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Stream: learning: reading & references

Topic: special properties of presheaf toposes

Notification Bot (Apr 07 2024 at 17:41):

A message was moved here from #learning: reading & references > Applications of sheaves by Mike Shulman.

Mike Shulman (Apr 07 2024 at 17:42):

One is that since representables are projective and every presheaf is a colimit of representables, presheaf categories satisfy the [[presentation axiom]] that every object is covered by a projective one.

Mike Shulman (Apr 07 2024 at 17:43):

Which, in turn, implies the axioms of [[countable choice]] and [[dependent choice]].

John Baez (Apr 07 2024 at 20:06):

Nice question and nice answer!

Chris Grossack (they/them) (Apr 07 2024 at 21:23):

From the perspective of geometric logic, a closely related question is when a [[geometric theory]] is classified by a presheaf topos (we know that every geometric theory is classified by a grothendieck topos, so this really is analogous to your original question). This question is deeply studied in Olivia Caramello's book Theories, Sites, and Toposes

Brendan Murphy (Apr 08 2024 at 00:45):

Presheaf categories are always locally finitely presentable while grothendieck topoi are in general only locally presentable

Brendan Murphy (Apr 08 2024 at 00:47):

Even locally presentable relative to (finite products, sifted colimits) rather than (finite limits, directed colimits), ie presheaf topoi are categories of models for a many sorted variety of algebras (take a sort for each object of the original category any a unary function symbol for each morphism in it) while this is very false for general grothendieck topoi

Brendan Murphy (Apr 08 2024 at 00:51):

Maybe more concretely: finite limits commute with filtered colimits (this is very much not about the internal logic, however! This is always going to be true internally, I believe, so the interesting bit is that it holds for external limits and colimits)

Amar Hadzihasanovic (Apr 08 2024 at 06:41):

Is there anything interesting that can be said from the point of view advocated by Anel and Joyal, i.e. are there non-obvious ways in which a presheaf category is "like" a free commutative ring?

Amar Hadzihasanovic (Apr 08 2024 at 06:42):

Possibly not in the obvious sense of free (i.e. free on a set)

Morgan Rogers (he/him) (Apr 08 2024 at 08:59):

Brendan Murphy said:

Maybe more concretely: finite limits commute with filtered colimits (this is very much not about the internal logic, however! This is always going to be true internally, I believe, so the interesting bit is that it holds for external limits and colimits)

This is true for all Grothendieck toposes, and actually all locally presentable categories, I think.

John Baez (Apr 08 2024 at 15:19):

There's more discussion of this here:

• Why do filtered colimits commute with finite limits?

Note also Zhen Lin's comment after Tim Campion's answer.

Brendan Murphy (Apr 08 2024 at 19:39):

Oh, my bad then

Brendan Murphy (Apr 08 2024 at 19:50):

Does anyone know the "usual argument" Zhen Lin is talking about? The thing I'm stuck on is that I thought a filtered colimit of sheaves is not generally going to be a filtered colimits of presheaves, in the case where the topos of sheaves is not lfp

Brendan Murphy (Apr 08 2024 at 19:54):

Ah nevermind, I looked at the nlab page and understand now

Brendan Murphy (Apr 08 2024 at 20:04):

For the question of whether it's true in locally presentable categories the paper "Localization of locally presentable categories" by Day & Street might be relevant. This says a category is a left exact localization of an lfp category iff it had all small colimits, has finite limits, has a small strongly generating set, and finite limits commute with filtered colimits. I'm not familiar with the concept of a strong generator and the paper doesn't explain it, but maybe general locally presentable categories have these?

