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Stream: deprecated: topos theory

Topic: morphisms of presheaf toposes

Christian Williams (May 11 2021 at 00:52):

@John Baez and I want to determine conditions on $F:C\to D$ which are necessary or sufficient for $F^*:[D^{op},Set]\to [C^{op},Set]$ to have certain topos-theoretic properties: logical, locally cartesian closed, etc.

Christian Williams (May 11 2021 at 00:53):

Let's start with the simplest, logical. As I understand, these are rare because they are so nice. What are useful necessary or sufficient conditions on $F$ for $F^*$ to be logical?

Nathanael Arkor (May 11 2021 at 00:55):

Is it just a coincidence that someone asked the same question for local cartesian closure on MathOverflow just an hour ago?

Christian Williams (May 11 2021 at 00:55):

that is quite a coincidence. I'm not sure who that is

Christian Williams (May 11 2021 at 00:56):

moving this to #theory: topos theory

Notification Bot (May 11 2021 at 00:56):

This topic was moved here from #theory: category theory > morphisms of presheaf toposes by Christian Williams

Christian Williams (May 11 2021 at 01:08):

I'm seeing one theorem which shows how special logical functors are: The Elephant A.2.3.8

Christian Williams (May 11 2021 at 01:09):

logical-functors.png

Christian Williams (May 11 2021 at 01:12):

so, a logical functor $F$ with a left adjoint that preserves pullbacks must be of the form "pullback along a morphism in a topos"

Christian Williams (May 11 2021 at 01:13):

that's a big restriction.

Christian Williams (May 11 2021 at 01:14):

if $F$ is cartesian, then the left adjoint of $F^*$ is cartesian. then if $F^*$ is logical, this theorem implies that it is actually an equivalence.

Christian Williams (May 11 2021 at 01:16):

so, basically we can't expect inverse image functors of presheaf toposes to be logical.

Christian Williams (May 11 2021 at 01:17):

but we don't have to focus the conversation only on $F^*$; we can also consider the extensions $\exists_F$ and $\forall_F$.

Christian Williams (May 11 2021 at 01:25):

from the topic on type theory of a topos, we made some progress toward if $F$ is cartesian closed then $\exists_F$ might be as well - subject to some condition of finitude

Christian Williams (May 11 2021 at 01:28):

of course demanding any functor to be logical is very strict, so we can also ask for "sublogical" or "subclosed", where the canonical morphisms are not isomorphisms but just monic.

Christian Williams (May 11 2021 at 01:30):

in general, this is a topic for conditions on $F:C\to D$ giving various properties of $\exists_F$, $F^*$, and $\forall_F$.

Morgan Rogers (he/him) (May 11 2021 at 09:07):

You could look at Corollaries 4.56 and 4.58 of Caramello's denseness conditions paper; these give conditions for the essential gm induced by $F$ to be locally connected, or equivalently for $F^*$ to be locally cartesian closed. The conditions are very fiddly, but more importantly don't appear to bear much resemblance to your situation.

It might be helpful to know that @Riccardo Zanfa and I have worked out that the first condition of Corollary 4.56 there can be reduced to the necessary and sufficient condition given by Johnstone for the induced geometric morphism to be open (equivalently, for $F^*$ to be sublogical) in C3.1.3 of the Elephant.

John Baez (May 11 2021 at 15:16):

All this is a bit hard for me to understand. Are there any nice easy examples of functors $F: C \to D$ such that $F_\ast : \widehat{D} \to \widehat{C}$ is a logical morphism?

John Baez (May 11 2021 at 15:17):

Any nice examples where $C$ and $D$ are finite categories, for example?

Jon Sterling (May 11 2021 at 15:34):

@John Baez Did you mean $F^*$? If so, I think we could potentially get some examples by looking at the relationship between $\mathbf{Set}^{\to}$ and $\mathbf{Set}$.

In particular, the codomain functor (which is logical) is the inverse image part of the essential geometric morphism determined by a functor $\mathbf{1}\to \{0\leq 1\}$.

