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Stream: deprecated: topos theory

Topic: translation dictionary: Category Theory ↔ Topos Theory

James Deikun (Mar 15 2022 at 17:59):

It's possible to use abstract category theory as practiced in a [[proarrow equipment]] to set up a correspondence between category theory and topos theory, so I started to do this. So far I have:

Category Theory	Topos Theory
category $C,D$	topos $\mathscr{E}, \mathscr{F}$
functor $F,G : C \rarr D$	geometric morphism $f,g : \mathscr{E} \rarr \mathscr{F}$
profunctor $P,Q : C ⇸ D$	lex functor $H,K : \mathscr{F} \rarr \mathscr{E}$ (note direction)
natural transformation $\alpha : P \rArr Q$	natural transformation $\alpha : K \rArr H$ (note direction)
$D(1,F)$	$f^{*}$
$D(F,1)$	$f_{*}$
$P(F,G)$	$g^{} \circ H \circ f_{}$
collage of $P$	Artin gluing of $H$
cocone under $C$	Freyd cover of $\mathscr{E}$
cone over $C$	Artin gluing of $LConst$ (sometimes called $\Delta$ )
$1$	$Set$
$2$ (walking arrow)	Sierpinski topos = $Set^{2}$
$!$	$\Gamma$ (global sections functor)
$0$	Initial topos
Cauchy completeness	(trivial)
presheaf	lex functor $C \rarr Set$
representable presheaf	lex left adjoint functor $C \rarr Set$
object $A,B : C$	point $p,q$ of $\mathscr{E}$
arrow $f : A \rarr B$	??? (geometric morphism from the Sierpinski topos mapping the open point to $p$ and the closed point to $q$ )
walking arrow classifies analyses of $C$ as collage	Sierpinski topos classifies analyses of $\mathscr{E}$ as Artin gluing
fully faithful functor	geometric embedding
$P$ -weighted limit of $F$	??? (a geometric morphism whose inverse image is $\mathrm{Lift}_{H} f^{*}$ )
conical limit of $F$	??? (a geometric morphism whose inverse image is $\mathrm{Lift}_{LConst} f^{*}$
terminal object	focal point (this should agree with the above but I haven't checked)
pointwise $\mathrm{Ran}_{g} f$	??? (a geometric morphism whose inverse image is $\mathrm{Lift}_{H_{}} f^{}$ )

Can anyone identify any of the "???" or anything left out?

Christian Williams (Mar 15 2022 at 18:35):

Nice! Are you getting this from Proarrows II?

James Deikun (Mar 15 2022 at 19:02):

Christian Williams said:

Nice! Are you getting this from Proarrows II?

I basically worked it out myself based on the definitions in the nLab ... on reading Proarrows II it mentions the interesting fact that lex functors are actually the codiscrete cofibrations in TOP ... which is a nice confirmation that they are actually the right thing to use as the analogue to profunctors, but that seems highly overdetermined at this point.

No doubt I'm getting some of that indirectly through the nLab too ...

James Deikun (Mar 15 2022 at 19:18):

Adding:

Category theory	Topos theory
sieve in $C$	open subtopos of $\mathscr{E}$
cosieve in $C$	closed subtopos of $\mathscr{E}$

James Deikun (Mar 15 2022 at 19:48):

BTW although little of this is in Proarrows II in particular I'd be surprised if much of it was truly original so references are welcome as well ...

Morgan Rogers (he/him) (Mar 15 2022 at 20:07):

How do you prove that the analogue of fully faithful functors are the embeddings? They're certainly internally fully faithful, but do you know how to prove the converse?

James Deikun (Mar 15 2022 at 20:16):

I'm using the definition of fully faithful from https://ncatlab.org/nlab/show/2-category+equipped+with+proarrows#fully_faithful_arrows and computing it out with the above dictionary, get that the definition is that $f^{*} \circ f_{*}$ is canonically isomorphic to $id$ via the counit of the adjunction $f^{*} \dashv f_{*}$ which is the second equivalent condition at https://ncatlab.org/nlab/show/geometric+embedding#definition ...

Mike Shulman (Mar 15 2022 at 20:20):

Right -- in particular, a "fully faithful arrow" in a proarrow equipment is a stronger condition than being representably fully faithful in its 2-category of arrows.

Mike Shulman (Mar 15 2022 at 20:21):

I think the ??? corresponding to $f:A\to B$ should be just "a morphism of points". The points of any topos form a category. (In the case of a classifying topos of some theory, the morphisms are the morphisms of models of that theory.)

