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Stream: deprecated: topos theory

Topic: Schanuel Topos

Morgan Rogers (he/him) (Oct 23 2020 at 10:53):

Does anyone know an accessible proof that the Schanuel topos (the topos of sheaves for the atomic topology on $\mathbf{FinSet}_{\mathrm{mono}}^{\mathrm{op}}$) is not a presheaf topos? I've just come up with a proof myself, but it's messy, and I would rather save myself from having to give perfect details if possible. It's definitely folklore; can it be deduced from some fact about theories classified by presheaf toposes?

Morgan Rogers (he/him) (Oct 23 2020 at 13:15):

Olivia Caramello described a nice proof to me which observes that the category of representables for a presheaf topos can be recovered up to idempotent completion as the (full) category of finitely presentable models in $\mathbf{Set}$ of the theory classified by the topos, but also that a presheaf topos is atomic if and only if that category is a groupoid. Since the Schanuel topos classifies infinite decidable objects and no non-empty subcategory of the category of infinite sets is a groupoid (under the weak assumption that every infinite set admits an injection from $\mathbb{N}$, say) it cannot be a presheaf topos :+1:

John Baez (Oct 23 2020 at 14:41):

Since the Schanuel topos classifies infinite decidable objects and no non-empty subcategory of the category of infinite sets is a groupoid (under the weak assumption that every infinite set admits an injection from $\mathbb{N}$, say) it cannot be a presheaf topos.

You seem to be using language a bit differently than I do, since I know plenty of non-empty subcategories of the category of infinite sets that are groupoids: for example, the category with $\mathbb{N}$ as its only object and bijections $f: \mathbb{N} \to \mathbb{N}$ as morphisms.

Jens Hemelaer (Oct 23 2020 at 14:59):

@John Baez I think subcategory means full subcategory here.

[Mod] Morgan Rogers said:

Olivia Caramello described a nice proof to me which observes that the category of representables for a presheaf topos can be recovered up to idempotent completion as the (full) category of finitely presentable models in $\mathbf{Set}$ of the theory classified by the topos, but also that a presheaf topos is atomic if and only if that category is a groupoid. Since the Schanuel topos classifies infinite decidable objects and no non-empty subcategory of the category of infinite sets is a groupoid (under the weak assumption that every infinite set admits an injection from $\mathbb{N}$, say) it cannot be a presheaf topos :+1:

Nice proof! For the last part, if you want to stay in ZF, you can also take an element $x \in X$ and then the constant endomorphism of $X$ sending all elements to $x$ is non-invertible.

Morgan Rogers (he/him) (Oct 23 2020 at 15:25):

@John Baez indeed, the category of finitely presented models is a full subcategory of the category of all models; I had said "(full)" earlier in the explanation, but I should have emphasised it.
@Jens Hemelaer the trouble is that the morphisms in the category of models are the monomorphisms (cf the classifying topos for decidable objects being presheaves on the category of finite sets and monomorphisms), so I need to know that there is a non-invertible mono, rather than any old endomorphism. I know that $\mathbb{N}$ has one of these, and so since I'm only dealing with decidable sets, an injection from $\mathbb{N}$ is sufficient to construct an such a non-invertible mono for any infinite set.

Jens Hemelaer (Oct 23 2020 at 16:24):

Ah right, I didn't think of that.