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Stream: learning: questions

Topic: on displayed categories vs. cats over C

fosco (Aug 06 2021 at 11:55):

I have used the "displayed category" construction many times, but I realised I have forgot some of the details of how it works. Which is shameful, because I wrote something about it in "Coend calculus", and promptly forgot about how to dot all the i's :grinning:
I guess this result is really doomed to be folklore...

The idea is simple, given a category $C$ and a normal lax functor $F : C \to {\sf Prof}$ one would like to perform a "generalised" Grothendieck construction to $F$ in order to obtain a category over $C$; the nLab (and I) claim that this can be obtained simply by considering the pullback

$\begin{array}{ccc} \Pi(F) & \to & Prof_* \\ \downarrow & & \downarrow \\ C & \to & Prof \end{array}$

where ${\sf Prof}_*$ is the bicategory of pointed profunctors, a pointed profuncton being a profunctor of pointed categories with a distinguished element in $p(obj_1, obj_2)$.

Now, a first question is: where is this pullback considered? I've always thought that one takes into consideration the categories ${\sf Prof}_*$ and ${\sf Prof}$, but now I re-did the math because I need to invoke a particular case of this and I'm unsure about whether this truncation procedure is effective.

It seems to erase precisely the information needed in order to restrict the morphisms of $\Pi(F)$ to a subclass of those of $C$!

Indeed, from the pullback(-like) description of $\Pi(F)$ it is quite easy to deduce that the objects of $\Pi(F)$ are pairs $(c, a)$ where $a$ is an object of the category $Fc$; now, what about morphisms $(c,a)\to (c',b)$? A morphism $\varphi : c \to c'$ in $C$ goes to a profunctor $F\varphi : (Fc)^\text{op} \times Fc' \to {\sf Set}$; this profunctor is in fact pointed, thus there is a distinguished element (so, at least one element) in $F\varphi(a,b)$. Here, I am stuck:

How to translate the condition that in a category of elements I want only the morphisms that preserve the distinguished element?

If $F$ was valued into $\sf Cat$, I would have just asked that the distinguished object $a$ goes into the distinguished object $b$; here, dealing not with a functor, but with a profunctor, I think the most I can ask is that the distinguished objects $a$ and $b$ are in a "generalised relation" expressed by $F\varphi$, i.e. that $F\varphi(a,b)$ is inhabited, which is ensured by the request that it is a pointed set.

However, now I am confused: what exactly is the condition I have put on $\varphi$ to be a morphism in $\Pi(F)$? And where did I need the normality of $F$, or even the laxity?

I tried to work out a particular case, without much success. Let's say that $F$ is in fact a functor; so, let's say that every $F\varphi : (Fc)^\text{op} \times Fc' \to {\sf Set}$ is a representable profunctor $\hom(\tilde F,1)$, in the sense that for every $(a,b)\in (Fc)^\text{op} \times Fc'$ is of the form $Fc'(\tilde Fa,b)$ for some functor $\tilde F : Fc \to Fc'$. (a,b) are now the distinguished objects of the categories $Fc, Fc'$, and the existence of a distinguished element in $F\varphi(a,b)$ means that there exists a distinguished arrow $\varphi^! : \tilde Fa\to b$.

Now, the natural guess would be that this arrows $\varphi^!$ are subject to a few coherence requests. Given composable morphisms $\psi,\varphi$ of $C$ for example, one can wonder what relates $\tilde F\psi(\tilde F\varphi a) \to b$ and $\tilde F(\psi\phi)a\to b$. They are equal, because $\tilde F$ is a functor (and thus $F$ was a pseudofunctor)!

Is this correct? And what is it saying about the general statement? In short,

What are the morphisms of $\Pi(F)$ "sending" the distinguished object $a$ in $Fc$ to $b\in Fc'$ via a profunctor $(Fc)^\text{op} \times Fc' \to {\sf Set}$?

Nathanael Arkor (Aug 06 2021 at 13:35):

Now, a first question is: where is this pullback considered? I've always thought that one takes into consideration the categories ${\sf Prof}_*$ and ${\sf Prof}$, but now I re-did the math because I need to invoke a particular case of this and I'm unsure about whether this truncation procedure is effective.

