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Stream: learning: questions

Topic: Pro-completion of N-indexed families of finite sets

Vincent Moreau (Jun 03 2024 at 14:23):

Dear category theorists,

Let $\mathbf{Pro}(\mathbf{C})$ be the pro-completion of a category $\mathbf{C}$. It is well-known that $\mathbf{Pro}(\mathbf{FinSet})$ is $\mathbf{Stone}$. Lately, I have been thinking about what the category $\mathbf{Pro}(\mathbf{FinSet}^{\mathbb{N}})$ looks like, where $\mathbb{N}$ is just a set / discrete category.

In general, we have an equivalence between $\mathbf{Pro}(\mathbf{C} \times \mathbf{D})$ and $\mathbf{Pro}(\mathbf{C}) \times \mathbf{Pro}(\mathbf{D})$, and from this we get interesting facts like

$$\mathbf{Pro}(\mathbf{FinSet}^{\mathbb{N}}) \times \mathbf{Pro}(\mathbf{FinSet}^{\mathbb{N}}) \quad\cong\quad \mathbf{Pro}(\mathbf{FinSet}^{\mathbb{N}}) \\ \mathbf{Pro}(\mathbf{FinSet}^{\mathbb{N}}) \times \mathbf{Stone} \quad\cong\quad \mathbf{Pro}(\mathbf{FinSet}^{\mathbb{N}})$$

but it does not help us to get an explicit description.

There is a canonical functor

$$\operatorname{Ev} \quad:\quad \mathbf{Pro}(\mathbf{FinSet}^{\mathbb{N}}) \to \mathbf{Stone}^{\mathbb{N}}$$

which sends a pro-object on its family of pointwise-limit Stone spaces. I like to think of $\operatorname{Ev}$ as sending a pro-object on its underlying extensional content, forgetting its intensional aspects, i.e. the form of the cofiltered diagram which computes the Stone spaces. As an example of pro-objects which have the same extensional behavior but that are not isomorphic, consider the representable family $(0)_{i \in \mathbf{N}}$ and the cofiltered diagram

$$\dots \to (0, 0, 1, \dots) \to (0, 1, 1, \dots) \to (1, 1, 1, \dots)$$

which are both sent by $\operatorname{Ev}$ on the constant family equal to the empty Stone space.

Question: is a "nice" description of $\mathbf{Pro}(\mathbf{FinSet}^{\mathbb{N}})$ known?

Vincent Moreau (Jun 03 2024 at 14:24):

PS: the ind-object page of the nlab says

Proposition 4.12. Let 𝒞 be a small category that has finite colimits and let ℐ be a small category. Then the canonical functor

$$\operatorname{Ind}(𝒞^ℐ) \to \operatorname{Ind}(𝒞)^ℐ$$

is an equivalence of categories.

In light of my counterexample above, do we agree that this statement is false?

Reid Barton (Jun 03 2024 at 14:26):

That theorem should have the hypothesis that ℐ is a finite category (only finitely many objects and morphisms in total).

Vincent Moreau (Jun 03 2024 at 14:27):

I see, thanks, that's what this document shows.

Vincent Moreau (Jun 03 2024 at 14:27):

in theorem 1.2

Reid Barton (Jun 03 2024 at 14:28):

Seems like the prefix "$\kappa$-" in "$\kappa$-small" was dropped from somewhere.

Vincent Moreau (Jun 03 2024 at 14:30):

I've changed the nlab accordingly!

Ivan Di Liberti (Jun 03 2024 at 16:07):

Reid Barton said:

That theorem should have the hypothesis that ℐ is a finite category (only finitely many objects and morphisms in total).

And finite is NOT even enough, please double read the assumptions in the theorem.

Vincent Moreau (Jun 03 2024 at 16:17):

If we consider only the categories which have finite colimits, isn't it OK?

Reid Barton (Jun 04 2024 at 22:15):

Yes. There is a second theorem, which drops this hypothesis, but with stronger conditions on $I$.

David Michael Roberts (Jun 07 2024 at 10:31):

See eg https://mathoverflow.net/a/442827/4177