the walking short exact sequence

alex and i tried googling on "free abelian category" yesterday, and this led to some ideas that seem pretty much simpler than what i was expecting, and which have apparently been understood pretty well for a long time. (for example one of the references was to a paper by freyd from the long-ago la jolla conference. that's not too surprising, of course; it's just the fact that the ideas seem so easy to understand that surprised me a bit.)

i didn't really get into the details, but if i'm not too badly confused here then apparently an abelian category is essentially just "an algebroid x equipped with operations [a,x]->[b,x] just like all the finitary such operations in the case where x is the algebroid of finitely presented abelian groups". (or something like that.) perhaps it's already near-obvious that something like that should be true, but actually understanding it could be very helpful.

there's a bunch of interesting variations on this idea depending on (roughly) just how we construe the concept of "operation" here. perhaps this amounts to investigating various meta-doctrines (or something like that).

anyway, it occurred to me that this might indicate that it would be interesting to try to calculate for example the "walking short exact sequence" abelian category. we know a nice easy way to describe the "walking epi" finitely cocomplete algebroid, and that it's not abelian, so that's part of it why it should be interesting to try to calculate the "walking short exact sequence".

i was just recalling a vague memory of freyd mentioning some ideas about "the doctrine of abelian categories" (in some vague sense of "doctrine", which word he probably didn't actually use), and offhand i don't think that i remember him emphasizing this line of thought (how to formalize the idea that an abelian category is one with "all" (in some interesting sense) of the structure that the category of abelian groups has). i could be wrong about that though.

anyway, i still have to think more about how these ideas might apply to algebraic geometry. on some vague level it reminds me of johnstone's work on comparing the idea of seeing a topos as a "geometric theory" to the idea of seeing it as a theory of a richer doctrine where for example exponentiation is also preserved (or something like that).