The nLab page says that it's a strict pullback in the 2-category of bicategories and lax functors. Note that it doesn't make sense to consider Prof and friends as categories, because a bicategory doesn't have an underlying category.

fosco (Aug 07 2021 at 10:14):

Sure, I probably explained it in a confused way

I guess I overlooked the fact that I don't have any idea

1. of a reference showing that the 2-category of bicategories and lax functors (with what 2-cells?) has pullbacks
2. Why $\Pi F$ is a category (well, I guess because $C$ is), and what exactly are its morphisms (a subclass of $\hom C$, but which ones?).

Mike Shulman (Aug 07 2021 at 15:17):

The 2-category of bicategories and lax functors doesn't have all pullbacks. Presumably the claim is that this particular pullback does exist, and is constructed by taking pullbacks of objects and of homsets (which tells you what the morphisms in $\Pi F$ are). It may not be obvious that this pullback exists or how to compose the thereby-defined morphisms, but I think the answer to both questions has something to do with $\rm Prof_* \to Prof$ being a strict functor and a local discrete opfibration.

Jade Master (Aug 07 2021 at 16:58):

@fosco for your morphism $\phi$ why is $F(\phi)$ pointed? Doesn't F land in ordinary profunctors not pointed ones?

Jade Master (Aug 07 2021 at 17:01):

@Joe Moeller and I actually wrote up some relatively detailed notes about the displayed category construction...Joe would you be okay with sharing them?

Graham Manuell (Aug 07 2021 at 17:03):

I don't really have anything to say about how the pullback works in particular, but incidentally I'm currently writing a paper aimed at an audience without extensive knowledge in category theory where I use the equivalence between functors into $C$ and normal lax functors $C \to {\rm Prof}$, so I am trying to include an explanation of that. It is unfortunate that all the details don't seem to have been written down anywhere (especially when it comes to the functoriality of the equivalence). I'm still deciding exactly how much detail would be appropriate me to include, but perhaps the paper could be at least a somewhat helpful resource for this when I am done.

I can tell you that the objects of $\Pi F$ are pairs $(c,c')$ where $c$ is an object of $C$ and $c'$ is an object of $F(c)$ and morphisms from $(d,d')$ to $(e,e')$ of $\Pi F$ are pairs $(f,f')$ where $f\colon d \to e$ is a morphism in $C$ and $f' \in F(f)(d',e')$.

Jade Master (Aug 07 2021 at 17:04):

Another question I have for Fosco is why the morphisms of $\Pi(F)$ should be a subclass of $C$...in general it seems to me that the total category could have many more morphisms.

Jade Master (Aug 07 2021 at 17:06):

@Graham Manuell your description seems exactly right to me...and the lack of a reference is why Joe and I wrote the notes. But the problem is they never made it out of the appendix of a paper we never finished :)

Graham Manuell (Aug 07 2021 at 17:07):

Jade Master said:

Joe Moeller and I actually wrote up some relatively detailed notes about the displayed category construction...Joe would you be okay with sharing them?

Perhaps I spoke too soon about them not being written down anywhere, though at least I couldn't find anything online :P. I do think having something like this online would be useful.

Jade Master (Aug 07 2021 at 17:07):

Haha yeah we haven't shared them with many people :sweat_smile:

Jade Master (Aug 07 2021 at 17:07):

We should!

Jade Master (Aug 07 2021 at 17:08):

Let me get Joe's permission first.

Graham Manuell (Aug 07 2021 at 17:16):

If you do make this publicly available, it does make me think maybe I should be citing your writeup in my paper instead of redoing it all myself, though on the other hand, having multiple accounts probably can't harm. Perhaps it would allow me to focus on the intuition and go into slightly less detail than I was planning on doing.

fosco (Aug 07 2021 at 18:51):

Jade Master said:

Another question I have for Fosco is why the morphisms of $\Pi(F)$ should be a subclass of $C$...in general it seems to me that the total category could have many more morphisms.

The morphisms of a category of elements are particular morphisms of the base: they are the point-preserving morphisms

fosco (Aug 07 2021 at 18:54):

Jade Master said:

fosco for your morphism $\phi$ why is $F(\phi)$ pointed? Doesn't F land in ordinary profunctors not pointed ones?

