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Stream: learning: questions

Topic: Why does he say arbitrary direct limits and finite inverse l

V Slicer (Apr 25 2026 at 14:33):

Going over some notes I was reminded of a question that Colins McLarty answered. He gives two reasons for why Grothendieck used (but I'm asking generally) arbitrary direct limits and finite inverse limits when arbitrary inverse limits also exist. Are there more reasons or perspectives missing or other ways it has since been clarified?

Colins McLarty says in Nonetheless one should learn the language of topos [27:15]:

Repeatedly [Grothendieck] says: a topos is a category with the exactness properties of sets, so far as it concerns arbitrary direct limits and finite inverse limits.

"It will turn out that arbitrary inverse limits [also] exist. And the arbitrary inverse limits are kind of generous. They go beyond what I said, that a topos should share the exactness properties of sets in finite inverse limits and arbitrary direct limits." - Grothendieck, Buffalo 1973

So there really is a problem of... Why does he say arbitrary direct limits and finite inverse limits when arbitrary inverse limits also exist?

The only answer I can see, and I mean I think it's a correct answer is: these are properties used in Tohoku if you drop additivity. He's saying: the Tohoku properties, those are the properties I'm going to use to define topos.

And this explains a peculiar of line in Récoltes Et Semailles where Grothendieck he spent some time when on topos he spends very little time on Abelian categories but he does say at one point "what is an Abelian category? well it's a it's a category that has essentially the properties of a topos." And if you come to this knowing the material you think: no, it doesn't have the properties of topos... ah, but it has the exactness properties. The same exactness properties that he used to axiomatize topos except it's also additive, which a topos is not.

He comments over and over that infinite inverse limits don't work as nicely in a topos as they do in sets in particular if you have an infinite product of epimorphisms, think of this as onto maps, an infinite product of them in sets is also enough an epimorphism, at least if you have the Axiom of Choice, but in a sheaf category very commonly not. A finite product will still be an epimorphism but an infinite product of epimorphisms will not be an epimorphism in general in a sheaf category.

So there's another reason to limit the inverse limits to the finite: only the finite ones work nicely but I think the original reason he did limit it was that's what he used in Tohoku.

Todd Trimble (Apr 25 2026 at 15:27):

I haven't watched the Grothendieck lectures (I should!), but on the off-chance that Grothendieck didn't mention it, one way of characterizing Grothendieck toposes is by Giraud's theorem: a Grothendieck topos is a category $\mathbf{E}$ that

• is locally small and has a small generating set,
• has finite limits,
• has small coproducts, and these are disjoint and stable under pullback, and
• has quotients of congruences, and these are effective and stable under pullback.

The latter three are the exactness properties, and they are all in the language of direct limits and finite inverse limits. Especially important is the stability: that pulling back along a morphism $f: X \to Y$, to give a functor between slices $\mathbf{E}/Y \to \mathbf{E}/X$, preserves all small direct limits (as well as finite inverse limits, but that's much easier).

There are no comparable exactness properties for general toposes that would relate, for example, general small inverse limits to finite direct limits.

There's another characterization of Grothendieck toposes in a similar spirit, attributed to Street, which throws this into sharp relief. Say that a locally small category $\mathbf{E}$ is lex total if its Yoneda embedding has a left exact left adjoint $[\mathbf{E}^{op}, \mathsf{Set}] \to \mathbf{E}$. (For convenience, let's make a mild foundational assumption that the collection of objects of $\mathsf{Set}$, i.e., of small sets, is itself a set -- just not a small one. If $\kappa$ is the cardinality of this collection, then $\kappa$ is a strongly inaccessible cardinal.)

• Theorem (Street): A Grothendieck topos is precisely a lex total category of size $\kappa$.

V Slicer (Apr 25 2026 at 18:31):

Thanks for the great answer Todd Trimble

Todd Trimble said:

I haven't watched the Grothendieck lectures (I should!)

I've tried but, (especially for me) it's very hard to follow without being able to see what's being written and pointed at. But since it's category theory, it shoulnd't be impossible to reconstruct.

Todd Trimble said:

but on the off-chance that Grothendieck didn't mention it

Funny you mention this, in the lecture there's a remark about this:

Colins McLarty says in Nonetheless one should learn the language of topos [29:43]:

Well eventually he gives Giraud axioms on a topos and he gives the sites definition of the topos and i'm not going to go into that here because he tells us in the lectures so here's the notion of a topos which is slightly technical he's not against it but well: "I think it's kind of intuitive though," he says, "to take the vague notion which intuitively makes more sense: the one where you have all direct limits and finite inverse limits".

He says "I have a tendency to forget which properties Giraud uses."

And then McLarty gives the Giraud axioms. :)
Later in the lecture McLarty also says:

thank Serre also for pointing us out to me would one of the first times I talked to him we talked about etale maps and he the first thing he wanted to tell me about a etale maps was they're like finite unramified maps but they're trivial in the fibers not in the base.

Grothedieck failed to find what he says a certain number of exactness axioms which imply all the others one were ought to deal with. He can't make that axiomatic approach work and in hindsight we'll see we'll see sort of why he couldn't get it to work well it's those Giraud properties that he tends to forget you know you didn't he didn't think of them in the first place

Thank you for your answer!

V Slicer (Apr 25 2026 at 20:29):

edit: in a previous version of the message I mentioned a Grothendieck quote. That was a misattribution, it's indeed a quote by McLarty, my mistake.

Todd Trimble (Apr 25 2026 at 21:32):

No problem; I listened to that bit of McLarty's lecture and understood that's what happened.

There are stronger conditions one can contemplate for toposes. Particularly, presheaf toposes are examples of Grothendieck toposes where the left adjoint to Yoneda, $c: [\mathbf{E}^{op}, \mathsf{Set}] \to \mathbf{E}$, is not only left exact (= preserves finite limits), but has itself a left adjoint. In particular, such a $c$, being a right adjoint, preserves arbitrary inverse limits, not just finite ones. These are called totally distributive toposes. These can be thought of as akin to completely distributive lattices, i.e., complete lattices in which arbitrary meets distribute over arbitrary joins.

In this line of thinking, [[frames]] are to Grothendieck toposes as [[completely distributive lattices]] are to totally distributive toposes. (A frame can be equivalently defined as a poset $P$ whose $\mathbf{2}$-enriched Yoneda embedding $\mathrm{prin}: P \to [P^{op}, \mathbf{2}]$ has a left adjoint $\mathrm{sup}: [P^{op}, \mathbf{2}] \to P$ that preserves finite meets.)

A good paper which explores this connection is by Lucyshyn-Wright.