Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: deprecated: topos theory

Topic: relationship of localic and representably faithful

James Deikun (Apr 11 2022 at 13:35):

I've been able to prove that every localic morphism in $\mathrm{Topos}$ is representably faithful, but I haven't been able to either prove the converse or find a counterexample as yet. My strategy so far has been to prove epicness of the top arrow in the pullback square

$$\begin{CD} P @>>> B\\ @VVV @VVV\\ f^{*}f_{*}\tilde{B} @>{\epsilon_{\tilde{B}}}>> \tilde{B} \end{CD}$$

However, I've been unable to find geometric transformations lying around that can probe the arbitrary $B$, never mind relating that back to the arrow from $P$. The Elephant in A4.6.2(b) claims that $\epsilon$ being epic is strictly stronger than $f$ being localic, so I can't hope to prove that; maybe I can prove that it's epic specifically on partial map classifiers though? Anyone have any ideas?

Tim Campion (Apr 11 2022 at 14:33):

James Deikun said:

I've been able to prove that every localic morphism in $\mathrm{Topos}$ is representably faithful, but I haven't been able to either prove the converse or find a counterexample as yet. My strategy so far has been to prove epicness of the top arrow in the pullback square

$$\begin{CD} P @>>> B\\ @VVV @VVV\\ f^{*}f_{*}\tilde{B} @>{\epsilon_{\tilde{B}}}>> \tilde{B} \end{CD}$$

However, I've been unable to find geometric transformations lying around that can probe the arbitrary $B$, never mind relating that back to the arrow from $P$. The Elephant in A4.6.2(b) claims that $\epsilon$ being epic is strictly stronger than $f$ being localic, so I can't hope to prove that; maybe I can prove that it's epic specifically on partial map classifiers though? Anyone have any ideas?

What does "representably faithful" mean?

My first guess was: a geometric morphism $X \to Y$ is representably faithful if $Geom(A, X) \to Geom(A,Y)$ is faithful for every topos $A$, where $Geom(A,Z)$ is the category of geometric morphisms from $A \to Z$, where a morphism is a natural transformation between the left adjoints. But then if $X$ is a localic topos, and $Y = Set$ is sheaves on a point, then taking $A = Set$ as well, we see that for the localic morphism $Y \to X$ to be representably faithful, the specialization order on $Y$ must be codiscrete. So for lots of $Y$ this fails.

My next guess was: a geometric morphism $X \to Y$ is representably faithful if $Geom(Y,A) \to Geom(X,A)$ is faithful for every topos $A$. Now taking $A = Fun(FinSet, Set)$ to be the object classifier, we see this condition implies that the left adjoint to $X \to Y$ must be faithful. Now, a left or right exact functor between complete / cocomplete categories is faithful if and only if it is conservative, so we see that the left adjoint to $X \to Y$ must be conservative, i.e. $X \to Y$ must be a surjective geometric morphism. So if $X \to Y$ is a geometric morphism between localic toposes which is not surjective, it is not representably faithful in this sense.

Reid Barton (Apr 11 2022 at 14:36):

Tim Campion said:

My first guess was: a geometric morphism $X \to Y$ is representably faithful if $Geom(A, X) \to Geom(A,Y)$ is faithful for every topos $A$, where $Geom(A,Z)$ is the category of geometric morphisms from $A \to Z$, where a morphism is a natural transformation between the left adjoints. But then if $X$ is a localic topos, and $Y = Set$ is sheaves on a point, then taking $A = Set$ as well, we see that for the localic morphism $Y \to X$ to be representably faithful, the specialization order on $Y$ must be codiscrete. So for lots of $Y$ this fails.

I think this is what it means--isn't every functor from a poset to the terminal category faithful?

Tim Campion (Apr 11 2022 at 14:42):

Reid Barton said:

Tim Campion said:

My first guess was: a geometric morphism $X \to Y$ is representably faithful if $Geom(A, X) \to Geom(A,Y)$ is faithful for every topos $A$, where $Geom(A,Z)$ is the category of geometric morphisms from $A \to Z$, where a morphism is a natural transformation between the left adjoints. But then if $X$ is a localic topos, and $Y = Set$ is sheaves on a point, then taking $A = Set$ as well, we see that for the localic morphism $Y \to X$ to be representably faithful, the specialization order on $Y$ must be codiscrete. So for lots of $Y$ this fails.