In the pullback, you have to consider pairs of objects $(c, \mathcal A) \in C \times {\sf Prof}_*$ (so $\mathcal A$ is a pointed category) and pairs of morphisms $(\varphi, \mathfrak p) \in C \times {\sf Prof}_*$ (so $\mathfrak p$ is a pointed profunctor), with the property that $\mathfrak p = F\varphi$, so the profunctor which is the image of $\varphi$ under $F : {\cal C} \to {\sf Prof}$ is in fact pointed.

Jade Master (Aug 07 2021 at 19:21):

fosco said:

Jade Master said:

fosco for your morphism $\phi$ why is $F(\phi)$ pointed? Doesn't F land in ordinary profunctors not pointed ones?

In the pullback, you have to consider pairs of objects $(c, \mathcal A) \in C \times {\sf Prof}_*$ (so $\mathcal A$ is a pointed category) and pairs of morphisms $(\varphi, \mathfrak p) \in C \times {\sf Prof}_*$ (so $\mathfrak p$ is a pointed profunctor), with the property that $\mathfrak p = F\varphi$, so the profunctor which is the image of $\varphi$ under $F : {\cal C} \to {\sf Prof}$ is in fact pointed.

Oh okay. I think more accurately you have that $U(\mathfrak{p})= F(\phi)$ where $U : Prof_* \to Prof$ is the forgetful functor, forgetting the points of categories and profunctors. In my opinion, the key to understanding this pullback is that you are thinking of the points of $\mathfrak{p}$ as all there is. So yes the morphisms of the pullback are pairs $(\phi, \mathfrak{p})$ with $U(\mathfrak{p})= F(\phi)$ but the part of $\mathfrak{p}$ that we really care about is the point of that profunctor call it f. So it's like the rest of profunctor is just along for the ride to make sure that f has the right type.

Jade Master (Aug 07 2021 at 19:29):

This is the same situation as with the ordinary Grothendieck construction by the way...for a functor $F: C \to Cat$ you can get its total category as the pullback of $F$ with the forgetful functor $Cat_* \to Cat$. The elements of this pullback are pairs (c,D) with c and object of C and with D a pointed category with U(D) = F(c). However most descriptions of the Grothendieck construction only care about the point x of D...and the condition for the pullback requires that this point lives in F(c). So people usually just write this pair as (c,x).

Jade Master (Aug 07 2021 at 19:29):

Am I misunderstanding your question?

Mike Shulman (Aug 07 2021 at 22:39):

In fact, this equivalence is an instance of a more general construction on double categories, which was written up very explicitly by Michael Lambert in Discrete Double Fibrations. The general statement is that just as discrete fibrations over a category are equivalent to functors into Set, discrete double fibrations over a double category are equivalent to lax double functors into Span. Now if a category is regarded as a double category with only identity tight arrows, discrete double fibration over it are just arbitrary functors, while lax double functors to the double category Span are the same as ordinary lax functors to the bicategory Span.

Jade Master (Aug 07 2021 at 23:08):

Cool!

Bryce Clarke (Aug 08 2021 at 02:46):

The unpublished note "Powerful functors" by Ross Street contains some details on the equivalence between normal lax functors into Prof and ordinary functors. http://science.mq.edu.au/~street/Pow.fun.pdf

Bryce Clarke (Aug 08 2021 at 02:52):

However, needing to add the condition that you restrict to lax natural transformations between normal lax functors $F, G : B \to Prof$ whose "component is actually a functor" is really just asking for us to use double categories rather than bicategories.

Bryce Clarke (Aug 08 2021 at 03:01):

In the 2-category of double categories, lax double functors, and "tight" natural transformations between them, we can take the comma object of a normal lax functor $B \to Prof$, where B is considered as a double category whose tight morphisms are identities as Mike states above, along the strict double functor $1 \to Prof$ which picks out the terminal category. The comma object should be a double category whose tight morphisms are also just identities, and the resulting projection to $B$ corresponds to a functor.

Mike Shulman (Aug 08 2021 at 03:06):

Right; it's just like for functors $B \to \rm Set$ in $\rm Cat$.

Mike Shulman (Aug 08 2021 at 03:07):

The analogy is even closer when you use Span and lax functors instead of Prof and normal lax functors.

