Not signed in (Sign In)

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

Discussion Tag Cloud

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.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 23rd 2018

    Just to drop a mention of this video I just discovered, a talk on allegories, by Bénabou. I have not watched it yet, but will do so in case there is something worth linking for, and will add it to relevant entries at that point.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 24th 2018

    I watched it; I’ll need either to watch parts of it again or work out on my own things I didn’t catch the first time (there may be lacunae in the recording). But one of the questions he focuses on is how to promote the 1-categorical equivalence between regular categories and unital tabular allegories to a 2-categorical equivalence, because it is not immediately clear what notion of 2-cell on the allegorical side corresponds to a natural transformation between regular functors. One of the key constructions he introduces is a double category of “admissible squares” attached to an allegory A\mathbf{A}, where the vertical arrows are “functions” (i.e. maps = left adjoints) f,gf, g in A\mathbf{A}, horizontal arrows are “relations” or arbitrary morphisms r,sr, s in A\mathbf{A}, and the 2-cells are inequalities of type grsfg r \leq s f (viewed as morphisms (r,s):fg(r, s): f \to g in the horizontal category). He says that the horizontal category is again an allegory, where the opposite of (r,s):fg(r, s): f \to g is (r *,s *):gf(r^\ast, s^\ast): g \to f. He also says this construction when applied to a (unital) tabular category is again (unital) tabular.

    I couldn’t quite catch all of it, but he generalizes this last construction to something he denotes A X\mathbf{A}^X where XX is a category. Apparently the objects are functors XMap(A)X \to Map(\mathbf{A}) and morphisms are families rx:FxGxr x: F x \to G x for which the “naturality squares” are not commutative, but are admissible (so in other words, lax natural). The horizontal category of the last paragraph would be the case X=2X = \mathbf{2}. It sounded like he was suggesting this construction was relevant to the problem of defining appropriate 2-cells between allegory maps, but as I say I didn’t catch how.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJun 24th 2018

    Excellent evidence that allegories should really be considered as double categories, not 2-categories. I rediscovered this notion of allegory transformation in exact completions and small sheaves (Definition 6.7). I think that “functor allegory” is also naturally a functor double category, with XX treated as a horizontally discrete double category.