I think this is what it means--isn't every functor from a poset to the terminal category faithful?

Oh of course you're right.

Tim Campion (Apr 11 2022 at 15:00):

I suppose the next thing to say is that by taking the hyperconnected-localic factorization of the terminal geometric morphism $A \to Set$, it suffices to consider the case where $A$ itself is a localic topos...

James Deikun (Apr 11 2022 at 15:15):

Tim Campion said:

What does "representably faithful" mean?

My first guess was: a geometric morphism $X \to Y$ is representably faithful if $Geom(A, X) \to Geom(A,Y)$ is faithful for every topos $A$, where $Geom(A,Z)$ is the category of geometric morphisms from $A \to Z$, where a morphism is a natural transformation between the left adjoints. [...]

This is in fact the meaning I am using, although I also cash it out in elementary terms:

$f : X \to Y$ is representably faithful if for each topos $A$ and geometric morphisms $g, h : A \to X$, one (hence both) of the following equivalent conditions obtain:

• for all $\eta, \lambda : g^{*} \rArr h^{*}$, whenever $\eta f^{*} = \lambda f^{*}$, then $\eta = \lambda$
• for all $\eta, \lambda : g_{*} \rArr h_{*}$, whenever $f_{*} \eta = f_{*} \lambda$, then $\eta = \lambda$.

James Deikun (Apr 11 2022 at 15:23):

The first elementary formulation above was particularly useful for proving that localic implies representably faithful.

Jens Hemelaer (Apr 11 2022 at 16:28):

Do you know whether every representably fully faithful morphism is an inclusion?

If this is the case, then we can maybe prove that representably faithful implies localic as follows.
Take a geometric morphism $f : \mathcal{F} \to \mathcal{E}$. To show that $f$ is localic, it is enough to show that the diagonal $\Delta_f : \mathcal{F} \to \mathcal{F}\times_{\mathcal{E}} \mathcal{F}$ is an inclusion (Elephant C2.4.14). We already know that $\Delta_f$ is localic (Elephant B3.3.8(i)), in particular representably faithful. It remains to show that $\Delta_f$ is representably full. However, by writing out the definitions, we see that $\Delta_f$ is representably full if and only if $f$ is representably faithful.

Jens Hemelaer (Apr 11 2022 at 16:30):

(I'm assuming $\mathrm{Topos}$ is the category of Grothendieck toposes and geometric morphisms between them, for elementary toposes we should be careful about boundedness assumptions.)

Jens Hemelaer (Apr 11 2022 at 16:33):

Johnstone gives a possible counterexample in the Elephant: the geometric morphism $f : \mathbf{Sets}/X \to \mathbf{Sh}(X)$ for any non-discrete topological space $X$. In this case, $f$ is not an inclusion, but maybe it is representably fully faithful.

James Deikun (Apr 11 2022 at 17:16):

Well, I do know that every inclusion (geometric embedding) is representably fully faithful, but I don't know the converse of that one either, I hadn't paid much attention to it since representably fully faithful is not the 'best' definition of fully faithful morphism in a lot of circumstances (e.g. enriched categories).

I think this mainly comes down to whether there are 'enough' geometric morphisms into a generic topos to show that a representably fully faithful geometric morphism has a fully faithful direct image.

Mike Shulman (Apr 11 2022 at 17:38):

One shouldn't expect every rep. ff geometric morphism to be a geometric inclusion, for the same reason not every monomorphism of topological spaces is a subspace inclusion.

James Deikun (Apr 11 2022 at 18:03):

It seems like this intuition leads to the same kind of counterexample for rep. ff -> geometric inclusion as was mentioned above: the surjection from the discrete topos on $X$'s set of points to the sheaf topos on $X$ itself, for a topological space $X$.

To turn this into a counterexample to rep. faithful -> localic, I'd guess would require an example of a (non-localic!) topos where the induced topology on the diagonal is coarser than the topology on the topos itself.