Jon Sterling (May 11 2021 at 15:36):

But this logical functor is just another one of those slice examples... That is, you can reconstruct $\mathbf{Set}$ as the slice of $\mathbf{Set}^{\to}$ over a certain representable subterminal object. This is the "generic" case of Artin gluing...

John Baez (May 11 2021 at 15:43):

Yeah, I meant $F^\ast$.

John Baez (May 11 2021 at 15:47):

Thanks for the example! Here are some followup questions:

1) So the functor $F: \{1\} \to \{0 \le 1\}$ sending $1$ to $1$ has the property that $F^\ast$ is logical? Just checking.

2) Is it correct that the functor $G: \{1\} \to \{0 \le 1\}$ sending $1$ to $0$ does not have the property that $G^\ast$ is logical?

3) Is the only functor $H : \{1\} \to \{0 \le 1 \le 2 \le \cdots \le n \}$ for which $H^\ast$ is logical the one mapping $1$ to $n$?

John Baez (May 11 2021 at 15:51):

I have some vague intuition about this involving the concept of "time to truth" mentioned in Mac Lane and Moerdijk's discussion of these sorts of presheaf categories. They consider presheaves on $\{0 \le 1 \le 2 \le 3 \le \cdots \}$ where "the final truth is never known", but in the example categories I just mentioned, "the full truth is known on the last day" - sort of like Judgement Day in the bible.

Jon Sterling (May 11 2021 at 15:54):

I am likely to get some things backwards as soon as I start talking about indices, but it is true that one of those inverse image functors is the codomain functor (and therefore clearly logical), but the other one is the domain functor (which is not logical). These correspond to the generic open and closed immersions of topoi respectively...

John Baez (May 11 2021 at 15:55):

Indeed, I probably got everything backwards due to the op in the definition of presheaves.

Jon Sterling (May 11 2021 at 15:56):

(I bet you got it forwards! But I don't want to claim it, becuase I always screw myself that way :laughing: )

John Baez (May 11 2021 at 15:57):

Okay, well the invariant formulation of my question 3) is: does just one of the functors from the terminal category to $[n] = \{1 \le 2 \le 3 \le \cdots \le n\}$ induce a logical functor between presheaf categories, via pullback?

John Baez (May 11 2021 at 15:57):

That's my conjecture: only one of them does.

Jon Sterling (May 11 2021 at 15:58):

i think there are more..

Morgan Rogers (he/him) (May 11 2021 at 16:19):

John Baez said:

All this is a bit hard for me to understand. Are there any nice easy examples of functors $F: C \to D$ such that $F^\ast : \widehat{D} \to \widehat{C}$ is a logical morphism?

Yes; take $F$ to be any discrete fibration. This identifies $\widehat{D}$ with the slice of $\widehat{C}$ over the corresponding presheaf, and the essential geometric morphism is the canonical one, whose leftmost adjoint is the forgetful functor, and whose inverse image is logical.

We precisely get the localic geometric morphisms into $\widehat{C}$ with logical inverse image this way.

Morgan Rogers (he/him) (May 11 2021 at 16:24):

John Baez said:

Thanks for the example! Here are some followup questions:

1) So the functor $F: \{1\} \to \{0 \le 1\}$ sending $1$ to $1$ has the property that $F^\ast$ is logical? Just checking.

2) Is it correct that the functor $G: \{1\} \to \{0 \le 1\}$ sending $1$ to $0$ does not have the property that $G^\ast$ is logical?

3) Is the only functor $H : \{1\} \to \{0 \le 1 \le 2 \le \cdots \le n \}$ for which $H^\ast$ is logical the one mapping $1$ to $n$?

Since all of these examples are faithful functors, so induce localic essential geometric morphisms, we can see that they induce geometric morphisms with logical inverse image if and only if they are discrete fibrations, which is false in the first case (there is no lifting of the morphism $0 \to 1$), true in the second case (so Jonathan did indeed get these mixed up), and true in the final case if and only if $H$ maps $1$ to $1$.

