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nLab: Derivators and 2-category-theory

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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeNov 5th 2010
    • (edited Nov 5th 2010)

    The page on derivators says that we only need 2-category theory to work with them. Does this mean strict 2-category theory or the much more difficult “bicategory theory”?

    Essentially, (since all of the 2-categories involved in derivator theory are strict) do we need to concern ourselves with pseudofunctors (which are substantially trickier to work with than strict 2-functors)?

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeNov 5th 2010

    Pseudofunctors aren’t really that difficult; you just need to get used to them. I’m not sure how much one needs them to work with derivators, though. I think you do need at least some 2-category theory in the common sense, which includes a fair amount of what one might call “bicategory theory” even if the only bicategories involved happen to be strict 2-categories — i.e. not just strict Cat-enriched category theory.

    • CommentRowNumber3.
    • CommentAuthorHarry Gindi
    • CommentTimeNov 5th 2010
    • (edited Nov 5th 2010)

    @Mike: Pseudofunctors themselves, sure, they’re not too hard to handle. The trouble is Pseudofunctor 2-categories (i.e. 2−Cat(A,B)2-Cat(A,B)). The compatibility conditions become really painful. Also, as I’m sure you know (this is merely rhetorical flourish, since I read it on your personal lab =p), slice bicategories are “wrong”, and so the right notions become substantially more subtle. Given how much of ordinary category theory relies on the slice concept, the full theory of bicategories is a lot of baggage (although it is still vanishingly small compared to the homotopical baggage of quasicategory theory).

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeDec 2nd 2010
    • (edited Dec 2nd 2010)

    Just a note to point out that Francois Métayer’ and the groupe de travail at Paris 7 are working through ”localisateur fondamental minimal” in the theory of derivators, so if anyone is in Paris on a Friday in the next few weeks drop in to their seminar (Chevalleret as usual). The web page is here

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