James Deikun (Apr 11 2022 at 18:07):

But this topos would still need to be 'thin' enough that the diagonal is representably ff.

James Deikun (Apr 11 2022 at 19:38):

James Deikun said:

It seems like this intuition leads to the same kind of counterexample for rep. ff -> geometric inclusion as was mentioned above: the surjection from the discrete topos on $X$'s set of points to the sheaf topos on $X$ itself, for a topological space $X$.

I'm having a bit of a hard time convincing myself that this is representably full tbh ...

Reid Barton (Apr 11 2022 at 21:51):

I don't think it is.

James Deikun (Apr 12 2022 at 00:16):

The functor $f: Set^{2} \to Set^{\to}$ of this type is given by: inverse image forgets the arrow, direct image is $(A_{0}, A_{1}) \mapsto (A_{0} \times A_{1}, A_{1}, \pi_{1})$. Maybe in this case we can at least find a counterexample to the supposed counterexample?

James Deikun (Apr 12 2022 at 02:26):

Anyone know what $\mathrm{Set}^{2}$ classifies? $\mathrm{Set}^{\to}$ classifies subterminal objects but it's on the wrong end of this geometric morphism for that to be especially useful here.

James Deikun (Apr 12 2022 at 12:13):

If I read Elephant B3.2.4(a) correctly, Set^2 classifies morphisms $1 \to 1 + 1$.

James Deikun (Apr 12 2022 at 12:25):

That means the above functor associates a subterminal object -- the preimage of one of the $1$ s, say the leftmost -- with each such morphism.

Reid Barton (Apr 12 2022 at 12:34):

So already in the case of Set-valued points, in one case we get a discrete category with two objects and in the other case we get two objects related by a morphism, right? That's not a fully faithful functor.

James Deikun (Apr 12 2022 at 12:41):

I think this doesn't completely dispose of the proposed counterexample, though, because the arrow that shows up corresponds to the fact that the Sierpinski space has a nontrivial specialization order. I think this means that finite topological spaces are kinda out as counterexamples though.

James Deikun (Apr 12 2022 at 12:52):

Next simplest example of this I can think of: $f : \mathrm{Set}/|X| \to \mathrm{Sh}(X)$ where $X$ is the one-point compactification of the countable discrete space.

Jens Hemelaer (Apr 12 2022 at 13:23):

I was thinking more in the direction of $X = [0,1]$ (the unit interval). Then $X$ is Hausdorff as a locale, so it follows that
$\mathbf{Geom}(\mathcal{E},\mathbf{Sh}(X))$ is a groupoid for each Grothendieck topos $\mathcal{E}$ (Elephant C3.6.9(b), C1.2.17(iii)). Points of localic toposes have no nontrivial automorphisms, so essentially $\mathbf{Geom}(\mathcal{E},\mathbf{Sh}(X))$ is just a set for each $\mathcal{E}$. The same holds for $\mathbf{Geom}(\mathcal{E},\mathbf{Sets}/X)$.

So to show that $\mathbf{Geom}(\mathcal{E},\mathbf{Sets}/X) \to \mathbf{Geom}(\mathcal{E},\mathbf{Sh}(X))$ is an equivalence, it remains to show that two maps $\mathcal{E} \to \mathbf{Sets}/X$ that define isomorphic maps $\mathcal{E} \to \mathbf{Sh}(X)$ are already isomorphic. I believe this is the case, but I'm a bit surprised about how difficult it is to write out a precise proof of this.

Jens Hemelaer (Apr 12 2022 at 13:27):

I think a possible proof would go like this: we can reduce to the case that $\mathcal{E}$ is localic, using the hyperconnected–localic factorization. Then we show that the inclusion $\mathcal{O}([0,1]) \to \mathcal{P}([0,1])$ is an epimorphism in the category of frames. Here $\mathcal{O}([0,1])$ is the frame of open subsets of $[0,1]$ and $\mathcal{P}([0,1])$ is the frame of all subsets of $[0,1]$, or in other words the frame of open subsets for the discrete topology.

Jens Hemelaer (Apr 12 2022 at 13:34):

So this should give an example of a representably fully faithful morphism that is not an inclusion. I hoped it would bring us closer to the construction of a representably faithful morphism that is not localic, but I don't see at the moment what the next step would be.