Mike Shulman (Mar 15 2022 at 20:22):

Although, since the proarrow equipment of topoi is not "well-pointed" the way Cat/Prof is, the points and their morphisms (with terminal domain) are generally less important.

James Deikun (Mar 15 2022 at 20:30):

Indeed; I similarly expect that for limits and colimits the ones that are properly proarrow-weighted rather than merely (co)presheaf-weighted are going to be important.

Mike Shulman (Mar 15 2022 at 20:36):

I look forward to hearing what those turn out to be!

Mike Shulman (Mar 15 2022 at 20:36):

It might be interesting already to ask what they are in the simpler proarrow equipment of locales.

Graham Manuell (Mar 15 2022 at 23:36):

You might also be interested in this paper by Susan Niefield.

Morgan Rogers (he/him) (Mar 16 2022 at 07:49):

James Deikun said:

Christian Williams said:

Nice! Are you getting this from Proarrows II?

I basically worked it out myself based on the definitions in the nLab ... on reading Proarrows II it mentions the interesting fact that lex functors are actually the codiscrete cofibrations in TOP ... which is a nice confirmation that they are actually the right thing to use as the analogue to profunctors, but that seems highly overdetermined at this point.

No doubt I'm getting some of that indirectly through the nLab too ...

Be patient with me, because this is my first time looking into proarrow equipments, but how is this overdetermined? Wood uses the (bi)functor $\mathrm{Top} \to \mathrm{TopLex}^{\mathrm{co}}$ which forgets the inverse image functor as his proarrow equipment. Something which would seem much more natural to me would be the (bi)functor $\mathrm{Top} \to \mathrm{TopLex}^{\mathrm{op}}$ which forgets the direct image functor. Why is the former the one you choose? Is it actually the one you use?

Morgan Rogers (he/him) (Mar 16 2022 at 07:52):

More generally, I'm sure there are other possible proarrow equipments for toposes. You can't present a table like the one above without saying which one you're using!

Nathanael Arkor (Mar 16 2022 at 12:09):

More generally, I'm sure there are other possible proarrow equipments for toposes.

In many cases, the proarrow equipment structure is essentially unique, and this is true in this example. See Rosebrugh–Wood's Proarrows and cofibrations.

Graham Manuell (Mar 16 2022 at 12:24):

Though Morgan's original point is right up to taking opposite categories! See Rosebrugh–Wood's Cofibrations in the bicategory of topoi. :P

Morgan Rogers (he/him) (Mar 16 2022 at 12:33):

Nathanael Arkor said:

More generally, I'm sure there are other possible proarrow equipments for toposes.

In many cases, the proarrow equipment structure is essentially unique, and this is true in this example. See Rosebrugh–Wood's Proarrows and cofibrations.

The conditions on the proarrow equipment $(-)_*: K \to M$ that they use in this paper which result in essential uniqueness are...

$M$ has all finite collages. All collage injections $i: A \to C$ are in $K$ . An arrow $C \to X$ is in $K$ if and only if all $A \to C \to X$ are in $K$ . Applying $(-)^*$ to a collage diagram yields an opcollage diagram.

That's a lot of non-trivial conditions, certainly enough to support my argument that you should specify which equipment you're using.

James Deikun (Mar 16 2022 at 12:50):

Morgan Rogers (he/him) said:

Be patient with me, because this is my first time looking into proarrow equipments, but how is this overdetermined? Wood uses the (bi)functor $\mathrm{Top} \to \mathrm{TopLex}^{\mathrm{co}}$ which forgets the inverse image functor as his proarrow equipment. Something which would seem much more natural to me would be the (bi)functor $\mathrm{Top} \to \mathrm{TopLex}^{\mathrm{op}}$ which forgets the direct image functor. Why is the former the one you choose? Is it actually the one you use?

I'm actually using the functor $\mathrm{Top}^{\mathrm{co}} \to \mathrm{TopLex}^{\mathrm{coop}}$ which forgets direct image, as it seems to be the most natural in this case, hence all the (note direction) above ... the inverse image functor is a left adjoint. I want it to be the 'forward' direction because it is more important in applications, and I want it to remain the left adjoint because that's what's proper for 'forward' in equipments. However in $\mathrm{TopLex}^{\mathrm{op}}$ it is a right adjoint, so the extra $^{co}$ is needed.

Now I guess I need to look into Proarrows and cofibrations to see what I might be missing by not using their orientation ...

