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    • CommentRowNumber1.
    • CommentAuthorHarry Gindi
    • CommentTimeJan 23rd 2011
    • (edited Jan 23rd 2011)

    Reposted from Math Overflow


    Preamble


    Let V=sSet with its cartesian monoidal structure.

    Let f:CDop be a V-functor, and let π:CC be another V-functor. Let pD be an object, and let hp:=D(,p):DopV.

    Let π! be the left adjoint of the pullback V-functor π*:VCVC induced by π (That is to say, for a V-functor h:CV, the functor π!(h) is the left Kan extension of h along π).

    Then g:=hpf is a functor CV.

    Question


    Why is the following true:

    In terms of the pushout diagram

    D:=colim(CopCopD)

    where ι1:CopD and ι2:DD are the canonical injections:

    We have an isomorphism of V-functors CV

    π!(g)()D(ιop1(),ι2(p)).

    Idea


    I think we can do something like this:

    We have a diagram

    CfDophpVπιop2Cιop1Dop

    Then we take the left Kan extension of hp by ιop2, but by one of the many variations of Yoneda’s lemma (computing via coends, this follows from Yoneda reduction), this is exactly D(,ι2(p)).

    Then it’s enough to show that the composite

    k:=D(,ι2(p))ιop1=D(ιop1(),ιop2(p))

    (with the natural map gkπ) has the required universal property, that is, that it is initial among fillers of the Kan extension diagram:

    CgVπC

    which is where I’m not sure how to proceed.

    • CommentRowNumber2.
    • CommentAuthorHarry Gindi
    • CommentTimeJan 23rd 2011
    • (edited Jan 23rd 2011)

    I just noticed that this is closely related to the calculus of exact squares and mates.

    I bet that there’s some theorem that proves that pushout squares in V-Cat are exact, since my question is equivalent to:

    Is the natural Beck-Chevalley transformation π!f*ι*1(ι2)! an isomorphism?

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJan 23rd 2011

    Well, arbitrary pushout squares aren’t exact even when V=Set. Consider the pushout of discrete categories 1←2→1, which is 1. I don’t think you can fix that by taking homotopy pushouts either. But cocomma squares are, I believe, always exact for any V, although I could be misremembering.

    • CommentRowNumber4.
    • CommentAuthorHarry Gindi
    • CommentTimeJan 23rd 2011
    • (edited Jan 23rd 2011)

    Ah, thanks! You’re right. My original question was too generalized from the original source (therefore false). I e-mailed Lurie, and the answer hinges on some additional information (i.e. that the point p in my first post is a weak strict sink (or whatever the right term is) in the following sense: D(p,p)=Δ0 and D(p,x)= for all xp (description is evil, but the point p is adjoined formally in a strict way as well (up to equality)).