Essential reflections versus minimal embeddings
This note addresses the following question. In a category C, with a full subcategory X which is reflective with each reflection morphism an embedding, and a C-object A: when does each X-object containing A contain the X-reflection of A? We prove, under mild hypotheses on C, the equivalence of (1) The reflection morphism for A is an essential embedding, (2) Any 'minimal' embedding of A in an X-object is the X-reflection of A, (3) 'Always' is the answer to the question above, (4) The reflection functor C → X carries embeddings of A to embeddings. We note that these conditions hold for every A, for every X, if, in C, every epic embedding is essential, or if C has the Amalgamation Property. Various examples are discussed. © 1985.
Publication Source (Journal or Book title)
Journal of Pure and Applied Algebra
Hager, A., & Madden, J. (1985). Essential reflections versus minimal embeddings. Journal of Pure and Applied Algebra, 37 (C), 27-32. https://doi.org/10.1016/0022-4049(85)90084-2