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Stream: learning: questions

Topic: Is binary gluing preserved by "localising" a presheaf?

Naso (May 14 2023 at 23:31):

I have two questions about a "binary gluing property" for presheaves.

(Binary gluing): a presheaf $F: \mathcal{T}^{op} \to \mathbf{Set}$ on a topological space $(X, \mathcal{T})$ has the binary gluing property if whenever $a \in F(A)$, $b \in F(B)$ and $a \mid_{A \cap B} = b \mid_{A \cap B}$ there exists $c \in F(A \cup B)$ such that $c \mid_A = a$ and $c \mid_B = b$.

First question: the usual gluing property for presheaves talks about arbitrary covers. I guess that for finite nonempty covers, the binary gluing property implies the usual gluing property by induction. But I think it could fail for an empty cover, where the usual gluing property forces the sheaf to take a nonempty value on the empty set but the binary gluing property does not.
But suppose $F(\emptyset) \neq \emptyset$. What is an example of a presheaf with the binary gluing property but not the usual gluing property? I think it should fail for some infinite cover, but I can't think of an example.

Second question: suppose that $(X,\mathcal{T})$ is locally connected (so that connected components are open) and also that $G$ is a presheaf. Define a new presheaf $F$ by

$F(U) =\prod_{C \in \pi_0 U} G(C)$

with restriction maps for $V \subseteq U$ sendind an element $(g_C)_{C \in \pi_0 U}$ to the element $(g_{\pi_0(C)})_{C \in \pi_0 V}$.

This is kind of like "localising" the presheaf $G$, for example if $G$ is a constant presheaf then $F$ is a constant sheaf.

Anyway, my question is: if $G$ is flasque and has the binary gluing property,are these proeprties inherited by $F$? I think flasqueness certainly is but I'm not sure about the binary gluing.

Also is there already terminology for this kind of "localisation" of a presheaf, or anything else I should know about it?

Naso (May 14 2023 at 23:40):

By the way, I just asked this to ChatGPT and here is its answer .

Morgan Rogers (he/him) (May 15 2023 at 06:50):

For your first question, if you take the discrete space on two points, any constant presheaf has the binary gluing property. Note that this gluing property is actually a sheaf condition for a smaller Grothendieck topology on the poset of open subsets of the space.

Morgan Rogers (he/him) (May 15 2023 at 06:53):

For your second question, I guess you meant to say that if G is constant then F is locally constant? Anyway, the latter procedure looks like sheafification for the Grothendieck topology in which each open is covered by its connected components.

Naso (May 15 2023 at 08:51):

Morgan Rogers (he/him) said:

For your first question, if you take the discrete space on two points, any constant presheaf has the binary gluing property. Note that this gluing property is actually a sheaf condition for a smaller Grothendieck topology on the poset of open subsets of the space.

Thank you, Morgan! For 1. Can we also have a counterexample if $F(\emptyset) = \{ * \}$ ? By the way, which smaller Grothendieck topology is that you refer to?

For 2. I did mean to say that if $G$ is a constant presheaf, then the $F$ we obtain is the constant sheaf with the same value, but maybe I am confused about this. The example I had in mind is this one on wikipedia, where we have the constant presheaf $G(U) = \mathbb{Z}$ on a discrete space on two points. Then doing the procedure I described gives us the sheaf of locally constant functions, i.e. the constant sheaf.

Morgan Rogers (he/him) (May 15 2023 at 10:44):

It's the Grothendieck topology where covering families are finite non-empty open covers. If $F(\emptyset)$ is already a singleton then $F$ is a sheaf for the topology where finite unions cover, so you do need and infinite space to get a counterexample. I'll see what I can think of.

Ah I suppose it does make sense to refer to that as a constant sheaf even though it isn't constant :rolling_on_the_floor_laughing: fair enough. I realise I didn't answer your flasque question, I'll also think about that.

Reid Barton (May 15 2023 at 15:28):

For the finite gluing question, one kind of example would be X an infinite discrete topological space, and F(A) = a singleton if A is finite, but empty if A is infinite.

Martti Karvonen (May 15 2023 at 15:59):

Another source of example (already where binary gluings don't imply finite gluings) comes from the sheaf-theoretic approach to contextuality I just mentioned in a different thread yesterday (this is the founding paper and this might be an easier entry-point).

Martti Karvonen (May 15 2023 at 16:01):

Here's an explicit small example in that direction. Consider the set $X:=\{x,y,z\}$ equipped with the discrete topology. Define a presheaf $F$ by sending a subset $U$ to the set of probability distributions on $\{0,1\}^U$. Now $F(\emptyset)$ is a singleton, and one can show that binary gluings exist (although aren't unique).

Reid Barton (May 15 2023 at 16:02):

Oh, I didn't notice that there was nothing about uniqueness.

Martti Karvonen (May 15 2023 at 16:03):

Here's a matching family for this presheaf that doesn't glue: given a subset of $X$ of size 2, consider the distribution that gives opposite outcomes to the two variables, both such options with equal probability.

Martti Karvonen (May 15 2023 at 16:04):

There's no way to glue these to a joint distribution on all three variables, basically because $x\leftrightarrow \neg y,y\leftrightarrow \neg z,z\leftrightarrow \neg x$ is inconsistent.

Naso (May 15 2023 at 23:41):

@Martti Karvonen I love this! I admit that I knew about contextuality already, but when asking the question I was led astray by the idea that binary gluing + induction would give us finite gluing... Your example shows that is completely wrong.

Is it something about the homotopy type of the (nerve of the) cover that makes contextual counterexamples like this possible? Like in your example, the nerve of the cover $\{\{X,Y\}, \{Y,Z\}, \{Z, X\} \}$ is a triangle. Is it right that on a homotopically trivial finite cover we could not have contextuality?

Martti Karvonen (May 16 2023 at 12:56):

So, the structure of "a set of measurements + which subsets of them are jointly measurable" is naturally viewed as an abstract simplicial complex. Vorob'evs theorem characterizes exactly those simplicial complexes that admit contextuality. However, the criterion is combinatorial and not topological - there are simplicial complexes whose geometric realizations are homotopy equivalent, but one admits contextuality and the other one doesn't.

Martti Karvonen (May 16 2023 at 12:58):

Here's one way of stating Vorob'evs theorem. If a vertex of a simplicial complex is in exactly one maximal face, remove it from the complex. Such a step is called a Graham reduction. Now, a simplicial complex does not admit contextuality iff it can be reduced to a point by a sequence of Graham reductions.

David Michael Roberts (Mar 19 2024 at 06:14):

To add an example developed in discussion elsewhere:

consider the presheaf $G$ on $\mathbb{R}$ of bounded (but otherwise arbitrary) functions. This is a sheaf for the pretopology consisting of finite open covers (hence satisfies the binary gluing property), and it is flasque. The construction $G\mapsto F$ Naso outlines is sheafification for something close to the infinitary extensive pretopology (the lattice of opens isn't an extensive category, but it has the properties of an extensive category for unions of disjoint open sets).

Then for the open subset $U$ of the real line that is the union $\bigcup_{n\in \mathbb{N}}(2n,2n+1)$, the restriction map $F(\mathbb{R}) \to F(U)$ is not surjective, so $F$ isn't flasque.

David Michael Roberts (Mar 21 2024 at 03:29):

To contrast, if one starts with a finite topological space, then under the same assumptions on a presheaf G (binary gluing, flasque, and G\emptyset = 1), then flasqueness and the value on the empty set are always preserved on passing to the construction of F.