On the canonical algebraic structure of a category
Let \({\mathcal A}\) be a locally small category and \(H=\Hom_{\mathcal A}: {\mathcal A}^{\text{op}}\times{\mathcal A}\to{\mathcal S}et\). Define the category \({\mathcal A}^*\) with objects the natural numbers and morphisms \(n\to m\) the natural transformations from \(H^n\) to \(H^m\). \({\mathcal A}^*\) is a Lawvere theory when the class of \(n\)-ary operations \(n\to 1\) is a (small) set for each \(n\). The first few pages of the paper are a useful review of the basic category theory of Lawvere theories. Commutative theories are those for which each operation \(\omega\) realizes as an homomorphism \(\omega_A:{\mathcal A}^n\to {\mathcal A}\) on the algebras. For a commutative \(T\), the category \(T\)-\({\mathcal A}lg\) of \({\mathcal S}et\)-models of \(T\) has a canonical symmetric monoidal closed structure. The authors point out that enrichments of a category \({\mathcal A}\) over such a \(T\)-\({\mathcal A}lg\) correspond to liftings of \(H\) through the forgetful functor \(T\)-\({\mathcal A}lg \to{\mathcal S}et\), and hence to theory-maps \(T\to A^*\). Conditions for \({\mathcal A}^*\) to be an honest theory (i.e., to be small) are also investigated, and the existence of finite powers and of a small cogenerating set in \({\mathcal A}\) (or in \({\mathcal A}^{\text{op}})\) is found to be a sufficient condition. Existence of finite powers in \({\mathcal A}\) (or \({\mathcal A}^{\text{op}})\) also implies that \({\mathcal A}^*\) is commutative, so that in this case \(\mathcal A\) has a canonical enrichment over \({\mathcal A}^*\)-\({\mathcal A}lg\). Among other results, it is shown that \((T\)-\({\mathcal A} lg)^* \cong T^*\cong\) the center of \(T\) (= the subtheory of \(T\) on the operations which commute with all the operations of \(T)\), and \({\mathcal A}^* \cong ({\mathcal A}_f)^*\) if \({\mathcal A}\) is a locally finitely presentable category and \({\mathcal A}_f\) the full subcategory on its finitely presentable objects.
scientific article; zbMATH DE number 1538803
scientific article