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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2011
    • (edited Apr 21st 2011)

    stub for internal site. Hope to expand this later, but am out of time now.

    Here is something I need to figure out in this context: over at the stub smooth super infinity-groupoid I thought better about what I should be doing there. I think I should take the idea that we are to be working over the base topos of sheaves over superpoints more seriously than I did so far.

    Here is what I currently think one needs to do:

    1. consider the base topos Sh(SuperPoint)Sh(SuperPoint) of presheaves on superpoints

    2. this has a canonical line object Sh(SuperPoint)\mathcal{R} \in Sh(SuperPoint) which is such that the internal ordinary \mathcal{R}-linear algebra is externally traditional superalgebra.

    3. Define an internal site sCartSpsCartSp whose objects are the n\mathcal{R}^n s and whose morphisms are those morphisms in Sh(SuperPoint)Sh(SuperPoint) that are smooth according to the Schwarz-Molotkov-version of supermanifolds, as discussed there.

    4. Then defined smooth super \infty-groupoids to be the internal \infty-sheaves on sCartSpsCartSp.

    I am beginning to think that this is the right way to go. But I need to understand better what this definition will unwind to in detail.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2011
    • (edited Apr 21st 2011)

    started adding the discussion of externalization of internal sites. But only got to the presheaf case so far. Need to run now.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 21st 2011

    now with a bit on the sheaf version

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeApr 28th 2011

    The definition of internal site is obvious and straightforward.

    Definition. For ℰ a topos, an internal site in ℰ is an internal category ℂ equipped with an internal coverage.

    As far as I looked into the Elephant, the definition of internal coverage is to me very non-obvious, and, in particular, it is not clear to me wheather it is fully necessary to have a topos or it makes sense in somewhat more general categories. Do you have an insight which makes the complicated definition of internal coverage easier to comprehend and remember ?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeApr 28th 2011

    I think the definition is a very straightforward (though lengthy) “internalization” of the usual notion of coverage. For instance, a usual coverage on a category consists of a set of covers, each of which comes with a target (the object it covers) and a set of morphisms (those in the cover). Thus, an internal coverage is an object TT of covers with maps TC 0T\to C_0 and TP(C 1)T \to P(C_1), where C 0C_0 and C 1C_1 are the objects and morphisms of the internal category. And so on. Strictly speaking, one doesn’t even need to do all this “by hand” – once we have the internal language of the topos developed, we just need to say “interpret the usual definition of coverage in the internal language”.

    The occurrence of P()P(-), the power-object functor, means that the definition as usually given only makes sense in an elementary topos. (This can be seen easily from the standard definition which says we have, for every object a set of covers, each consisting of a set of morphisms.) However, there are variants that make sense in pretoposes as well, which demand only a family of covers and a family of morphisms. What is more difficult in the latter situation is if one wants a notion of “sieve” and to demand that covering families consist of sieves.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeApr 29th 2011

    I understand that the internal language is the general principle here, but I hoped it is not that complicated and lengthy (Elephant book has a relatively long treatment and still most of the details are omitted). Thanks for the second paragraph insight!

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 18th 2013
    • (edited Apr 18th 2013)

    I’ve put in some of the missing details at internal site.

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