Not signed in (Sign In)

Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homology homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory itex k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes science set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 8th 2010

    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?

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 9th 2010
    • (edited May 9th 2010)

    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.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 10th 2010

    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 XX to YY is a map XYX \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!

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMay 10th 2010

    Todd, do you know about Goodwillie calculus? Shouldn’t this have something interesting to say on this topic here?

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 10th 2010

    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.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 10th 2010

    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.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 11th 2010

    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 (,1)(\infty, 1)-category”. :-)

Add your comments
  • Please log in or leave your comment as a "guest post". If commenting as a "guest", please include your name in the message as a courtesy. Note: only certain categories allow guest posts.
  • To produce a hyperlink to an nLab entry, simply put double square brackets around its name, e.g. [[category]]. To use (La)TeX mathematics in your post, make sure Markdown+Itex is selected below and put your mathematics between dollar signs as usual. Only a subset of the usual TeX math commands are accepted: see here for a list.

  • (Help)