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Stream: theory: algebraic topology

Topic: sheaf condition on sites

Matteo Capucci (he/him) (Oct 19 2020 at 10:21):

The sheaf condition on top spaces can be phrased in a rather succint way, namely that for every open covering $\{U_i\}_i$ (of an open set $U$), $F(\operatorname{colim} U_i) \cong \operatorname{lim} F(U_i)$.
I was wondering... is this true for any site? So let 'open covering' be replaced by 'covering sieve' in the above sentence. I'm quite convinced $F(U) \cong \operatorname{lim} F(U_i)$ for a sheaf, not so much that $\operatorname{colim} U_i \cong U$.

Morgan Rogers (he/him) (Oct 19 2020 at 10:34):

Indeed, the coincidence relies on the Grothendieck topology being canonical, I think

Matteo Capucci (he/him) (Oct 19 2020 at 10:44):

:thinking: It seems a reasonable characterization.

Jens Hemelaer (Oct 19 2020 at 11:04):

The condition $\mathrm{colim} \,U_i \simeq U \Rightarrow F(\mathrm{colim} \, U_i ) \simeq F(U)$ does not depend on the Grothendieck topology. So, for an arbitrary Grothendieck topology, it is not equivalent to the sheaf condition.

You can modify your criterion a bit:
$\mathrm{colim} \, \mathbf{a}(U_i) \simeq \mathbf{a}(U) \Rightarrow \mathrm{lim} \, F(U_i ) \simeq F(U)$.
Here $\mathbf{a}$ denotes first taking Yoneda embedding and then sheafifying.

I think this is equivalent to the sheaf condition for an arbitrary Grothendieck topology. The reason is that a covering sieve $S$ of $U$ can be interpreted as a sub-presheaf of $\mathbf{y}(U)$, and sheafification makes sure that the inclusion $S \subseteq \mathbf{y}(U)$ becomes an isomorphism.

Matteo Capucci (he/him) (Oct 19 2020 at 12:28):

I'm a bit puzzled, why $\operatorname{colim} F(U_i)$ and not $\operatorname{lim}$?

Matteo Capucci (he/him) (Oct 19 2020 at 12:28):

Also I don't understand the implication. If $\operatorname{colim} U_i \neq U$ I still want $F$ so satisfy a sheaf condition on that covering sieve

Matteo Capucci (he/him) (Oct 19 2020 at 12:29):

Finally, why would I want the inclusion $S \subseteq yU$ to be an iso?

Kenji Maillard (Oct 19 2020 at 12:32):

Matteo Capucci said:

Finally, why would I want the inclusion $S \subseteq yU$ to be an iso?

You can describe sheaves as the local objects with respect to the maps $S \subseteq yU$, so after sheafification these are isos.

Matteo Capucci (he/him) (Oct 19 2020 at 12:33):

:thinking: I see

Matteo Capucci (he/him) (Oct 19 2020 at 12:34):

Btw using sheafification to define the sheaf condition kind of defeats the point of my question, since then it's easier to say $F$ is a sheaf if $F \cong aF$

Matteo Capucci (he/him) (Oct 19 2020 at 12:36):

I think it's fine to stick with $F(U) \cong \operatorname{lim} F(U_i)$

Matteo Capucci (he/him) (Oct 19 2020 at 12:37):

Jens Hemelaer said:

You can modify your criterion a bit:
$\mathrm{colim} \, \mathbf{a}(U_i) \simeq \mathbf{a}(U) \Rightarrow \mathrm{colim} \, F(U_i ) \simeq F(U)$.
Here $\mathbf{a}$ denotes first taking Yoneda embedding and then sheafifying.

Still I'd be interested to understand this @Jens Hemelaer

Matteo Capucci (he/him) (Oct 19 2020 at 12:39):

Mmmh maybe I get it, provided that the right hand side of the implication is actually $\operatorname{lim}F(U_i) \cong F(U)$

Matteo Capucci (he/him) (Oct 19 2020 at 12:39):

Then I see that the left hand side is always true, since as you and @Kenji Maillard remarked, sheafification makes it work

Fabrizio Genovese (Oct 19 2020 at 13:49):

Matteo Capucci said:

Jens Hemelaer said:

You can modify your criterion a bit:
$\mathrm{colim} \, \mathbf{a}(U_i) \simeq \mathbf{a}(U) \Rightarrow \mathrm{colim} \, F(U_i ) \simeq F(U)$.
Here $\mathbf{a}$ denotes first taking Yoneda embedding and then sheafifying.