Morgan Rogers (he/him) (Mar 16 2022 at 13:14):

James Deikun said:

However in $\mathrm{TopLex}^{\mathrm{op}}$ it is a right adjoint, so the extra $^{co}$ is needed.

Ah I see, variance being tricky as usual :joy: I'm glad I took the time to ask!

Mike Shulman (Mar 16 2022 at 15:11):

Morgan Rogers (he/him) said:

That's a lot of non-trivial conditions, certainly enough to support my argument that you should specify which equipment you're using.

It looks like a lot of conditions when expressed in terms of a bo-lff functor $(-)_* : K\to M$ , but from an enriched perspective it's really just a cocompleteness condition.

Morgan Rogers (he/him) (Mar 16 2022 at 18:56):

So @Mike Shulman is it the case that the difference between the proarrow equipments we've been discussing is that they satisfy a completeness and cocompleteness condition respectively? Are they "the same" equipment in some sense, up to suitably dualizing the bicategories involved? This is still unclear to me.

Graham Manuell (Mar 16 2022 at 19:05):

I don't know for certain, but I think the notion of proarrow equipment is not self-dual and the one is a proarrow equipment for the category and the other is for its opposite category. I don't think they are 'equivalent' equipments.

Mike Shulman (Mar 16 2022 at 21:03):

Let's see. Both kinds of 2-categorical duality reverse the handedness of internal adjoints, i.e. a left adjoint in $C$ is a right adjoint in $C^{op}$ and $C^{co}$ , and a left adjoint in $C^{coop}$ . Thus, if $K\to M$ is a proarrow equipment, then $K^{op} \to M^{op}$ and $K^{co} \to M^{co}$ are not, but $K^{coop} \to M^{coop}$ is.

On the other hand, if $(-)_* : K\to M$ is an equipment, i.e. a bo+lff functor such that each $f_*$ has a right adjoint $f^*$ , then since adjoints point in the opposite direction and passage to adjoints reverses 2-cells, we also have a functor $(-)^* : K\to M^{coop}$ . By the above, each $f^*$ is still a right adjoint in $M^{coop}$ , so this is not an equipment. But applying one more duality of either sort fixes that, so $(-)^* : K^{op} \to M^{co}$ and $(-)^* : K^{co} \to M^{op}$ are both equipments (and, of course, one is the coop of the other).

In other words, the Klein 4-group acts on equipments, but the action isn't simply lifted from the action on 2-categories: it acts differently on $K$ and $M$ .

Now there is a "tautological" equipment $\rm Adj \to Cat$ , where $\rm Adj$ is the 2-category of categories and adjunctions pointing in the direction of their left adjoints. Thus, we also have equipments $\rm Adj^{coop} \to Cat^{coop}$ , $\rm Adj^{op} \to Cat^{co}$ , and $\rm Adj^{co} \to Cat^{op}$ , where the first two send an adjunction to its left adjoint, and the second two send an adjunction to its right adjoint.

It sounds like in this conversation we're defining $\rm Top$ as a subcategory of $Adj^{op}$ , with geometric morphisms pointing in the direction of their right adjoint, but with 2-cells defined as going between their left adjoints. Thus, we have equipments $\rm Top^{op} \to TopLex$ , $\rm Top^{co} \to TopLex^{coop}$ , $\rm Top \to TopLex^{co}$ , and $\rm Top^{coop} \to Cat^{op}$ . Since they constitute an orbit of the Klein-4 action, they all carry the same information.

Graham Manuell (Mar 16 2022 at 21:39):

Hmm. This is interesting. How this this square with Roseburgh and Wood's comment on the second page of "Cofibrations in the bicategory of topoi" that "the proarrow equipment $\mathrm{TOP} \to \mathrm{LEX}^\mathrm{co}$ enjoys more of the properties of $\mathrm{CAT} \to \mathrm{Prof}$ than does $\mathrm{TOP}^\mathrm{op} \to \mathrm{LEX}$ "? Does this mean they were mistaken here or am I misunderstanding something?

Mike Shulman (Mar 16 2022 at 21:48):

"Carrying the same information" doesn't mean that they're the same or have the same properties. $\rm Set$ and $\rm Set^{op}$ carry the same information, but one is a topos and one isn't.

Graham Manuell (Mar 16 2022 at 21:49):

I see. Fair enough.

