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[edit: removed, sorry]
[edit: removed, sorry]
Maybe a silly comment, I didn’t think about it too hard yet, but is it possible that the 2-functor $Set^-: Poset \to Topos_{ess}$ to the 2-category of Grothendieck toposes and essential geometric morphisms is reflective (has a left 2-adjoint)?
Thanks, Todd, for offering help. I had been doing too many things at once, and got myself mixed up here.
Maybe I still am. But I hope it’s getting better. So what about this “fix” here:
the functor $[-,Set] : Poset \to Topos$ factors through $TopSpace$
$[-,Set] : Poset \stackrel{Alexandrov}{\to} TopSpace \hookrightarrow Locale \stackrel{Sh}{\hookrightarrow} Topos \,.$The trouble with my original suggestion is that of course the second morphisms does not preserve all limits (it is left adjoint, instead) but the first one seems to.
The functor $TopSpace \hookrightarrow Locale$ isn’t really an inclusion (and essentially so, being not faithful), but you should be able to replace $TopSpace$ here with the category of sober spaces if that’s important. (This works because you start with $Poset$ instead of with $Proset$.)
Right, yes. Just the other day I have added that discussion to locale (or expanded on the discussion that had been there).
A while back Todd kindly made the following suggestion:
is it possible that the 2-functor $[-,Set] : Poset \to Topos_{ess}$ to the 2-category of Grothendieck toposes and essential geometric morphisms is reflective (has a left 2-adjoint)?
It took me a while to come back to this. But I seem to believe now that this suggestion is going exactly in the right direction.
A key step to see this while using standard facts about localic reflection should be theorem 4.2 in this article (a discussion of which I have by now added to Alexandrov space).
This says that a morphism $Alex P \to Alex Q$ of Alexandrov locales comes from a morphism $P \to Q$ of posets precisely if its inverse image has a left adjoint.
Now, one has to be a bit careful with the 2-category structures on Locale and Topos, but unless I am mixing it up, I think this means under the 2-fully faithful 2-functor $Sh : Locale \to Topos$ that:
a geometric morphism $[P, Set] \to [Q,Set]$ comes from a morphisms $P \to Q$ of posets precisely if it is essential.
Right? (I mixed up my variances and 2-variances in all possible ways in the last hour and am getting a bit tired, so I won’t mind if you give me a sanity check.)
(Unless I am wrong, there is probably also a much more direct way to see this. )
a geometric morphism $[P, Set] \to [Q, Set]$ comes from a morphism $P \to Q$ of posets precisely if it is essential.
Yes, that is precisely what I was thinking. (And yes, getting mixed up in variances here is as easy as falling off a log.) More exactly, my thinking was that
If $C$ and $D$ are small categories, then there is a contravariant equivalence between the category of essential geometric morphisms $Set^C \to Set^D$ and the category of functors $\widebar{C} \to \widebar{D}$ where $\widebar{C}$ denotes Cauchy completion, and
Posets are already Cauchy-complete.
To break down the first point a little more: the bicategory of profunctors between (Cauchy-complete) categories is equivalent to the bicategory of presheaf categories and cocontinuous maps. In more detail, a functor $F: C^{op} \to D^{op}$ induces a profunctor or bimodule $\hom_{D^{op}}(-, F-): D \times C^{op} \to Set$ that has a right adjoint bimodule $\hom_{D^{op}}(F-, -): C^{op} \times D \to Set$, so that we get an induced adjoint pair
$(F_!: Set^C \to Set^D) \dashv (F^\ast: Set^D \to Set^C)$in the bicategory of cocontinuous maps between presheaf categories, which gives an adjoint string
$F_! \dashv F^\ast \dashv F_\ast$in the bicategory of presheaf categories and functors. This $F_\ast$ is the essential geometric morphism. To go the other direction, starting with an adjoint string
$F \dashv G \dashv H: Set^C \to Set^D,$(i.e., starting with an essential geometric morphism $H$), the bicontinuous functor $G$ induces a functor
$Bicont(G, Set): Bicont(Set^C, Set) \to Bicont(Set^D, Set)$where $Bicont(Set^C, Set)$ is practically the definition of Cauchy completion $\widebar{C}$. So an essential geometric morphism $Set^C \to Set^D$ induces a functor $\widebar{C} \to \widebar{D}$. However, a morphism $\theta: H \to H'$ between essential geometric morphisms induces a mate $\theta^\dagger: G' \to G$ running in the opposite direction, so that what we get is an equivalence
$EssGeom(Set^C, Set^D) \to [\widebar{C}, \widebar{D}]^{op}$induced by $H \mapsto Bicont(G, Set)$. Of course, the codomain here is equivalent to $[\widebar{C}^{op}, \widebar{D}^{op}]$.
