Abstract:
I have given two apparently different proofs of the regular category embedding theorem. The first, gotten by adapting Lubkin's argument for abelian categories, is rather opaque. The second, gotten by adapting Mitchell's proof is much more elegant. Mitchell used Grothendieck's theorem that an AB5 category with a generator has an injective cogenerator. However, the analogous result for regular categories fails. It turns out that full injectivity is not needed. Surprisingly, it turns out that ``under the hood'' the two proofs are really doing much the same thing. It is using functors rather than representing diagrams that makes the difference.