Mike Shulman (Aug 08 2021 at 03:10):

In fact, Pare and his school often refer to the double category Span as "the double category of sets".

Mike Shulman (Aug 08 2021 at 03:11):

And just as in Cat, the "pointed category" $\rm Prof_*$ or $\rm Span_*$ or $\rm Set_*$ is the commal object of the terminal object over the identity functor (of Prof, Span, or Set). The pasting law relating comma squares and pullback squares then implies the original question about pullbacks of the "universal bundle" $\rm Prof_* \to Prof$ and so on.

fosco (Aug 08 2021 at 09:44):

Uff, there is so much I still have to learn on profunctors!

fosco (Aug 08 2021 at 10:22):

@Mike Shulman thanks, I agree that one must the result through a double categorical lens to appreciate that the universal bundles $Prof_* \to Prof$ and $Set_* \to Set$ share many features. For what I have in mind an explicit description of $\Pi F$ is better, but I will keep the double cat perspective as a guiding light :smile:

Mike Shulman (Aug 08 2021 at 16:12):

I don't see any conflict between a double-categorical lens and an explicit description of $\Pi F$.

fosco (Aug 08 2021 at 19:35):

I see a conflict in resorting to double categories to explain something to people who asked "what exactly is a comonad?".
I can try telling them what is a double category. It rarely works.

Nathanael Arkor (Aug 08 2021 at 20:53):

Why do you need to define double categories to explain what a comonad is?

Mike Shulman (Aug 09 2021 at 04:18):

Sure, that makes sense. I didn't realize that what you had in mind was an expository purpose rather than a mathematical one.

fosco (Aug 09 2021 at 08:07):

Yeah, I should have said "for what I have in mind and for the intended target", but that remained in my fingers while typing. Anyway, I can tell you more precisely what I'm after. Consider the functor $C\mapsto {\cal C}/C$ from a fixed category $\cal C$, and the associated fibration $\Gamma[{\cal C}] \to \cal C$ having fibers all the slices at once.

The category $\Gamma[{\cal C}]$ can be seen as obtained from a Grothendieck construction "done twice", because each ${\cal C}/C$ is the category of elements of $yC$, the representable in C, thus $\Gamma[{\cal C}]=Elts(F)$ where $F : C\mapsto Elts(y(C))$.

The following mathematical gadgets are interesting for a problem I started thinking about with @Fabrizio Genovese and @Daniele Palombi

1. a generalization of this picture where instead of a functor to Cat, one takes a displayed category
2. a specialization of this construction when $\cal C$ is (the free category on) a graph

3. (linked to 1, evidently) classify all (or some) profunctors $({\cal C}/C)^\text{op} \times {\cal C}/B \to {\sf Set}$
   3b. a generalization of the above picture where instead of a functor into Cat, one considers a functor ${\cal C} \to {\sf Par}$, still sending an object $C$ to the objects of the slice, but sending a morphism to a partial function between those sets.

4. classify all (or some) profunctors $({\cal C}/C)_o \times ({\cal C}/B)_o \to {\sf Set}$, where now I'm just considering the set of objects of the slice, without worrying about morphisms over $B,C$, etc.

3b should encode a "block/allow" policy a post-composition map $f_* : {\cal C}/A \to {\cal C}/B$ applies to objects of a slice. In 4, when C is free on a graph G, one wants to classify "matrices of sets", indexed by paths in G, relative to $A,B$; indeed such a profunctor is a matrix whose entries are sets, and whose rows/columns are in bijection with objects of ${\cal C}/A, {\cal C}/B$. I suspect some of the matrix graph invariants can be categorified and make sense for this profunctor.

Now, this is pretty elementary. Do you know any reference where 1,2,3,4 are addressed (and used for a specific purpose)?

My goal was to post a separate question for each of these points. Stay tuned!

fosco (Aug 09 2021 at 12:17):

Just to go back in topic: Street's note says (page 2) that given a normal lax functor $N : B^\text{op} \to {\sf Prof}$ one can consider its lax bicolimit ("collage") and from its universal property obtain a functor $E \to B$; I guess one can deduce that this is a category from the fact that the lax cocone inducing $E \to B$, as well as the universal lax cocone is made of functors?