James Deikun (Apr 12 2022 at 13:38):

It seems like this argument should work for any Hausdorff topological space.

James Deikun (Apr 12 2022 at 13:39):

(and show something interesting whenever the space is not discrete)

James Deikun (Apr 12 2022 at 13:52):

Maybe it would help to prove something like 'grouplike localic geometric morphisms are representably f.f.' ... assuming that is true.

James Deikun (Apr 12 2022 at 13:55):

Then what would remain would be to find a topos whose diagonal is grouplike.

James Deikun (Apr 12 2022 at 13:55):

(but not an embedding)

James Deikun (Apr 12 2022 at 15:11):

James Deikun said:

Maybe it would help to prove something like 'grouplike localic geometric morphisms are representably f.f.' ... assuming that is true.

I think it isn't true, since $\mathrm{Set}^{2}$ is a grouplike localic $\mathrm{Set}$-topos but its global sections geometric morphism is not representably fully faithful.

Jens Hemelaer (Apr 12 2022 at 15:37):

James Deikun said:

It seems like this argument should work for any Hausdorff topological space.

Yes, but the topological spaces should be Hausdorff as locale, which is different from Hausdorffness as topological space. In both cases, $X$ is Hausdorff if and only if the diagonal $X \to X \times X$ is closed, but products are computed differently in the category of locales and the category of topological spaces. The two notions agree if $X$ is locally compact and sober, or if $X$ is completely metrizable, see Picado and A. Pultr, Notes on the products of locales, Proposition 4.4.1 here. I played it a bit too safe by taking $[0,1]$, for example the real line would work as well.

Ivan Di Liberti (Apr 12 2022 at 19:12):

The fact that if f is localic, then pt(f) is faithful appears as 4.4 in General facts on the Scott adjunction and I do not expect the inverse implication to be true. May I ask for what motivates your question?

James Deikun (Apr 12 2022 at 19:42):

The motivation is the considerable analogies between faithful functors in $\mathrm{Cat}$ and localic geometric morphisms, e.g.:

• their stability properties
• their place as the right class of an important factorization
• their place in a "tower" of conditions on 1-morphisms and their diagonals (Elephant C2.4.14)
• being a property that indicates relative 'thinness' of the preimage
• one actually being taken to the other by the $\mathrm{Psh}$ 2-functor

This all made me wonder if localic morphisms can be expressed as a kind of faithful morphism (like, for example, embeddings of topological spaces are 'a kind' of monomorphism) and if so what kind of extra condition, if any, is needed.

James Deikun (Apr 12 2022 at 19:55):

Also I certainly don't expect faithfulness on $\mathrm{Set}$-valued points alone to detect if a morphism is localic, but I thought faithfulness on all of the $\mathrm{Hom}$-categories going in just might be enough... I now think it's not enough but I'm still not 100% sure, and I still want to see what a "not quite localic" morphism would actually look like.

Jonas Frey (Apr 13 2022 at 15:59):

I discussed a very similar question with Mathieu Anel once (but in the $(2.1)$-category of toposes, not the $(2,2)$-category) and I think the étendue topos obtained by quotienting $Sh(\R)$ by the translation action of the (discrete) group of additive real numbers is a counterexample: it is "representably discrete" but not localic. But I don't know how to show it from the top of my head.

(Connes uses a very similar example in his intro to non-commutative geometry where he quotients $\R$ by $\mathbb{Q}$, that's where I got the example from. Toen discusses the same example in a sheaf-theoretic setting in section 1.4 here).

Mike Shulman (Apr 13 2022 at 16:24):

A $V$-enriched functor $f:C\to D$ can be defined to be "$V$-faithful" if each map on hom-objects $C(x,y) \to D(fx,fy)$ is a monomorphism in $V$. This is in general a stronger condition than the underlying ordinary functor being faithful (though it's equivalent if $V(I,-) : V\to \rm Set$ reflects monomorphisms, which happens quite often), and it can be expressed internally in the equipment $V \rm Prof$. If you take that latter expression and interpret it instead in the equipment of topoi, what do you get?