Morgan Rogers (he/him) (Mar 17 2022 at 07:42):

Great, thanks for the analysis Mike! To sum up, once we've decided which of $K$ , $K^{op}$ , $K^{co}$ or $K^{coop}$ we're attaching the equipment to, if we insist that the equipment satisfies a completeness property, it is essentially unique, and we can obtain the other equipments via the transformations Mike described. But you do still need to specify the variance (which would maybe make it easier to parse the directions of the arrows you describe @James Deikun)

James Deikun (Mar 17 2022 at 13:49):

A few more things:

Category theory	Topos theory
$D(-,F=)$	$f^{*}$
$D(F-,=)$	$f_{*}$
bijective-on-objects functor	geometric surjection [Cofibrations in the bicategory of topoi]
faithful functor	localic geometric morphism (conjectured)
conservative functor	??? (maybe locally connected g.m.?)
fibration	???
discrete fibration	???
cofunctor	???

I'm still digesting the Niefield, there's a lot of implications but it's hard for me to nail them down, thanks for that though @Graham Manuell ...

I'm conjecturing "localic" for "faithful" as it makes sense intuitively, fits into the existing factorization system, and $\mathrm{Psh}$ takes faithful functors to localic geometric morphisms. And anyway I'm not aware of a definition of "faithful" in equipments.

The equivalent of conservative functors is fairly important as it helps nail down what discrete objects look like, it has to be something that is true of the global sections functor of a Boolean topos but not for general presheaf toposes. (So in particular "essential" is out.) Locally connected has the right "feel" to me but I can't yet justify it beyond that.

James Deikun (Mar 17 2022 at 14:13):

Actually "totally disconnected" feels even better except the definition of "totally disconnected geometric morphism" confuses me as it looks more like what I'd expect for totally path-disconnected ...

Mike Shulman (Mar 17 2022 at 17:50):

There's a sense in which localic morphisms are analogous to faithful functors, but I'm pretty sure it's only an analogy, that not every representably faithful morphism in Top is localic. And, as you said, it's not clear whether there is a way to use equipments to define a better notion of "faithful", the way there is for "fully faithful".

Mike Shulman (Mar 17 2022 at 17:50):

I think a better analogy is that localic morphisms are to faithful functors the same way that subspace inclusions are to injective functions.

Mike Shulman (Mar 17 2022 at 17:52):

I don't know what your goals are with fibrations, but ordinary representable fibrations in Top are already interesting. Peter Johnstone did some work with them, e.g. his paper "Fibrations and partial products in a 2-category" I think.

Mike Shulman (Mar 17 2022 at 17:53):

It's not clear to me whether to expect a good analogue of conservative functors in Top.

James Deikun (Mar 17 2022 at 18:39):

Hm, looks like Johnstone did a lot of good work on characterizing the representable fibrations and opfibrations in "Fibrations and partial products in a 2-category" but it didn't lead to the kind of nice punchine that Rosebrugh and Wood got for cofibrations. It seems most of the further work on this is by Sina Hazratpour and focuses on reproducing the existing sufficient characterizations by improved methods rather than improving the results themselves...

Mike Shulman (Mar 17 2022 at 18:44):

Well, not everything has an equally nice punchline...

James Deikun (Mar 17 2022 at 18:47):

True. I think the "faithful" case does though. Adding to the evidence: eso+full functors give rise to hyperconnected morphisms under Psh and (hyperconnected,localic) is an OFS echoing (eso+full,faithful).

James Deikun (Mar 17 2022 at 18:49):

It leads me to think there is a nice abstract definition of "faithful" even if I don't know what it is ...

Mike Shulman (Mar 17 2022 at 18:55):

If there is an abstract notion that specializes to the localic maps, I wouldn't call it "faithful" but rather something like "strongly faithful", analogously to how the subspace inclusions in spaces are not the monomorphisms. In that regard, I suppose you could try to generalize notions like "strong monomorphism" or "regular monomorphism" to 2-categories. I don't know if anyone's tried that.

James Deikun (Mar 17 2022 at 19:28):

Example 4 in [[generalized kernel]] maybe?

James Deikun (Mar 17 2022 at 19:31):

I don't know if $\mathrm{Top}^{\mathrm{co}}$ has something as nice as coequifiers though!

Mike Shulman (Mar 17 2022 at 19:42):

I this it probably does. Section B3.4 in Sketches of an Elephant constructs lots of colimits in Top, and I don't know if coequifiers are among them but I expect a similar method would work (or they could be constructed out of the colimits already mentioned).

Morgan Rogers (he/him) (Mar 18 2022 at 11:42):

I'd like to say before this thread goes quiet that in spite of my questions which may have come across as very sceptical, I'm very interested in this effort, and I look forward to seeing the results recorded more formally somewhere!