To continue with my train of thought when I first made that proposal to Urs: here is my current guess as to what is going on.
First, for $\mathcal{B}$ a 2-category, let $\mathcal{B}^{op}$ denote the 2-category we get by reversing 1-cells but not 2-cells; let $\mathcal{B}^{co}$ denote the 2-category we get by reversing 2-cells but not 1-cells, and let $\mathcal{B}^{coop}$ denote the 2-category that we get by reversing both.
I think I want to start with the observation or guess that $Set$ plays a role of ambimorphic object which in one role is a Cauchy complete category and which in another role is a bicomplete category. The idea is that we have an adjunction determined by this ambimorphic object, giving a local equivalence of categories
$\frac{B \to Cat_{Cauchy}[C, Set]}{C \to Bicont(B, Set)}$where the objects on top are bicontinuous functors $B \to Set^C$ ($B$ bicomplete), and the objects on the bottom are ordinary functors $C \to Bicont(B, Set)$. This means we have a biadjunction
$(Cat_{Cauchy} \stackrel{Set^{(-)}}{\to} Bicomp^{op}) \dashv (Bicomp^{op} \stackrel{Bicont(-, Set)}{\to} Cat_{Cauchy})$We can specialize this biadjunction by restricting bicomplete $B$ (as above) to toposes, noting the fact that $Set^C$ lands in toposes. In other words, if $Topos_{bicont}$ denotes the category of toposes and bicontinuous functors between them, the biadjunction above specializes to a second “ambimorphic biadjunction”
$(Cat_{Cauchy} \stackrel{Set^{(-)}}{\to} Topos_{bicont}^{op}) \dashv (Topos_{bicont}^{op} \stackrel{Bicont(-, Set)}{\to} Cat_{Cauchy})$Next: the process of taking right adjoints at the 1-cell level and mates at the 2-cell level should give us a biequivalence
$Topos_{bicont}^{coop} \to Topos_{ess}$and so, applying “co” to the second ambimorphic adjunction, we ought to get a biadjunction
$(Cat_{Cauchy}^{co} \to Topos_{ess}) \dashv (Topos_{ess} \to Cat_{Cauchy}^{co})$hoping here I haven’t made variance/level slips. (The “co” affects the directions of 2-cells but I suspect not the direction of biadjunctions, because the triangulator 2-cells in a biadjunction are isomorphisms and I think “co” here just amounts to inverting them. Even though I’m getting slightly nervous about variance, it does look right that the functor from toposes and essential geometric morphisms back to categories should be the right adjoint part, since this should be the category of essential points $EssGeom(Set, -)$, and this looks to be limit preserving.)
I am guessing that the process described as
$Poset \stackrel{Alex}{\to} Locale \to Topos$first of all factors through $Topos_{ess} \hookrightarrow Topos$, but could be alternatively described as
$Poset \to Cat_{Cauchy}^{co} \to Topos_{ess}$where the first functor is actually the assignment $P \mapsto P^{op}$. In other words, the 2-functor which takes a poset $P$ to its opposite $P^{op}$ is a 2-functor which preserves directions of 1-cells but reverses 2-cells, so gives
$Poset \stackrel{(-)^{op}}{\to} Poset^{co} \hookrightarrow Cat_{Cauchy}^{co}$which we follow with the aforementioned $Cat_{Cauchy}^{co} \to Topos_{ess}$.
Maybe I should pause for a sanity check myself: the thing you (Urs) proposed takes a poset $P$ to the locale whose frame is that of upward-closed subsets of $P$, and then we take sheaves on that locale. The thing I suggested takes $P$ to $P^{op}$ to $Set^{P^{op}}$. Hm… this seems a little twisted around; sheaves on the locale would be particular presheaves on the frame of up-sets $[P, 2]$, which would be functors $[P, 2]^{op} \to Set$, which would induce functors $P \to Set$, not $P^{op} \to Set$. Hm - what would be your take on using down-sets instead of up-sets here?
The more I look at this, though, the less sure I am of my original suggestion that there is an adjunction along the lines of what I suggested in comment 3., and I can’t remember what might have been proposed in comments 1. and 2.
Thanks a whole lot, Todd!
I need to think about some of the things you are saying here. For the moment I have started (just started) to record some statements at essential geometric morphism in a new section Properties – Relation to morphisms of (co)sites.
