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Stream: deprecated: topos theory

Topic: Every elementary topos concretizable?

Jens Hemelaer (Nov 02 2020 at 09:16):

Does every elementary topos admit a faithful functor to the category of sets?

This is the case for every Grothendieck topos, and also for the few examples of non-Grothendieck toposes that I know (like the topos of finite sets or the topos of $M$-sets where $M$ is a "large" category with one object).

Maybe there are even elementary toposes that are not locally small?

Reid Barton (Nov 02 2020 at 12:20):

I think the category of small sheaves on a site (probably we need the site to have finite limits) is supposed to be an elementary topos, and it doesn't seem obviously concretizable, so maybe that would be a place to look.

Morgan Rogers (he/him) (Nov 02 2020 at 12:47):

Jens Hemelaer said:

This is the case for every Grothendieck topos.

What functor are you referring to here? A functor represented by some "bound" (separating object) for the topos?
An unbounded topos over $\mathbf{Set}$ does seem like the way to go.

Morgan Rogers (he/him) (Nov 02 2020 at 12:48):

(does a single separating object always exist if the topos is bounded? That seems like a very strong result, since it would surely imply that every topos is a subtopos of a topos of actions of a monoid)

Reid Barton (Nov 02 2020 at 12:53):

You can equip the category of presheaves on any small category $A$ with the faithful functor $\coprod_{a \in A} \mathrm{Hom}(a, -)$ to Set.

Reid Barton (Nov 02 2020 at 12:55):

So to be concretizable it's enough to have a small dense (full) subcategory, which every Grothendieck topos does.

Morgan Rogers (he/him) (Nov 02 2020 at 12:55):

Ah, so that's true for any locally small category with a separating set of objects.

Jens Hemelaer (Nov 02 2020 at 14:16):

Reid Barton said:

I think the category of small sheaves on a site (probably we need the site to have finite limits) is supposed to be an elementary topos, and it doesn't seem obviously concretizable, so maybe that would be a place to look.

Thanks! So maybe an example is the category $\prod_{i \in I} \mathbf{Sets}$, where $I$ is some proper class? It seems that the subobject classifier, exponential objects and finite limits can be defined pointwise here.

There seem to be some size issues here: if you work in von Neumann--Bernays--Gödel set theory, then maybe the collection of objects in the category $\prod_{i \in I} \mathbf{Sets}$ can fail to form a 'class'. But this problem appears also for the category of $M$-sets where $M$ is a large monoid, I think.

Morgan Rogers (he/him) (Nov 02 2020 at 14:28):

As soon as you have a topos which fails to be locally small with respect to some fixed universe of Sets, like in your example, it certainly can't be concrete in a faithful way, but I had hoped there would be a more concrete example of a locally small topos over Set that failed to be bounded. Or is local smallness (i.e. Set-enrichment) the same as being faithfully concretizable?

Jens Hemelaer (Nov 02 2020 at 14:45):

Regarding your last question: the category of topological spaces and homotopy classes of maps between them is locally small but not concretizable (see here). If I recall correctly, another example is the category of small categories and functors up to natural equivalence.

Morgan Rogers (he/him) (Nov 02 2020 at 14:58):

That's a relief...! So there should be some locally small but not concrete topos, I expect.

Reid Barton (Nov 02 2020 at 15:09):

I'm not sure whether the product of a class-sized family of copies of Set really makes sense at all in ZFC, but it's definitely not locally small, as Morgan said. That's why I suggested working with small (pre)sheaves, which we can regard as formal small colimits of representables. They form an ordinary, class-sized locally small category which is however not locally presentable.

Reid Barton (Nov 02 2020 at 15:11):

I'm not sure exactly when the category of small presheaves on $C$ is an elementary a topos. At a minimum it has to have finite limits, which is a nontrivial condition on $C$. For example if $C$ is a class-sized discrete category then the terminal object of the presheaf category is not a small presheaf.

Reid Barton (Nov 02 2020 at 15:15):

I think the condition on $C$ is that every finite diagram has a set of weakly universal cones (i.e., a set of cones such that every cone factors through one of them in at least one way). For example, $C$ could itself have finite limits.

Reid Barton (Nov 02 2020 at 15:16):

The rest of the Giraud axioms seem fine (the category of presheaves on $C$ in a bigger universe is definitely a topos, and things like small coproducts and coequalizers and pullbacks don't leave the full subcategory of small presheaves) so I guess the above condition on $C$ is enough for the presheaf category to be an elementary topos.

Reid Barton (Nov 02 2020 at 15:22):

I also don't know when exactly the small sheaves for a topology on such $C$ form what we could call a "Giraud topos" (= Giraud axioms minus a set of generators), but I understand that it's supposed to be true for condensed sets, for example.

Jens Hemelaer (Nov 02 2020 at 15:27):

Ah, I misunderstood what small presheaves are. Thank you!

David Michael Roberts (Nov 03 2020 at 04:56):

A Giraud topos is a cocomplete locally small pretopos, no? This is rather easy to arrange, and we can do better. Say $R^{op} \to Topos_b$ is a cofiltered diagram with $R$ well-founded (maybe we don't need this bit). Then the colimit of $R\to Topos_b^{op} \to LEX_{cocont}$ (lextensive categories with finite-limit- and colimit-preserving functors) is a cocomplete Heyting pretopos with a subobject classifier and finitary W-types. It might not be locally small, but if the diagram of inverse image functors is eventually full it will be. Further, if powerobjects are eventually constant, it will be an unbounded Set-topos.

David Michael Roberts (Nov 03 2020 at 04:57):

I believe if the diagram arises from a diagram of (small) sites, then the colimit looks something like small sheaves on a large site. Compare with Moerdijk's _Continuous fibrations and inverse limits of toposes_ http://www.numdam.org/item/CM_1986__58_1_45_0/