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touched five lemma
I added some remarks, one pertaining to the category of groups, and another to topological abelian groups. Feel free to stick those remarks elsewhere in the article, if you think they’re not well-placed.
Thanks!
I rearranged the proof very slightly to make it constructively acceptable and added a remark on how to avoid the use of the embedding theorem.
Of course, part of the beauty of the embedding theorem, either for abelian categories or for regular categories (Freyd-Scedrov, p. 77), is that it implies that you don’t have to worry about making proofs constructive – they assure one that proofs can be made constructive. (Here we mean proofs of Horn sentences written in the defining predicates of the theory.)
That’s a nice way to put it! Of course, your point is even stronger, considering that it’s not enough to make proofs constructive; they need to be formulated in such a way as to only use regular logic.
I still appreciate an explicitly constructive proof, since then I know that the theorem also holds in constructive mathematics.
At five lemma I have now made also the short five lemma explicit.
added (here) the statement of the five-lemma in the generality of homological categories.
One would think this is stated in Borceux + Bourn 2004, but apparently they forgot to do so. It is made explicit as Prop. 1.3.3 in the PhD thesis T. Janelidze 2009, so I have added pointer to that. But if anyone has a more canonical pointer to add (or, better yet, the energy to type out the proof) please do.
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