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: learning: questions

Topic: transporting structure across morphisms

David Egolf (Nov 26 2023 at 22:16):

Sometimes we can induce additional structure on an object using a morphism from that object. Here are some examples where structure can "flow" across a morphism:

Assume we have a function $f: A \to U(B)$ in $\mathsf{Set}$, where $B$ is a topological space and $U(B)$ is its underlying set. Then we can induce a topology on $A$ using $f$. We can do this by using the "initial topology", which places the coarsest topology on $A$ so that $f$ becomes a continuous function between topological spaces.

David Egolf (Nov 26 2023 at 22:16):

As another example, assume we have a function $g: C \to U(D)$ in $\mathsf{Set}$, where $D$ is a set that has an equivalence relation placed on it and $U(D)$ is the underlying set. Then we can induce an equivalence relation on $C$ using $g$. We can do this by setting $c \sim c' \iff g(c) \sim g(c')$ for all $c,c' \in C$.

I seem to recall we can also do a similar things with quivers. Let $h: S \to U(G)$ in $\mathsf{Set}$ be a function that maps from $S$ to the set of vertices of a quiver $G$. Then I think we can induce a quiver on $S$ by adding in an arrow from $s$ so $s'$ exactly if there is an arrow from $h(s)$ to $h(s')$ in $G$.

David Egolf (Nov 26 2023 at 22:17):

Can we organize these examples, and view them as special cases of some general procedure for transporting structure across morphisms?

As a start, we might consider this setup:

• we have a functor $F: C \to D$
• we have a morphism $f:d \to F(c)$ for some $d \in D$ and $c \in C$

In this context, maybe we want to get a morphism $F^*(f):F^*(d) \to c$ in $C$, where $F^*$ is supposed to indicate some procedure that moves us from $D$ to $C$. The idea is that $F^*(d) \in C$ is a version of $d$ after we've added on some structure to $d$, and that this added structure allows us to promote $f$ to $F^*(f)$, which is a morphism in the more "structured" category $C$.

Ralph Sarkis (Nov 26 2023 at 22:32):

Look into cartesian morphisms. We also had a discussion about them (under a different name) here.

David Egolf (Nov 26 2023 at 22:33):

Ralph Sarkis said:

Look into cartesian morphisms. We also had a discussion about them (under a different name) here.

Thanks for linking those resources!

Chris Grossack (they/them) (Nov 27 2023 at 03:41):

It sounds like you want a topologically concrete category. I have an old blog post about them here, if you're interested in an overview

David Egolf (Nov 27 2023 at 05:18):

Chris Grossack (they/them) said:

It sounds like you want a topologically concrete category. I have an old blog post about them here, if you're interested in an overview

Thanks for sharing - that looks cool as well! I'll plan to take a look.

Matteo Capucci (he/him) (Nov 27 2023 at 09:12):

Ralph Sarkis said:

Look into cartesian morphisms. We also had a discussion about them (under a different name) here.

To spell Ralph's remark out loud, the phenomenon you talk about, David, happens when $U$ is a [[fibration]] (this allow to induce 'final' structures) or an [[opfibration]] (giving 'initial' ones).

What I don't know is if there is a good understanding of sufficient conditions to make $U$, right adjoint of an Eilenberg-Moore adjunction, a fibration :thinking:

John Baez (Nov 27 2023 at 09:37):

Gray gave sufficient conditions for an opfibration to be a rali - a right adjoint left inverse:

• J. Gray, Fibred and cofibred categories, in Proceedings of the Conference on Categorical Algebra: La Jolla 1965, eds. S. Eilenberg et al, Springer, Berlin, 1966, pp. 21-83.

John Baez (Nov 27 2023 at 09:39):

He probably studied fibrations too. But I forget if he studied the converse question, of when a right adjoint is an (op)fibration.

Mike Shulman (Nov 27 2023 at 09:46):

A result that I don't think is widely-enough known is that if $U:D\to C$ is an isofibration with a fully faithful left adjoint (I guess that is a "rali"?), and $D$ has pushouts preserved by $U$, then $U$ is an opfibration. I don't know if that's in Gray's paper.

