Categories of binars and the associated representations (Q1918678)

Let \({\mathcal C}\) be a category with choosen finite products. Then a lot of canonical isomorphisms are built up by universal properties. A binar in \({\mathcal C}\) is an object \(A\) equipped with a binary operation, i.e., a morphism \(\mu : A \times A \to A\). The usual properties of associativity, commutativity, neutral morphisms, inversion, etc., are defined and lead to 12 different kinds \(T\) of binars. The category \({\mathcal C}^T\) of binars of type \(T\) is constructed. The paper states a theorem which tolds us under which conditions the canonical functor: \({\mathcal F} ct [{\mathcal D}, {\mathcal C}^T] \to {\mathcal F} ct [{\mathcal D}, {\mathcal C}]^T\) from the functor category \({\mathcal F} ct [{\mathcal D}, {\mathcal C}^T]\) to the category \({\mathcal F} ct [{\mathcal D}, {\mathcal C}]^T\) of binars of type \(T\) in the functor category \({\mathcal F} ct [{\mathcal D}, {\mathcal C}]\) is an isomorphism of categories.