Concerning the very last bit:
Hm - what would be your take on using down-sets instead of up-sets here?
The equivalence $[P,Set] \simeq Sh(Alex P)$ really proceeds via
$Op(Alex P) \simeq UpperSets(P)$as far as I can see.
The more I look at this, though, the less sure I am of my original suggestion that there is an adjunction along the lines of what I suggested in comment 3., and I can’t remember what might have been proposed in comments 1. and 2.
The proposal in 1. and 2. was too naive as that I will repeat it here :-)
For the moment I’d be happy to postpone the discussion of preservation of limits for a bit and concentrate on the full and faithfulness of $[-,Set] : Poset \to Topos_{ess}$. One thing I need to sort out for myself still is how its action on morphisms works out through the factorization $[-,Set] \simeq Sh \circ Alex$.
The equivalence $[P,Set] \simeq Sh(Alex P)$ really proceeds via $Op(Alex P) \simeq UpperSets(P)$ as far as I can see.
Sorry, I didn’t make myself clear. I agree with this statement.
What I was trying to do was put together your chain of arrows from $Poset$ to $Topos_{ess}$ with my chain of arrows from $Poset$ to $Topos_{ess}$ and see whether there was agreement (i.e., a commutative diagram). I was coming to the conclusion that there wasn’t agreement, because my chain does this:
$P \mapsto P^{op} \mapsto Set^{P^{op}}$(the first arrow goes from $Poset$ to $Cat_{Cauchy}^{co}$, and the second goes from $Cat_{Cauchy}^{co}$ to $Topos_{ess}$). So I was trying to feel out if you were willing to use the other convention for the Alexandrov topology, where you take down-sets instead, and for which you’d have $Sh(Alex P) \simeq Set^{P^{op}}$ (which would then agree with my chain).
But now that I think it over further, I think it probably doesn’t matter much: there is a 2-equivalence $Cat_{Cauchy}^{co} \simeq Cat_{Cauchy}$. So for now you can ignore this particular remark, and I’ll mull over it privately.
For the moment I’d be happy to postpone the discussion of preservation of limits for a bit and concentrate on the full and faithfulness of $[-, Set]: Poset \to Topos_{ess}$.
I think that’s going to work out just fine. Using my chain at least (and I repeat that I don’t think any discrepancies between our chains will be a serious matter), it’s clear to me that $i: Poset \to Cat_{Cauchy}^{co}$ is full and faithful (i.e., the local functors $\hom(P, Q) \to \hom(i P, i Q)$ will be equivalences), and the map $j: Cat_{Cauchy}^{co} \to Topos_{ess}$ will be similarly “full and faithful” because it has a 2-coreflector $EssGeom(Set, -)$, i.e., the unit of the 2-adjunction, whose components look like
$C \to EssGeom(Set, Set^C),$is an equivalence for every Cauchy-complete $C$.
What I am worried about though is the fact that $i$ is a right 2-adjoint, where $j$ is a left 2-adjoint, as far as I can make out. That makes me worried about preservation of limits.
Hi Todd,
okay, I see. I am still a bit behind with following some details of what you describe, though I do follow the general strategy.
Right at the beginning, you may have to help me here: you say
$Bicont(Set^C,Set)$ is practically the definition of Cauchy completion
Why is that?
Why is that?
Ah, never mind, I see it now in the description of “points of the Cauchy completion”
Okay, I think I follow all the constructions now. What I do not quite see yet is how you deduce that some of them actually constitute equivalences.
For instance, how do you see that $j : Cat^{co}_{Cauchy} \hookrightarrow Topos_{ess}$ is co-reflective?
Meanwhile, I am trying to get hold of the book by Borceux and Dejan Cauchy completion in category theory . Is that a good idea? Or would you recommend something else?
Sorry: for someone who seems to talk about Cauchy completion a lot, I really don’t know the literature on it too well, in particular the Borceux-Dejan reference. I’m sure it can’t hurt.
So I was basically claiming that $Topos_{ess}(Set, -)^{op}$ is right adjoint to $j$. In more detail, if $E$ is a Grothendieck topos and $C$ is Cauchy complete, that there is an equivalence
$\frac{C \to Topos_{ess}(Set, E)^{op}}{Set^C \to E}$where the arrows on top are ordinary functors, and the arrows on bottom are essential geometric morphisms. This may look more recognizable if I rewrite it like this:
$\frac{C \to LRAdj(E, Set)}{E \to Set^C}$where $LRAdj(E, Set)$ denotes the category of functors $E \to Set$ which are simultaneously left and right adjoints, and the arrows on the bottom are functors that are simultaneously left and right adjoints.
