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I had started an article on AT category (which I originally mis-titled as “AT categories” – thank you Toby for fixing this!), but getting a little stuck here and there. I’m using the exchange between Freyd and Pratt on the categories mailing list (what else is there?) as my reference, but as is so often the case, Freyd’s discussion is a little too snappy and terse for me to follow it down to all the nitty-gritty details.
There’s a minor point I’m having trouble verifying: that coproducts are disjoint (as a consequence of the AT axioms that Freyd had enunciated thus far where he made that claim, in his main post), particularly that the coprojections are monic. Presumably this isn’t too hard and I’m just being dense. A slightly less than minor point: I’m having trouble verifying Ab-enrichment of the category of type A objects. I believe Freyd as abelian-categories-guru implicitly – I don’t doubt him. Can anyone help?
I’m actually tempted to not follow Freyd in his particular choice of axioms for AT categories, at least not all of them. There is in particular one axiom of his that looks a bit hack-y: the one that says that coproducts in the category of T objects are universal. IMO it would be more elegant to state all the exactness axioms so that they do not refer specifically to A objects or T objects.
Thus I would replace this axiom by: pushouts are universal (i.e., the pullback of a pushout square along any map to the pushout object is again a pushout square). This is true for abelian categories (because it checks out for module categories, unless I’m badly in error); it does not of course imply that coproducts are universal since the initial object as colimit of empty diagram need not be universal. But it does lead to Freyd’s axiom that coproducts of T objects are universal, since it’s clear that the initial object is universal when we restrict to the category of T objects (because there the initial object is strict: anything mapping to it is initial).
My proposed axiom also leads to a neat resolution of the minor point that I was stuck on (see my previous comment). I haven’t yet resolved the trouble with getting Ab-enrichment for type A objects.
I’m now pretty close to being done with AT category. In particular, I’m no longer stuck :-) and basically all the proofs are written up. There’s some topos stuff at the end which I quickly jotted down and will get back to soon, but it’s clear what Freyd’s up to there.
The underlying idea of all this is pretty neat, but I feel there’s definitely room for greater elegance and/or concision in presentation. One idea of course is that abelian categories and pretoposes have a lot of exactness properties in common, but the other idea is basically that due to the behavior of 0, the type A objects and type T objects are sequestered in their own worlds and “speak” to one another only in the most trivial way. (If a “way of speaking” from $X$ to $Y$ is a map $X \to Y$, then A objects speak to T objects in exactly one way, through 0, and T objects don’t speak to A objects at all unless the T object is 0.) Freyd is able to exploit this fact with a variety of clever “hacks” so that the stuff specific to pretoposes automatically gets killed off (trivialized) for A objects, and vice-versa.
Having worked through this now, I feel I see through all the tricks and while it’s sort of nice, I’m not quite gushing with excitement the same way Pratt is. :-) Hope that’s not too impolitic to say!
Todd, do you know about Goodwillie calculus? Shouldn’t this have something interesting to say on this topic here?
I think you mentioned this once before, and sorry not to have responded earlier, but I had no idea what to make of it, and still don’t. To answer your first question: not really; I had sat in on some lectures by Greg Arone years ago where he talked about Goodwillie calculus, but it didn’t really sink in.
Could you tell me why you think this is relevant? So far the page is mainly about exactness properties.
Could you tell me why you think this is relevant?
I have only a vague idea of it, so I am just asking. But since it’s all about the study of taking a category and seeing what happens as we “freely abelianize” it, or taking an (oo,1)-category and free stabilizing it, it ought to say also something about the process of the abelianization/stabilizaiton of a (higher) topos.
I like this conjunction “abelianization/stabilization”: it gives me a glimmer of what you have in mind. Passing thought is to consider what might be meant by an “AT $(\infty, 1)$-category”. :-)
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