Ivan Di Liberti (Apr 09 2024 at 04:50):

Morgan Rogers (he/him) said:

Brendan Murphy said:

Maybe more concretely: finite limits commute with filtered colimits (this is very much not about the internal logic, however! This is always going to be true internally, I believe, so the interesting bit is that it holds for external limits and colimits)

This is true for all Grothendieck toposes, and actually all locally presentable categories, I think.

locally finitely* (!) presentable categories

Morgan Rogers (he/him) (Apr 09 2024 at 05:29):

Hmmm it's interesting that I was corrected on this, since it may mean that a proof I wrote recently is flawed. (It's nice when that happens before it's in the public eye hahaha)

Kevin Carlson (Arlin) (Apr 09 2024 at 13:56):

I do think you get all Grothendieck toposes as well as all lfp categories, but for different reasons: finite limits and filtered colimits in an lfp category are taken in some presheaf supercategory, while this isn’t true in every Grothendieck topos but you can apply sheafification, which preserves filtered colimits and finite limits, to both sides of the equation saying these commute in a presheaf supercategory. This is just fleshing out Zhen Lin’s comment that John linked. And neither argument applies in an lp category that’s not lfp or a Grothendieck topos, so probably the property is false.

Morgan Rogers (he/him) (Apr 09 2024 at 14:37):

Indeed. My error was going from the observation that limits and colimit of a diagram $F$ in a reflective subcategory $U:C' \to C$ (with reflector $L$) can be respectively computed as $L \lim UF$ and $L \mathrm{colim} UF$ (where the limit and colimit are taken in $C$) to concluding that the comparison morphism from colim-lim to lim-colim must coincide. In fact, assuming that finite limits commute with filtered colimits in $C$, it suffices that either:

• $U:C' \to C$ is conservative and preserves finite limits and filtered colimits (we do not need an adjoint at all), or
• There is a `natural retraction' $L$ of $U$ (it need not be an adjoint but we need a natural isom $LU \cong \mathrm{id}_{C'}$) and $L$ preserves finite limits and filtered colimits.

These correspond to the two cases that @Kevin Carlson (Arlin) pointed out, and they are indeed different in flavour; they aren't at all particular to the finite limit - filtered colimit case.

Kevin Carlson (Arlin) (Apr 09 2024 at 14:40):

If anybody has an example of filtered colimits not commuting with finite limits in non-topos non-finitely lp categories I’d be curious to see it. It doesn’t seem like my go-to $\aleph_1$-presentable case of Banach spaces provides a counterexample.

Morgan Rogers (he/him) (Apr 09 2024 at 14:41):

Indeed, finding a counterexample is tricky if one wants filtered colimits and finite colimits to exist in the category :thinking:

Kevin Carlson (Arlin) (Apr 09 2024 at 14:42):

Wait, but those will exist in any lp category, yes?

Morgan Rogers (he/him) (Apr 09 2024 at 14:43):

Oops, indeed, I was confused by the example that Tim Campion told me about recently of Banach + bounded maps!

Kevin Carlson (Arlin) (Apr 09 2024 at 14:44):

Something about the fact that the reflector for Banach involving closing under certain sequences, which are themselves filtered diagrams (!) seems to preclude a counterexample there.

Mike Shulman (Apr 09 2024 at 22:05):

Locally non-finitely presentable categories do seem to be a bit thin on the ground.

Todd Trimble (May 25 2024 at 16:29):

Kevin Carlson said:

If anybody has an example of filtered colimits not commuting with finite limits in non-topos non-finitely lp categories I’d be curious to see it. It doesn’t seem like my go-to $\aleph_1$-presentable case of Banach spaces provides a counterexample.

What's wrong with this example, if anything? Take the sup-lattice of closed subsets of the one-point compactification $\mathbb{N} \sqcup \{\infty\}$ of $\mathbb{N}$ with the discrete topology, considered as a thin category. This is accessible (because it is a small category in which idempotents split) and cocomplete, hence locally presentable. Now consider the filtered diagram $\{B_i\}$ of finite subsets of $\mathbb{N}$, and take $A = \{\infty\}$. Then

$\emptyset = \mathrm{colim}_i A \cap B_i = \mathrm{colim}_i A \times B_i$

whereas

$\{\infty\} = A \cap (\mathbb{N} \sqcup \{\infty\}) = A \times \mathrm{colim}_i B_i.$

Kevin Carlson (May 25 2024 at 21:22):

Yeah that looks right to me, thanks! I can’t remember how I’d talked myself out of that one.