Preservation of bi-endomorphic composition by categorical reflection
Given a concrete category \({\mathcal C}\) over Set (with forgetful functor U), a bi-endomorphic composition on a \({\mathcal C}\)-object A is a binary operation on the underlying set U(A) such that both exponential transposes \(U(A) \rightrightarrows [U(A),U(A)]\) factor through the hom- set \({\mathcal C}(A,A)\). If \(\bar {\mathcal C}\) is a (full) reflective subcategory of \({\mathcal C}\), one may ask whether the two compositions \(U(A) \rightrightarrows {\mathcal C}(A,A)\to{\mathcal C}(\bar A,\bar A)\) factor through the underlying map of the universal morphism \(A\to \bar A\), and if they do, whether the resulting functions \(U(\bar A) \rightrightarrows {\mathcal C}(\bar A,\bar A)\to [U(\bar A),U(\bar A)]\) are in fact the exponential transposes of some bi-endomorphic composition on \(\bar A.\) The author uses this approach to compare reflections in categories \({\mathcal C}\) and \({\mathcal C}'\) of partial algebras and domain-preserving homomorphisms, where the \({\mathcal C}'\)-objects are \({\mathcal C}\)-objects with an additional total bi-endomorphic composition. In good cases, the reflection in \({\mathcal C}'\) reduces to the reflection in \({\mathcal C}\), thus eliminating the need for a separate construction. As an example, the relatively free \(\Sigma\)- ring can simply be obtained by forming the corresponding relatively free \(\Sigma\)-group.
reflections in categories of partial algebras and domain-preserving homomorphisms
scientific article; zbMATH DE number 4103157