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I had created coshape of an (infinity,1)-topos
Mike, what should we do? If we rename that entry to something else we would also need to rename shape of an (infinity,1)-topos.
I’d rather suggest that we proceed entirely in parallel to the dual shape theory and instead create now entries global sections of an (infinity,1)-topos, global sections in an (infinity,1)-topos etc, dual to fundamental infinity-groupoid of a locally infinity-connected (infinity,1)-topos, etc.
I’m okay with having coshape as a page, although I would express it rather differently than you have. If the shape is an object of Pro-∞Gpd, then the coshape should be an object of Ind-∞Gpd, right? rather than the huge topos. And since it’s always representable, even saying that is overcomplicating things a bit. I really don’t see the point of introducing the huge topos, since we already have an adjunction between (∞,1)Topos and ∞Gpd.
should be an object of $Ind \infty Grpd$, right?
Right, and that’s $Sh(\infty Grpd)$.
I really don’t see the point of introducing the huge topos, since we already have an adjunction between $(\infty,1)Topos$ and $\infty Topos$.
Okay, then let’s sort this out, because I don’t see this. Maybe I am mixed up. But to me it looks like currently the discussion at “coshape” is formulated correctly, while that at “shape” is wrong as stated:
there is no Yoneda embedding $(\infty,1)Topos \to Func((\infty,1)Topos, \infty Grpd)^{op}$. It goes instead $(\infty,1)Topos \to Func((\infty,1)Topos, \infty \hat Grpd)^{op}$ (with a little “hat” over the $Grpd$ :-)
Right, and that’s $Sh(\infty Grpd)$.
That’s not at all obvious to me. Is that something in HTT?
that at “shape” is wrong as stated
Yes, that’s right. Although I don’t like putting hats on things to mean “a bigger version thereof” — to me a hat looks like some kind of completion. I would probably write ∞GPD to mean the category of possibly large ∞-groupoids.
Of course we know, by its identification with $\Gamma \circ \Delta$, that the shape actually lands in the smaller functor category. I think this is connected with the comment I made at the Cafe about $Lex(\infty Gpd, \infty Gpd)^{op}$ not really being the “right” definition of $Pro\infty Gpd$, since $Lex(\infty Gpd, \infty GPD)^{op}$ would be the “right” definition of $PRO\infty Gpd$, i.e. the pro-completion of $\infty Gpd$ considered as a small category one universe up.
Anyway, I don’t have a problem with the very large category ∞GPD of large ∞-groupoids. What I’m objecting to is the category of large sheaves on the large category of (∞,1)-topoi. You’re right that we don’t actually have an adjunction between (∞,1)Topos and ∞Gpd at the same level of size—I misspoke. But I would prefer to say what we actually do have: a functor $(\infty,1)Topos \to \infty GPD$ which has a partial left adjoint defined on $\infty Gpd$. Or equivalently, a functor $\infty Gpd \to(\infty,1)Topos$ whose “ind-right-adjoint” is representable by a functor into $\infty GPD$. I think the introduction of the huge sheaf topos just makes this simple statement sound more complicated.
Sorry, Mike, I only now realize that I had forgotten about this thread here and your last reply, after last week’s wildly busy weekend (I was helping some friends move appartments).
That’s not at all obvious to me. Is that something in HTT?
Yes, corollary 5.3.5.4 used in remark 6.3.5.18.
Here is a question to which you probably already know the answer from 1-topos theory:
as we discussed, we have -for the shape of a locally $\infty$-connected $\infty$-topos that it is represented by $\Pi(*)$. So I’d expect that the coshape of a local $\infty$-topos is somehow analogously encoded by the extra right adjoint. Can you see this?
Yes, corollary 5.3.5.4 used in remark 6.3.5.18.
Thanks, but now I see that I asked the wrong question, because I assumed a different definition of SH(∞Gpd) from the one that Lurie gives. The question should have been: If C is a large (∞,1)-category with small limits, why is it reasonable to call small-limit–preserving functors $C^{op}\to \infty GPD$ “sheaves”? Specifically, are they a left-exact–reflective subcategory of $[C^{op},\infty GPD]$? (I’m fairly sure that’s not true in the 1-dimensional case, but it seems like one of those things that may be better in the ∞-case.)
I’d expect that the coshape of a local ∞-topos is somehow analogously encoded by the extra right adjoint.
But the coshape of any ∞-topos is “large-representable”, so I don’t know what additional sort of “encoding” you’d want.
Specifically, are they a left-exact–reflective subcategory of […]
Of presheaves with values in the very large category of large $\infty$-groupoids, yes.
That’s the claim of HTT Remark 6.3.5.17 a). But let’s see, let’s sort this out. The proof relies on prop. 5.5.4.20:
for $L: C \to S^{-1}C$ a localization of a locally presentable $\infty$-cat, then for any $\infty$-cat $D$ precomposition with $L$ identifies
$Fun^{leftadj}(S^{-1}C , D) \to Fun^{leftadj}(C,D)$as the full subcat on functors that invert $S$ in $D$.
Using this (and the standard loclization story) one should find that the very large sheaf topos $\hat Sh(\mathcal{X})$ is for any site $C$ of definition $\mathcal{X} \simeq S^{-1}PSh(C)$ just $S^{-1} \hat PSh(C)$, hence in particular itself a left exact localization of $\hat PSh(C)$.
(I put hats to indicate size. We can change that on the Lab if you want, but for the moment, while we straighten this out, let me stick to the HTT notation as used there.)
[edit: oh, now I see what you mean: it is not a localization of $\hat PSh(\mathcal{X})$, though. Right. So maybe the notation is bad. Not sure.]
I don’t know what additional sort of “encoding” you’d want.
I wanted to encode the coshape of a local $\infty$-topos in terms of properties of its terminal object or so.
But I should ask directly question that I am after: what’s the coshape of a cohesive $\infty$-topos?
I guess it should be (corepresented by) the point and I guess it should be very easy to see. But I have trouble seeing it right now.
Wait a minute. Suppose $C$ is a locally presentable (∞,1)-category, $C\simeq Ind_\kappa D$ for a small $D$ that has $\kappa$-small colimits. Then in particular, every object of $C$ is a small colimit of objects of $D$, which means that any functor $C^{op}\to \infty GPD$ which preserves small limits must be uniquely determined by its restriction to $D$, which will be a functor preserving $\kappa$-small limits. So it seems like the category Lurie calls $\hat{Sh}(C)$, of small-limit–preserving functors $C^{op}\to \infty GPD$, ought to be equivalent to what I would call $IND_\kappa D$, the category of $\kappa$-small-limit–preserving functors $D^{op}\to \infty GPD$. And that, in turn, ought always to be the obvious extension of the large category $C$ to its analogous version in a higher universe—t’s the category of “large models” of the same essentially-algebraic theory. In particular, if $C$ is a topos, then $\hat{Sh}(C)$ should be, as you said (right?), the category of large-∞-groupoid–valued sheaves on the same site—and in particular we should have $\hat{Sh}(\infty Gpd) \simeq \infty GPD$. Did I make a mistake somewhere?
what’s the coshape of a cohesive ∞-topos?
I think it depends on the cohesive ∞-topos. It’s the category of points of that topos, and even a big topos can have have a complicated category of points—otherwise known as the category of models in Set/∞Gpd of the geometric theory classified by that topos. For instance, the category of points of “the Zariski topos” is the category of local rings in Set. The topos being local just tells you that its category of points has an initial object.
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