Universalities
Let \({\mathcal V}\) be a cocomplete monoidal category such that \(A \otimes - \) and \(- \otimes A\) preserve colimits for any \({\mathcal V}\)-object \(A\). Then there exist a \({\mathcal V}\)-category \({\mathcal U}\) and a \({\mathcal V}\)-functor \(F : {\mathcal U} \to {\mathcal U}\) such that for every \({\mathcal V}\)-functor \(H : {\mathcal H}_ 1 \to {\mathcal H}_ 2\) between \({\mathcal V}\)-categories \({\mathcal H}_ 1\) and \({\mathcal H}_ 2\) there exist strict \({\mathcal V}\)-isomorphisms \(\varphi_ i : {\mathcal H}_ i \to {\mathcal U}\), \(i = 1,2\), with \(\varphi_ 2 \circ H = F \circ \varphi_ 1\). As a consequence we obtain that there exist a quotient category \({\mathcal U}\) of the category of all topological spaces and all open continuous mappings (or of the category of all metric spaces and all open uniformly continuous mappings, respectively) and a functor \(F : {\mathcal U} \to {\mathcal U}\) such that for every functor \(H : {\mathcal H}_ 1 \to {\mathcal H}_ 2\) between arbitrary categories there exist full embeddings \(\varphi_ i : {\mathcal H}_ i \to {\mathcal U}\), \(i=1,2\), with \(\varphi_ 2 \circ H = F \circ \varphi_ 1\).
scientific article; zbMATH DE number 617343
scientific article