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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeFeb 4th 2014
• (edited Feb 4th 2014)

The entry (infinity,1)-Kan extension is still a sad stub which you shouldn’t look at if you have better things to do. But I have now briefly added at least a few more specific pointers to HTT, in particular to the pointwise-ness issue. But just pointers, essentially no text for the moment. (If you feel energetic, be invited to turn the entry into something prettier!)

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeJul 26th 2016

At super formal smooth infinity-groupoid, how do I know what the pattern of adjoints on $(\infty, 1)$-sheaves will look like from the diagram of sites?:

$\ast \stackrel{\longleftarrow}{\hookrightarrow} CartSp \stackrel{\hookrightarrow}{\longleftarrow} CartSp\rtimes InfPoint \stackrel{\longleftarrow}{\stackrel{\hookrightarrow}{\longleftarrow}} CartSp \rtimes SuperPoint \,.$

Any arrow there induces a map in the opposite direction on sheaves, and this has left and right adjoints by (infinity,1)-Kan extension. But what more can I tell from the kinds of maps between sites?

E.g., when there is an adjunction between sites, does this make one of the induced maps on sheaves coincide with one of the adjoints of the induced map from the adjoint?

And does a map of sites being an inclusion make a difference to the induced maps?

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeJul 26th 2016

The page restriction and extension of sheaves is relevant but rather isolated.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJul 26th 2016
• (edited Jul 26th 2016)

E.g., when there is an adjunction between sites, does this make one of the induced maps on sheaves coincide with one of the adjoints of the induced map from the adjoint?

Yes! If $i \dashv p$ then $i_! \dashv p_! \simeq i^\ast \dashv i_\ast \simeq p_\ast$ and so on.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeJul 26th 2016

So I guess there should be a further right adjoint generated from

$CartSp\rtimes InfPoint \stackrel{\longleftarrow}{\stackrel{\hookrightarrow}{\longleftarrow}} CartSp \rtimes SuperPoint \,,$

a right Kan extension of the map induced by the lower arrow.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJul 26th 2016

So the Kan extension is a functor not between the sites, but between the toposes over these sites. A single morphism of sites induces an adjoint triple on presheaves over the sites. An adjoint pair of morphisms between sites induces an adjoint quadruple on presheaves, by the relation in #4. Under certain conditions then some of these adjoints descend also to sheaves.

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeJul 26th 2016

Yes, I know that. I was just saying that each of those three arrows in #5 generates an adjoint triple on presheaves, which amounts to an adjoint quintuple between toposes.

But maybe we neglect the rightmost adjoint as not descending to sheaves?

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 26th 2016

Oh, sorry, I didn’t read properly. Yes, that’s right. There might be a further adjoint.