Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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 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
which makes manifest that the “little topos” of is that of sheaves over the big site of .
(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 lies at the crossover point. It’s big relative to , 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 and . It is not trivial in general, is it?
For instance when one speaks of the topos associated to the modul stack of elliptic curves, one considers the subcategory of etale morphisms of the slice category, and then sheaves/-sheaves on that. Presumeably this is equivalent to , 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
and any -sheaf we get an adjunction
One would want to combine this with and deduce that is a reflective localization of .
So one needs to think about when the left adjoint 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 a quasi-category an object and a diagram, the limit is the initial object in . But now
But since the joint preserves colimits in both arguments, we have
so in total
And therefore
with the evident meaning of the notation on the right, and hence the limit over in the over-category of is the limit over in 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 -categories – at over -topos – as -sheaves on the big -site of an object.
The discussion there is currently just for representables, though.
I am stuck: can we extend the equivalence to non-representables ?
1 to 11 of 11