James Deikun (Apr 13 2022 at 18:39):

I was thinking of trying something like that "$V$-faithful" definition, didn't realize it was already in the literature. What I get out of it is that the counit of the adjunction between the inverse and direct images is epic, which is mentioned to be strictly stronger than localic in Elephant A4.6.2(b). I wonder how much stronger it is.

James Deikun (Apr 13 2022 at 18:40):

(Still digesting that étendue but it seems promising...)

James Deikun (Apr 14 2022 at 13:00):

A simple example of a geometric morphism that is localic but not "$V$-faithful" is the etale geometric morphism from $\mathrm{Set}/2$ to $\mathrm{Set}$. The components of the counit of $!^{*} \dashv \Pi_{!}$ on $\iota_{1}$ and $\iota_{2}$ (the coproduct injections into $2 \cong 1 + 1$) are not epic.

Morgan Rogers (he/him) (Apr 15 2022 at 09:11):

If the original question hasn't been answered in the next few days, I'll have a proper crack at it.

James Deikun (Apr 19 2022 at 10:55):

Just want to point out that while I think @Jonas Frey may have given a counterexample I'm having a pretty hard time understanding the maps into this thing well enough to confirm or deny ... the combination of the discrete topology on the continuum-sized group and the continuous topology on the space it's acting on is confusing my intuitions.

Jonas Frey (Apr 19 2022 at 18:00):

Regarding intuition, here's a point that is maybe helpful. Suppressing the canonical functor from top. spaces to toposes to lighten notation, I think we have
$\hom(X,\R/\Delta\R)=\hom(X,\R)/\sim$
for connected spaces $X$, where $\sim$ is the equivalence relation identifying functions that differ only by a constant, so e.g. continuous functions from $\R$ to $\R/\Delta\R$ are continuous functions to $\R$ mod translation.

To show that $\R/\Delta\R$ is representably discrete, the idea should be that $\R$ is representably discrete, and the group action is "free", so I suppose one could try to prove a more general statement that such a quotient is always representably discrete?

Morgan Rogers (he/him) (Apr 19 2022 at 18:54):

Why is $\mathbb{R}$ representably discrete?

Morgan Rogers (he/him) (Apr 19 2022 at 18:55):

This is true for morphisms from $\mathrm{Set}$ because the ordering on points is the refinement ordering, so the category of points is discrete for (sheaves on) any Hausdorff space. Similarly, if I consider functions to $\mathbb{R}$ from any other space, the ordering will be discrete because it lifts the pointwise ordering... to convince myself that it's true for toposes in general, I would need to do more work. Any hint?

Morgan Rogers (he/him) (Apr 19 2022 at 19:08):

I can probably use orthogonality. If I take any old Grothendieck topos, any morphism to $\mathrm{Sh}(\mathbb{R})$ must factor through the localic reflection of that topos, so the discreteness lifts as one would hope. Or dually, I use the fact that hyperconnected morphisms are "representably cofull and cofaithful" to arrive at the conclusion ;)

Morgan Rogers (he/him) (Apr 19 2022 at 19:11):

But I'm not yet convinced of @Jonas Frey's example, since as soon as we consider a topos which is not localic, the lifting fails. We could have morphisms to $\R/\Delta\R$ from non-localic toposes which do not factor through $\R$.

James Deikun (Apr 19 2022 at 20:34):

Pretty sure the identity is one such morphism.

Jens Hemelaer (Apr 19 2022 at 21:35):

Morgan Rogers (he/him) said:

Why is $\mathbb{R}$ representably discrete?

For each Grothendieck topos $\mathcal{E}$, we have that $\mathbf{Geom}(\mathcal{E},\mathbf{Sh}(\mathbb{R}))$ is both a poset, because $\mathbf{Sh}(\mathbb{R})$ is localic, and a groupoid, because $\mathbf{Sh}(\mathbb{R})$ is grouplike (being a Hausdorff locale). So then $\mathbf{Geom}(\mathcal{E},\mathbf{Sh}(\mathbb{R}))$ is always a discrete category.