Mike Shulman (Aug 09 2021 at 17:14):

Your 1-4 don't immediately ring a bell.

Mike Shulman (Aug 09 2021 at 17:14):

fosco said:

I guess one can deduce that this is a category from the fact that the lax cocone inducing $E \to B$, as well as the universal lax cocone is made of functors?

Do you mean "deduce that this is a functor"? If so, then yes, that's the "tight projections detect tightness" property of collages in $\rm Prof$.

fosco (Aug 09 2021 at 18:17):

"tight" as in ...a Street's paper I can't remember the title of? Or was it Kelly's "and so on"?

fosco (Aug 09 2021 at 18:18):

I have random thoughts on 3. I will type them sparsely in a while.

fosco (Aug 09 2021 at 18:43):

A profunctor from ${\cal C}/A$ to ${\cal C}/B$ is a left adjoint $\widehat{{\cal C}/B} \to \widehat{{\cal C}/A}$ (presheaf categories)

But now, $\widehat{{\cal C}/B} = \widehat{{\cal C}}/yB$ and similarly for $A$, so I'm (probably) trying to classify left adjoints $L: \widehat{{\cal C}}/yB \to \widehat{{\cal C}}/yA$ that are functors over $\widehat{\cal C}$.

fosco (Aug 09 2021 at 18:43):

(possibly the request that L is fibered over the slices is additional, I don't mind asking it if it simplifies the problem)

fosco (Aug 09 2021 at 18:50):

...and nothing, from here I don't know what to say. L can be anything. I guess I'd be content with seeing some L's that are genuine profunctors, maybe also expressed in terms of a normal lax functor ${\cal C} \to {\sf Prof}$ that on objects sends $A$ to the slice over $A$.

I'll keep thinking.

Nathanael Arkor (Aug 09 2021 at 20:14):

fosco said:

"tight" as in ...a Street's paper I can't remember the title of? Or was it Kelly's "and so on"?

Tight as in the paper Enriched categories as a free cocompletion of Mike and Richard Garner.

fosco (Aug 09 2021 at 20:55):

very good(TM)!

fosco (Aug 11 2021 at 21:02):

fosco said:

...and nothing, from here I don't know what to say. L can be anything. I guess I'd be content with seeing some L's that are genuine profunctors, maybe also expressed in terms of a normal lax functor ${\cal C} \to {\sf Prof}$ that on objects sends $A$ to the slice over $A$.

I'll keep thinking.

Let me add something on this.

A profunctor $\mathfrak{p} : {\cal C}/A \rightsquigarrow {\cal C}/B$ is a functor ${\cal C}/B \to \widehat{{\cal C}/A}$ (i.e. a left adjoint $\widehat{{\cal C}/B} \to \widehat{{\cal C}/A}$). Now: $\widehat{{\cal C}/B}$ is just the slice topos $\widehat{{\cal C}}/yB$. So, a profunctor $\mathfrak{p} : {\cal C}/A \rightsquigarrow {\cal C}/B$ is a functor ${\cal C}/B \to \widehat{{\cal C}}/yB$. This, in turn (given the universal property of a slice category, i.e. of a comma object), corresponds to a 2-cell like

$\begin{array}{ccc}{\cal C}/B &\to& \widehat{\cal C} \\ \downarrow &\Downarrow& || \\ \cal C &\to & \widehat{\cal C}\end{array}$

which in turn is just a cocone $\Delta : P \Rightarrow yA$ to the representable on $A$ (with domain the slice: so, for every object $h : X \to B$ of ${\cal C}/B$ one has a natural transformation $\Delta_h : P(h, \_) \Rightarrow yA : {\cal C}^\text{op} \to Set$ of presheaves.

Now, one can take different paths, according to how it's more convenient to transform this last functor; $P$ can be uncurried into a presheaf

$\hat P : {\cal C}/B \times {\cal C}^\text{op} \to Set$ equipped with a cocone to $yA$; or, $\Delta$ induces a unique arrow $\bar\Delta : \text{colim}_h P_h \Rightarrow yA$ with components $\bar\Delta_{A'} : .\text{colim}_h P_h(A') \to {\cal C}(A', A)$.

So, ultimately, the profunctors I'm interested in correspond bijectively to pairs $(P,\Delta)$ as above.

How to go further from here?