John Baez (May 11 2021 at 16:24):

Morgan Rogers (he/him) said:

John Baez said:

All this is a bit hard for me to understand. Are there any nice easy examples of functors $F: C \to D$ such that $F^\ast : \widehat{D} \to \widehat{C}$ is a logical morphism?

Yes; take $F$ to be any discrete fibration.

Great! That gives me a large supply of of examples I can try to understand in detail.

John Baez (May 11 2021 at 16:26):

And you're saying that a faithful $F$ induces a logical morphism $F^\ast$ only if $F$ is a discrete fibration! That's great too, since together these facts settle questions like 1)-3).

Morgan Rogers (he/him) (May 11 2021 at 16:28):

Yep! For another supply of examples, you can take $C$ and $D$ to be groups; any group homomorphism is a functor which turns out to induce an atomic geometric morphism (i.e. one with logical inverse image). The geometric morphism is (hyper)connected if and only if the morphism is surjective, and localic if and only if the morphism is injective, which is pleasant.

John Baez (May 11 2021 at 16:34):

Great - I like presheaves on groups, aka G-sets, because I'm basically an old-fashioned mathematician who likes stuff like G-sets and representations of groups and such.

Morgan Rogers (he/him) (May 11 2021 at 16:43):

I should also refer @Christian Williams to the fact that the direct image of a geometric morphism is cartesian closed or sublogical if and only if the geometric morphism is an inclusion, cf A4.2.9 and C3.1.8 in the Elephant, and the direct image functor is logical if and only if the morphism is an equivalence, so that's something of a dead end.
However, as I mentioned in previous discussion, there is a reasonable chance of the left adjoint which you're calling $\exists_F$ of having nice properties in the situations you're interested in.

Christian Williams (May 11 2021 at 17:13):

Morgan Rogers (he/him) said:

You could look at Corollaries 4.56 and 4.58 of Caramello's denseness conditions paper; these give conditions for the essential gm induced by $F$ to be locally connected, or equivalently for $F^*$ to be locally cartesian closed. The conditions are very fiddly, but more importantly don't appear to bear much resemblance to your situation.

It might be helpful to know that Riccardo Zanfa and I have worked out that the first condition of Corollary 4.56 there can be reduced to the necessary and sufficient condition given by Johnstone for the induced geometric morphism to be open (equivalently, for $F^*$ to be sublogical) in C3.1.3 of the Elephant.

yes, you mentioned this before, thanks. it's very good to know that fibrations are sufficient for their inverse to be LCC. good to hear that you've made new progress in this area too!

Christian Williams (May 11 2021 at 17:31):

Morgan Rogers (he/him) said:

I should also refer Christian Williams to the fact that the direct image of a geometric morphism is cartesian closed or sublogical if and only if the geometric morphism is an inclusion, cf A4.2.9 and C3.1.8 in the Elephant, and the direct image functor is logical if and only if the morphism is an equivalence

okay, I see the first fact in the book. how are you getting the second? I thought "logical iff equivalence" was only if it had a cartesian left adjoint

Morgan Rogers (he/him) (May 11 2021 at 18:03):

Direct image functors of geometric morphisms do have cartesian left adjoints (don't forget that that's what the right adjoint is called!)

John Baez (May 11 2021 at 18:05):

"that that's what" is one of those freaky English constructions...

Christian Williams (May 11 2021 at 19:27):

oh yes of course, thanks

Christian Williams (May 11 2021 at 19:31):

so there isn't much hope for $F^*$ to be logical, fine. I'll think about those weakenings involving inclusions. right now the main thing I'll think about is conditions on $F$ for $\exists_F$ to be locally cartesian closed.

John Baez (May 11 2021 at 20:51):

No, there's not much hope for $F_\ast$ to be logical.

John Baez (May 11 2021 at 20:51):

Am I confused here?

John Baez (May 11 2021 at 20:51):

I thought we'd just seen a bunch of fun examples where $F^\ast$ is logical.

Christian Williams (May 12 2021 at 02:41):

yeah, sorry I was mixed up about something. those F* are logical, so they must not have cartesian left adjoints, because otherwise as Morgan says they would be equivalences.