I was furthermore claiming that for any (small) category $C$, that left-right adjoints $E \to Set^C$ are the same as bicontinuous functors $E \to Set^C$. It could be a rash claim, but let’s see. I know that Grothendieck toposes are total categories, and this would mean that left adjoints $E \to Set^C$ coincide with cocontinuous functors $E \to Set^C$. I guess I was also hoping that Grothendieck toposes are cototal. This I am less sure about, but it would mean dually that right adjoints $E \to Set^C$ coincide with continuous functors. If that is true, then left-right adjoints $E \to Set^C$ are the same as bicontinuous functors $E \to Set^C$.
So let’s see: total category informs me (by dualizing) that any complete, well-powered category with a cogenerator is cototal. Well, Grothendieck toposes $E$ are complete and well-powered, and if $S$ is a set of objects that generates $E$, then I assume that $\prod_{c \in S} \Omega^c$ cogenerates $E$. So for now that’s my argument that $E$ is cototal.
So now we are down to checking that functors $C \to Bicont(E, Set)$ are equivalent to bicontinuous functors $E \to Set^C$. I was hoping this was obvious: to check whether a functor $f: E \to Set^C$ preserves limits and colimits, it is enough to check that $f \circ ev_c: E \to Set$ preserves limits and colimits for every object $c$ of $C$. This means that given bicontinuous $f: E \to Set^C$, we can construct a functor $C \to Bicont(E, Set)$ that sends $c$ to $f \circ ev_c$. And so on.
Summarizing: there should be an equivalence
$\frac{functors: C \to LRAdj(E, Set)}{left-right-adjoints: E \to Set^C}$and this was supposed to justify the adjunction
$\frac{C \to Topos_{ess}(Set, E)^{op}}{Set^C \to E}$where $C \mapsto Set^C$ is left adjoint to $E \mapsto Topos_{ess}(Set, E)^{op}$.
The unit of the adjunction $C \to Topos_{ess}(Set, Set^C)$ is an equivalence precisely when $C$ is Cauchy complete. That gives the coreflection.
Hi Todd,
thanks again, that does help.
I was offline for a few hours (had to go to the cinema ;-). I’ll try to come back to this now, but maybe won’t get back to you before tomorrow morning.
Todd,
after having managed to get myself distracted from this discussion here in all possible kinds of ways (as you will have seen) I would like to finalize this now.
After having included much standard stuff from Borceux-Dejean into Cauchy complete category the other day I am now working on writing up the section In terms of essential geometric morphisms. I am doing this in a very explicit and pedestrian manner, since this is not just for me (not that I myself would not benefit from explicit and pedestrian discussion, of course, I am just saying this in case you are wondering why I am expanding your half-line remarks into lengthy arguments! :-).
So far I have mainly the statement and proof that
$\overline{C} \simeq Topos_{ess}(Set, Set^C)$Or do I? Because I get
$\overline{C} \simeq Topos_{ess}(Set, Set^C)^{op} \,.$Could you maybe check that? Because I get that the equivalence proceeds by sending $F \in \overline{C}$ to $f^* := [C,Set](F,-)$ and that means that a morphism $F \to G$ is sent to a tranformation $[C,Set](G,-) \to [C,Set](F,-)$ which is a geometric transformation $g \to f$.
(Sorry if I am falling off the variance-log here once again :-)
It’s $Topos_{ess}(Set, Set^C)^{op}$. I’m pretty sure I wrote that at some point, but I might have later forgotten to put in the $^{op}$. Edit: yes I included it in the third sentence of comment 15, and elsewhere in that comment.
Okay, good. Thanks.
Okay, I have now typed up the remainded of the argument: still in the subsection In terms of essential geometric morphisms.
By the way, I think I gave a needlessly complicated argument back in 15, because I didn’t need to get into all that cototality stuff. (Although I’m glad I did, because I learned something from it.)
