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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2010

    I expanded some entries related to the Café-discussion:

    • at over-(infinity,1)-topos I expanded the Idea-section, added a few remarks on proofs and polished a bit,

      and added the equivalence Grpd/XPSh (X)\infty Grpd/X \simeq PSh_{\infty}(X) to the Examples-section

    • at base change geometric morphism I restructured the entry a little and then included the proof of the existence of the base change geometric morphism

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2010

    I added at over-(infinity,1)-topos the example

    PSh(C/p)PSh(C)/y(p). PSh(C/p) \simeq PSh(C)/y(p) \,.

    which makes manifest that the “little toposH/X\mathbf{H}/X of XX is that of sheaves over the big site of XX.

    (Is that state of the terminology convention now a cause of concern, by the way? Not sure.)

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeNov 17th 2010

    My conclusion from the discussion was that there are two different uses of big/little, and H/XH/X lies at the crossover point. It’s big relative to Sh(X)Sh(X), but it’s “little” in the sense that it represents a single space rather than a category of spaces. I’m becoming more inclined to call it “big,” however, since its objects are not modeled on X.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 17th 2010

    I would call Sh(X) the little topos of X, and Sh(Top/X) the big topos of X (in parallel with little and big sites), whereas Sh(Top) is a big topos, and not a little topos. But, in all seriousness, could we say Sh(Top/X) is a medium topos?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2010
    • (edited Nov 17th 2010)

    We might simply decouple “big/little” from “gros/petit” and would probably have a consistent set of terms:

    • gros topos = regarded as a category of spaces / petit topos = regarded as a space

    • big topos = sheaves on a big site / little topos = sheave son open subsets

    But more urgently for me is: I need to better understand the relation between Sh(C)/XSh(C)/X and Sh(C/X)Sh(C/X). It is not trivial in general, is it?

    For instance when one speaks of the topos associated to the modul stack ell\mathcal{M}_{ell} of elliptic curves, one considers the subcategory (Aff/ ell) et(Aff/\mathcal{M}_{ell})_{et} of etale morphisms of the slice category, and then sheaves/\infty-sheaves on that. Presumeably this is equivalent to (Sh(Aff)/ ell) et(\infty Sh(Aff)/\mathcal{M}_{ell})_{et}, but it would seem to me that there is a little bit to be thought about here.

    What i know is (using HTT, section 5.2.5) that starting with the adjunction

    (Fi):Sh(C)lexPSh(C) (F \dashv i): \infty Sh(C) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} \infty PSh(C)

    and any \infty-sheaf XX we get an adjunction

    (F/Xi/X):Sh(C)/XPSh(C)/X (F/X \dashv i/X) : \infty Sh(C)/X \stackrel{\leftarrow}{\hookrightarrow} \infty PSh(C)/X

    One would want to combine this with PSh(C)/XPSh(C/X)\infty PSh(C)/X \simeq \infty PSh(C/X) and deduce that Sh(C)/X\infty Sh(C)/X is a reflective localization of PSh(C/X)\infty PSh(C/X).

    So one needs to think about when the left adjoint F/XF/X is left exact, and how that is stable under passingto full subcategories, such as on etale morphisms (where available).

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 17th 2010

    We might simply decouple “big/little” from “gros/petit” and would probably have a consistent set of terms:

    Actually I thought of something similar, but was less certain of myself to suggest it.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeNov 17th 2010

    I think decoupling big/little from gros/petit isn’t a great plan. If, as seems to be the case, people already use gros/petit in both of the two slightly different ways, I think it would create more confusion to try to use the translated words in only one of the cases. But “medium” is kind of appealing to me.

    At least for 1-categories, any slice of a left exact functor is left exact, since limits in a slice category are computed by limits in the original category with the base object thrown in at the bottom.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2010
    • (edited Nov 17th 2010)

    At least for 1-categories, any slice of a left exact functor is left exact, since limits in a slice category are computed by limits in the original category with the base object thrown in at the bottom.

    Ah, of course And this has an immediate generalization to quasi-categories:

    for CC a quasi-category XCX \in C an object and F:KC/XF : K \to C/X a diagram, the limit is the initial object in (C/X)/F(C/X)/F. But now

    (C/X)/F:[n]Hom F([n]K,C/X) (C/X)/F : [n] \mapsto Hom_F( [n] \star K, C/X)

    But since the joint preserves colimits in both arguments, we have

    Hom([n]K,C/X)Hom X([n]K[0],C) Hom( [n] \star K, C/X ) \simeq Hom_X( [n] \star K \star [0], C)

    so in total

    (C/X)/F:[n]Hom F,X([n]K[0],C) (C/X)/F : [n] \mapsto Hom_{F,X}( [n] \star K \star [0], C)

    And therefore

    (C/X)/F=C/(F/X) (C/X)/F = C/(F/X)

    with the evident meaning of the notation on the right, and hence the limit over FF in the over-category of CC is the limit over F/XF/X in CC itself.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2010

    have written this out at limits in over (oo,1)-categories

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2010
    • (edited Nov 17th 2010)

    I have now written out my above argument – patched by Mike’s remark generalized as above to \infty-categories – at over (,1)(\infty,1)-topos – as (,1)(\infty,1)-sheaves on the big (,1)(\infty,1)-site of an object.

    The discussion there is currently just for representables, though.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeNov 17th 2010

    I am stuck: can we extend the equivalence PSh(CX)PSh(C)/X\infty PSh(C\downarrow X) \simeq \infty PSh(C)/X to non-representables XPSh(C)X \in \infty PSh(C)?