Jens Hemelaer (Apr 19 2022 at 21:52):

I'll write $\mathbf{Sh}_{\mathbb{R}}(\mathbb{R})$ for the topos-theoretic quotient of $\mathbf{Sh}(\mathbb{R})$ by the translation action of the discrete group $\mathbb{R}$. Then there is an étale surjection $\mathbf{Sh}(\mathbb{R}) \to \mathbf{Sh}_{\mathbb{R}}(\mathbb{R})$, associated to some object $X$ in $\mathbf{Sh}_{\mathbb{R}}(\mathbb{R})$.

For every Grothendieck topos $\mathcal{E}$, the induced map
$\xi : \mathbf{Geom}(\mathcal{E},\mathbf{Sh}(\mathbb{R})) \to \mathbf{Geom}(\mathcal{E},\mathbf{Sh}_{\mathbb{R}}(\mathbb{R}))$
is then a discrete fibration (or maybe Street fibration, since we are working with categories up to equivalence). A morphism $g : \mathcal{E} \to \mathbf{Sh}(\mathbb{R})$ is the same thing as a pair $(f,x)$, with $f$ a geometric morphism $f : \mathcal{E} \to \mathbf{Sh}_{\mathbb{R}}(\mathbb{R})$, and $x$ a global element of $f^*X$.

For $\mathcal{E}=\mathbf{Sets}$, the functor $\xi$ is essentially surjective, and you can compute that
$\mathbf{Geom}(\mathbf{Sets},\mathbf{Sh}_{\mathbb{R}}(\mathbb{R}))$
is a discrete category with just one object. However, this strategy doesn't seem to work for more general toposes $\mathcal{E}$.

It would be great to have a description of $\mathbf{Geom}(\mathcal{E},\mathbf{Sh}_{\mathbb{R}}(\mathbb{R}))$, and not just for the current problem!

Jens Hemelaer (Apr 19 2022 at 21:56):

James Deikun said:

Pretty sure the identity is one such morphism.

Yes, I agree. Since $\mathbf{Sh}_\mathbb{R}(\mathbb{R})$ is hyperconnected, any geometric morphism from $\mathbf{Sh}_\mathbb{R}(\mathbb{R})$ to $\mathbf{Sh}(\mathbb{R})$ factors through $\mathbf{Sets}$. So if the identity factors through $\mathbf{Sh}(\mathbb{R})$, then it would also factor through $\mathbf{Sets}$, a contradiction.

Morgan Rogers (he/him) (Apr 20 2022 at 07:30):

It's a shame that Johnstone doesn't provide a concrete characterization of grouplike toposes in the Elephant; it's a very hard property to verify directly unless you can construct a theory classified by your topos.

James Deikun (Apr 21 2022 at 14:36):

Grouplike morphisms/toposes seem like a very odd creature that was defined just because it's easy to prove they are right orthogonal to other things, it wouldn't be surprising if they didn't have a really nice characterization. The best I've been able to come up with myself is that a morphism $f : \mathcal{E} \to \mathcal{S}$ is grouplike iff it's representably conservative as a (terminal) morphism in $\bold{Geom}/\mathcal{S}$ but that's basically just restating the definition.

Morgan Rogers (he/him) (Apr 21 2022 at 15:05):

Interesting... that reformulation reminds me of conditions for fibrations. So that's just as hard as the representably faithful case! Maybe @Jonas Frey has a conclusive proof in mind that will prove that this example is representably discrete.

Morgan Rogers (he/him) (Apr 21 2022 at 15:14):

In case it helps to indicate why finding counter-examples is hard, the equivalence of localic and representably faithful does hold for the global sections morphism of presheaf toposes, since being representably faithful means that for any theory $\mathbb{T}$ classified by such a topos, there is at most one morphism between any pair of models in any topos. In particular, since we can recover the category over which we're taking presheaves up to idempotent completion as the category of finitely presentable models in $\mathrm{Set}$, we conclude that the topos is equivalent to the category of presheaves on a preorder, which is necessarily localic.

The classifying topos perspective is interesting here. For a theory to be classified by a topos which is representably faithful over Set, we in particular need rigidity, which is to say that the only endomorphism of any model is the identity. I don't know of a systematic way to produce examples of such theories, but maybe a logician can chime in ;)