Namely, we can see the equivalence
$\frac{functors: C \to LRAdj(E, Set)}{left/right-adjoints: E \to Set^C}$directly. Given a left/right-adjoint $f: E \to Set^C$, we get for each object $c$ of $C$ a left/right-adjoint
$E \stackrel{f}{\to} Set^C \stackrel{ev_c}{\to} Set$and so given such $f$, we get a functor $C \to LRAdj(E, Set)$ which sends $f$ to $ev_c \circ f$. And given a functor $g: C \to LRAdj(E, Set)$, we get for each object $e$ of $E$ a functor
$C \stackrel{g}{\to} LRAdj(E, Set) \stackrel{ev_e}{\to} Set$and so given such $g$, we get a functor $E \to Set^C$ that sends $e$ to $ev_e \circ g$. This has both a left adjoint $Set^C \to E$ and a right adjoint $Set^C \to E$. The left adjoint takes an object $F$ of $Set^C$ to the following object of $E$:
$\int^c l(c)(F(c)),$where $l(c) \dashv g(c)$. For a morphism $\int^c l(c)(F(c)) \to e$ corresponds to a wedge $l(c)(F(c)) \to e$ that is extranatural in $c$, which corresponds to a family $F(c) \to g(c)(e)$ that is natural in $c$, i.e., to a transformation $F \to ev_e \circ g$, as desired. Similarly, the right adjoint takes an object $F$ of $Set^C$ to the following object of $E$:
$\int_c r(c)(F(c)),$where $g(c) \dashv r(c)$. For a morphism $e \to \int_c r(c)(F(c))$ corresponds to a wedge $e \to r(c)(F(c))$ that is extranatural in $c$, which corresponds to a family $g(c)(e) \to F(c)$ that is natural in $c$, i.e., to a transformation $ev_e \circ g \to F$, as desired.
But isn’t it even simpler? Both sides are evidently equivalent to functors $C \times E \to Set$ that preserve all limits and colimits in the second argument.
But isn’t it even simpler?
But see, that brings us back to my first argument. We would have to convince ourselves that a functor $F: C \times E \to Set$ has left and right adjoints for each $F(c, -)$ if and only if each $F(c, -)$ preserves limits and colimits. To get that equivalence, we need co/totality of $E$ (or something similar).
But we know that the adjoint functor theorem applies to functors between toposes by standard facts?! Sorry, I may be missing your point.
Well, maybe I’m forgetting which standard facts. Could you tell me?
My verification of cototality amounted to a verification of one set of hypotheses for an adjoint functor theorem to apply (that guarantees when a limit-preserving functor has a left adjoint). If you have something shorter in mind, please remind me.
Just to add another comment to that: my second argument is purely conceptual, i.e., doesn’t rely on the technicalities of adjoint functor theorems. But I don’t mind – whichever way you’d prefer to do it is fine by me.
The adjoint functor theorem is known to hold for locally presentable categories (and accessible functors between them, I should add, but that’s okay for our case).
I think this is Theorem 1.66 of J. Adamek, J. Rosicky, “Locally Presentable and Accessible Categories”. But let me try to look it up again.
Also corollary 5.5.2.9 in HTT
Yes, but: you probably have in mind the version that guarantees existence of a right adjoint for a colimit-preserving functor. It is not in general true that a limit-preserving functor between locally presentable categories has a left adjoint. (Mike gave an example recently at MO of a limit-preserving functor $Grp \to Set$ that is not a right adjoint. I’ve fallen into this trap several times myself.) That’s why I went to the trouble of proving that toposes are cototal.
Here is the link to Mike’s example.
By the way, I have been chatting about this ($Cat_{Cauchy} \hookrightarrow Topos_{ess}$) also with Benno, and he remarks:
In the meantime I found “my own” proof: it identifies the Cauchy-completion $\overline{C}$ as the full subcategory of indecomposable projectives in $[C^op, Set]$ and one check that the further left adjoint $f_!$ sends indecomposable projectives to indecomposable projectives
It is not in general true that a limit-preserving functor between locally presentable categories has a left adjoint.
I am thinking: it is if it is accessible. Which it is in our case, since it preserves even all colimits.
Ah, okay, that works. Thanks.
As I say: anyway you want to do it is fine. I still like my second argument, because the other way requires some bigger guns (which you call ’standard facts’ – you do have to argue, as you’ve just done).
Okay. But quite generally we need to record these things better on the $n$Lab.
I have added to adjoint functor theorem
the version for locally presentable categories;
in the Examples-section: the statement about totality and cototality of toposes. There I say: See the discussion at topos. We should write out the arguments there.
Sorry, Grothendieck toposes.
I added a proof for the assertion of totality and cototality in Grothendieck topos.
Thanks! I have added some hyperlinks. We should really move this discussion to a more dedicated entry, though.
I wasn’t really following this discussion, unfortunately, but it seems to have come to a satisfactory conclusion. I like the observation that toposes are both total and cototal; it hadn’t entered my consciousness in that language before.
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