Algebras over variable theories. (Q1771899)

By a theory is meant a small category \(\mathcal T\) equipped with a choice of finite products. A functor \(A : {\mathcal T} \rightarrow \mathcal {SET}\) preserving the chosen products is called an algebra of \(\mathcal T\) and by \(\text{ALG}({\mathcal T})\) is denoted the category of algebras of \(\mathcal T\) and their homomorphisms. Further, \(\text{ALG}_{\mathcal T}\) is the category whose objects are pairs \((F,A)\) where \(F : {\mathcal T} \rightarrow {\mathcal S}\) is a concrete theory morphism and \(A : {\mathcal S} \rightarrow \mathcal{SET}\) an algebra of \(\mathcal S\); morphisms \((F_{1},A_{1}) \rightarrow (F_{2},A_{2})\) are pairs \((H,\varphi )\) where \(H : {\mathcal S}_{1} \rightarrow {\mathcal S}_{2}\) is a concrete theory morphism such that \(H \cdot F_{1} = F_{2}\) and \(\varphi : A_{1} \rightarrow A_{2} \cdot H\) is a homomorphism of \(S_{1}\)-algebras. To each theory \(\mathcal T\) is found a theory \(\alpha {\mathcal T}\) whose category of algebras is isomorphic to \(\text{ALG}_{\mathcal T}\). Iterating this processing, it is found \(\xi {\mathcal T}\) such that \(\text{ALG}(\xi {\mathcal T})\) is isomorphic to \(\text{ALG}_{\xi {\mathcal T}}\) for each \(\mathcal T\) of \(\underline{\mathcal T}\), where \(\underline{\mathcal T}\) is the category of theories.