The n-Category Café

I’ll settle for E^2-Theory. The Theory of Everything Else.

I’m not sure how to fix:

[…]

or I’d do it myself.

Ah, right. Thanks for catching that. I had begun going over the text and adding more cross references to other existing entries. I messed up the wording here apparently.

Have now changed this to

there is a very general abstract nonsense way to understand [[sheaf|sheaves]] as generalized spaces in the context of a very general abstract [[duality]] between the notions of [[space and quantity]]. The following is a heuristic way to understand this.

Also added a bit more cross references the other way to the old entry [[space and quantity]].

But I am in a hurry, so this was done a bit hastily again. Please feel free to improve.

got a bit lost in the one section where you apply the sheaf to another sheaf.

Good point, I didn’t properly introduce the notation used at this point, even though it’s meant to be the very intuitive notation:

For $X$ and $Y$ two generalized spaces (realized as sheaves), we simply write $X(Y)$ for the collection of morphisms from $Y$ to $X$!

To make this clear, I have now created a subsection at that entry titled maps between generalized spaces which discusses this in detail.

The point here is to make the notation blur the distinction between “probes” in $X(U)$ and maps $U \to X$. The Yoneda lemma says that these are canonically identified. So we just write $X(Y)$ even if $Y$ is not a test space but a generalized space itself, to denote the collection of “probes” of $X$ by $Y$.

The gain is that this makes the “descent condition for $\infty$-stacks”, which otherwise tends to look like an intimidating concept, become conceptually the easiest thing in the world:

just like an isomorphism $U \stackrel{\simeq}{\to} V$ of ordinary test spaces must induce an isomorphism $Z(V) \stackrel{\simeq}{\to} Z(U)$ of collections of probes of the generalized space $Z$ by probe spaces $U$ and $V$ (this is just functoriality of the presheaf $Z$), so a weak equivalence $X \to Y$ of generalized spaces must induce a weak equivalence $Z(Y) \to Z(X)$ of probes by those. Otherwise the entire “probe” interpretatioon were inconsistent.

Let me know if this helps.

(Unfortunately the formatting of the entry is a bit rough now: I stopped using bulleted lists at some point as the software was choking on somthing I couldn’t identify. But we’ll fix that eventually).

This implies that the stable analog of $(\infty,1)$-topoi are just the stable left Bousfield localizations of $(\infty,1)$-categories of presheaves of spectra by a small set of maps. Such $(\infty,1)$-categories can be characterized as the accessible stable $(\infty,1)$-categories.

Thanks for the reply!

So the story is entirely analogous to the way Grothendieck $(\infty,1)$-topoi are precisely the localizations of $(\infty,1)$-categories of presheaves with values in topological spaces.

That’s what i would have thought. But given that one can write a book on these (unstable case), I was wondering if somebody found it worthwhile to consider those (stable case) explicitly.

There is another question in the same vein I have:

it should be interesting to consider (localizations of) $(\infty,1)$-categories of presheaves with values in rational topological spaces.

There should be maps back and forth between Grothendieck $(\infty,1)$-topoi and such rational Grothendieck $(\infty,1)$-topoi, and this should essentially be $\infty$-Lie theory.

Has anyone considered “rational Grothendieck $(\infty,1)$-topoi” explicitly? Or is this also considered too straightforward to deserve explicit mentioning?

In his above comment Denis-Charles Cisinki kindly emphasized that

- a “stable $(\infty,1)$-topos” is nothing but an arbitrary stable $(\infty,1)$-category,

- while a “stable Grothendieck $(\infty,1)$-topos” is just an accessible stable $(\infty,1)$-category.

Now I am wondering about the following:

in the Kontsevich-style noncommutative geometry program people try to characterize noncommutative spaces in terms of their derived category of coherent sheaves and then more abstractly in terms of triangulated (dg-)categories etc. Given that all these are models for stable $(\infty,1)$-categories, it seems that one could characterize the Kontsevich-style ncg programe by the slogan:

conceive noncommutative spaces as stable $(\infty,1)$-categories.

In light of the fact we can just as well say then:

conceive noncommutative spaces as stable $(\infty,1)$-toposes.

In this form the slogan would have the advantage that it fits nicely into the (petit-)topos theoretic way of thinking about genralized spaces.

Now I am wondering: a big driving force of much of the Kontsevich-style ncg literature is that the standard examples that one want to model are not exactly sheaves/stacks on the category of formal duals of non-commutative algebras: the sieves which they regard as covering do not satisfy all the axioms of a Grothendieck topology.

Kontesvich and Rosenberg have proposed, therefore, to replace in the context of ncg the notion of sheaves with respect to a grothendieck topology by something more general, called sheaves with respect to a “$Q$-category”. See Zoran Škoda’s entry $n$Lab:Q-categories.

I am wndering how this might fit into the pigger picture of “stable $(\infty,1)$-toposes”. A naive first guess would be simply that this is telling us that non-commutative geometric spaces correspond to non-Grothendieck stable $(\infty,1)$-toposes (i.e. those not arising as localizations of a presheaves of spectra).

But I don’t know. Has anyone an idea about this?

I am talking with Zoran about this, who knows vastly more about this than I do, but I thought I’d check if anyone reading this here might have some comment.

Hi David,

interesting points. I am not sure if there is a first-principle symmetry breaking between sheaves and co-sheaves or if it just always boils down to the fact that one is more used to working from one perspective than from the dual.

For instance, one important point of the in-plot perspective is that it matches so well with the idea that you want some structure (a bundle-like structure) $P \to U$ sitting over your test space, and that you want to think of that as arising from pullback along a classifying map $U \to B something$.

At that “heuristic introduction” entry I kind of try to make the point that it is this perspective which leads one naturally to $\infty$-stacks aka $\infty$-sheaves.

(By the way, I spent the morning polishing and expanding that entry, still not perfect, though, so let me hear your comments).

Now, one might have just as well maybe arrived there from a dual point of view, maybe one is just not used to that.

Notice, however, that usually it is an iteraded back and forth: consider for instance the Moerdijk-Reyes model for synthetic differential geometry:

- they start with smooth rings, defined a co-presheaves;

- then they take generalized synthetic smooth spaces to be sheaves on the category of the co-preshesheaves.

Maybe it’s really a matter of going back and forth between presheaves and co-presheaves sufficiently often such as to obtain a required approximation to Isbell self-duality.

Ultimately, one can look at $C^op$ just as easily as at $C$, and then sheaves become cosheaves, etc.

However, if one already has developed intuition for $C$ rather than for $C^op$, then there is good reason to use sheaves (on $C$) rather than cosheaves (which are sheaves on $C^op$). The reason is that $C$ embeds in the category of sheaves, whereas $C^op$ embeds in the category of cosheaves. That is, morphisms between representable sheaves go in the same direction as morphisms between their representing objects.