Still I'd be interested to understand this Jens Hemelaer

If I'm not wrong this is in Sheaves in Geometry and Logic, somewhere between Chapters 1 and 3

Fabrizio Genovese (Oct 19 2020 at 13:50):

(Most likely chapter 3)

Jens Hemelaer (Oct 19 2020 at 13:54):

Matteo Capucci said:

I'm a bit puzzled, why $\operatorname{colim} F(U_i)$ and not $\operatorname{lim}$?

This was a mistake, I edited it now.

Jens Hemelaer (Oct 19 2020 at 13:58):

Matteo Capucci said:

Then I see that the left hand side is always true, since as you and Kenji Maillard remarked, sheafification makes it work

Yes, for a sieve $S = \{ U_i \to U \}_{i \in I}$, the left hand side is equivalent to $S$ being a covering sieve. So instead of the criterion I wrote down, you can just say instead that $F$ is a sheaf if and only if $F(U) = \mathrm{lim}\, F(U_i)$ for each covering sieve.

Matteo Capucci (he/him) (Oct 19 2020 at 14:03):

:thumbs_up:

Matteo Capucci (he/him) (Oct 19 2020 at 14:06):

Fabrizio Genovese said:

If I'm not wrong this is in Sheaves in Geometry and Logic, somewhere between Chapters 1 and 3

I couldn't find M&M explicitly using this characterization. They provide a different definition of sheaf on a site, which is one such that for every covering sieve $S$ on $U$ (= open covering) and any $\alpha :S \to F$ (= a compatible family of sections on the covering), there is a unique extension of $\alpha$ along the inclusion $S \rightarrowtail yU$ (= a global section)

Reid Barton (Oct 19 2020 at 14:09):

By definition/Yoneda, "$\mathrm{colim}\, \mathbf{a}(A_i) = \mathbf{a}(A)$" means exactly "for every sheaf $F$, $FA = \lim F(A_i)$". This doesn't use anything about sheaves other than that they are a reflexive subcategory of presheaves with reflector $\mathbf{a}$.
In particular the phenomena discussed here don't have anything to do with sheaves per se--the same happens in any localization.

Jens Hemelaer (Oct 19 2020 at 14:13):

Yes, the M&M definition says in other words that the natural transformation $Hom(yU,-)\to Hom(S,-)$ is an isomorphism, when the second argument takes values in the subcategory of sheaves. So by the adjunction and Yoneda, you get that $\mathbf{a}(S) \to \mathbf{a}(yU)$ is an isomorphism.

Jens Hemelaer (Oct 19 2020 at 14:16):

It is important here that you work with sieves, not just with covering families.

For example, the map $\mathrm{Spec}(\mathbb{C}) \to \mathrm{Spec}(\mathbb{R})$ is a covering map for the étale topology. But the colimit of $\mathrm{Spec}(\mathbb{C})$, as a diagram with one object, is equal to $\mathrm{Spec}(\mathbb{C})$ itself.

On the other hand, if you take the sieve generated by $\mathrm{Spec}(\mathbb{C}) \to \mathrm{Spec}(\mathbb{R})$, then you get that the colimit of this sieve is equal to $\mathrm{Spec}(\mathbb{R})$, in the étale topos.

Reid Barton (Oct 19 2020 at 14:16):

The sieve stuff is more subtle--the map $S \to yU$ is the "inclusion of the image" part of the map of presheaves $\mathrm{colim}\,y U_i \to y U$ and for some reason which I don't recall at the moment, but I think is rather specific to the situation of (1-)topoi, it's equivalent to impose the locality condition with respect to the map from the sieve.

Reid Barton (Oct 19 2020 at 14:17):

(Probably not on a per-map basis, but for the whole coverage at once, or something like that.)

Reid Barton (Oct 19 2020 at 14:20):

I guess everyone has been a bit vague about what the index category in "$\mathrm{colim}\,yU_i$" is exactly.

Jens Hemelaer (Oct 19 2020 at 14:24):

Ah right, this criterion
$\mathrm{colim} \, \mathbf{a}(U_i) \simeq \mathbf{a}(U) \Rightarrow \mathrm{lim} \, F(U_i ) \simeq F(U)$
is equivalent to the sheaf condition by @Reid Barton's argument, and you don't even need that $\{U_i \to U\}_{i \in I}$ is a sieve.
The only problem is that a "covering map" like $\mathrm{Spec}(\mathbb{C})\to\mathrm{Spec}(\mathbb{R})$ does not necessarily satisfy the criterion on the left hand side. A covering sieve always does though.