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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 $\infty Grpd/X \simeq PSh_{\infty}(X)$ 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
I added at over-(infinity,1)-topos the example
$PSh(C/p) \simeq PSh(C)/y(p) \,.$which makes manifest that the “little topos” $\mathbf{H}/X$ of $X$ is that of sheaves over the big site of $X$.
(Is that state of the terminology convention now a cause of concern, by the way? Not sure.)
My conclusion from the discussion was that there are two different uses of big/little, and $H/X$ lies at the crossover point. It’s big relative to $Sh(X)$, 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.
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?
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 $Sh(C)/X$ and $Sh(C/X)$. It is not trivial in general, is it?
For instance when one speaks of the topos associated to the modul stack $\mathcal{M}_{ell}$ of elliptic curves, one considers the subcategory $(Aff/\mathcal{M}_{ell})_{et}$ of etale morphisms of the slice category, and then sheaves/$\infty$-sheaves on that. Presumeably this is equivalent to $(\infty Sh(Aff)/\mathcal{M}_{ell})_{et}$, 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
$(F \dashv i): \infty Sh(C) \stackrel{\overset{lex}{\leftarrow}}{\hookrightarrow} \infty PSh(C)$and any $\infty$-sheaf $X$ we get an adjunction
$(F/X \dashv i/X) : \infty Sh(C)/X \stackrel{\leftarrow}{\hookrightarrow} \infty PSh(C)/X$One would want to combine this with $\infty PSh(C)/X \simeq \infty PSh(C/X)$ and deduce that $\infty Sh(C)/X$ is a reflective localization of $\infty PSh(C/X)$.
So one needs to think about when the left adjoint $F/X$ is left exact, and how that is stable under passingto full subcategories, such as on etale morphisms (where available).
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.
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.
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 $C$ a quasi-category $X \in C$ an object and $F : K \to C/X$ a diagram, the limit is the initial object in $(C/X)/F$. But now
$(C/X)/F : [n] \mapsto Hom_F( [n] \star K, C/X)$But since the joint preserves colimits in both arguments, we have
$Hom( [n] \star K, C/X ) \simeq Hom_X( [n] \star K \star [0], C)$so in total
$(C/X)/F : [n] \mapsto Hom_{F,X}( [n] \star K \star [0], C)$And therefore
$(C/X)/F = C/(F/X)$with the evident meaning of the notation on the right, and hence the limit over $F$ in the over-category of $C$ is the limit over $F/X$ in $C$ itself.
have written this out at limits in over (oo,1)-categories
I have now written out my above argument – patched by Mike’s remark generalized as above to $\infty$-categories – at over $(\infty,1)$-topos – as $(\infty,1)$-sheaves on the big $(\infty,1)$-site of an object.
The discussion there is currently just for representables, though.
I am stuck: can we extend the equivalence $\infty PSh(C\downarrow X) \simeq \infty PSh(C)/X$ to non-representables $X \in \infty PSh(C)$?
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