*}\times {\mathcal M}act\to{\mathcal M}act\) is an internal lift of \(H_{\alpha}\) if \( | -| \circ K=H_{\alpha}\) and if the inclusion \(i:K\to(-)^{| -| }\) is a monomorphism then it is a functional internal lift. Characterizations of functional internal lifts of \(H_{\alpha}\) are given. A connection to tensor product is discussed.