David Egolf (Nov 27 2023 at 21:51):

Chris Grossack (they/them) said:

It sounds like you want a topologically concrete category. I have an old blog post about them here, if you're interested in an overview

I am very much enjoying this blog article so far! One highlight so far: if $U: \mathsf{Top} \to \mathsf{Set}$ or $U: \mathsf{Meas} \to \mathsf{Set}$ (where $\mathsf{Meas}$ is the category of measure spaces) is the forgetful functor, then the fibre of $U$ over a set $X$ is a complete lattice! Very roughly, the different ways that we can augment $X$ to form a topological space or measure space are related to one another in a very structured way.

Maybe the blog article gets to this question later on, but I wonder: What determines the structure of a fibre of a functor? What kinds of different structures can appear as a fibre of a functor?

John Baez (Nov 27 2023 at 22:06):

If by "fibre" you mean what the nLab calls the essential fiber of a functor, I wrote up a bunch of stuff saying when that fiber is a groupoid, a preorder, or (equivalent to) a discrete category.

Kevin Arlin (Nov 27 2023 at 22:24):

For isofibrations like the functors David mentioned, I guess the strict fiber (the pullback of $U$ along some object-selecting functor $X:1\to\mathsf{Set}$) is equivalent to the essential fiber.

John Baez (Nov 27 2023 at 22:27):

That sounds right. So in that case David can sort of relax and not worry about "fiber" versus "essential fiber".

John Baez (Nov 27 2023 at 22:29):

I give a simple condition for the (essential) fiber to be a poset, and then maybe the "topologically concrete category" idea gives further conditions under which this poset is a complete lattice. (I haven't really understood this "topologically concrete" stuff yet.)

Kevin Arlin (Nov 27 2023 at 23:00):

Yeah, so the idea is that if you have a bunch of maps from a set $X$ to the underlying sets of a bunch of spaces, $f:X\to U(S),$ then we can always topologize $X$ so that all the $f$'s just barely become continuous, by making the inverse images of all the opens in all the $S$'es open in $X.$ (This is how you construct limits in the category of topological spaces, for instance--take the limit in sets and then apply this so-called "initial" topology.)

Kevin Arlin (Nov 27 2023 at 23:00):

In particular, if all the $S$'s are copies of $X$, with a bunch of different topologies, and all the $f$'s are the identity, then you're just taking the union of a bunch of topologies on $X$! This is all there is to topologically concrete categories; the far-out thing is that since it's perfectly fine to let there be a large number of $f$'s, this makes the set of topologies on $X$ into a complete lattice, with intersections as well as unions.

David Egolf (Nov 28 2023 at 19:44):

I finished a first read of @Chris Grossack (they/them) 's blog article! I wanted to share a very cool construction mentioned there (which is described in more detail in "The Joy of Cats").

We start with any functor $T: C \to \mathsf{Set}$. Then we form $\mathrm{Spa}(T)$, a "functor-structured category" as follows:

• each object is a pair $(c, \alpha)$, where $c \in C$ and $\alpha \subseteq T(c)$
• each morphism from $(c, \alpha)$ to $(c', \alpha')$ is a morphism $f: c \to c'$ in $C$ such that $T(f)(\alpha) \subseteq \alpha'$.

David Egolf (Nov 28 2023 at 19:44):

According to the blog article, you can get ANY (fibre-small) topological category over $C$ by taking some concretely reflective subcategory of $\mathrm{Spa}(T)$ for some $T: C \to \mathsf{Set}$. If I'm reading the article correctly, you can get a bunch of cool categories in this way, including:

• $\mathsf{Top}$, the category of topological spaces
• $\mathsf{Meas}$, the category of measure spaces
• $\mathsf{DiGph}$, the category of directed graphs, with self-loops allowed but no multiple "parallel" arrows (and also several subcategories of this!)
• $\mathsf{PreOrd}$, the category of preorders
• the category of sets with equivalence relationships placed on them

All of these categories are, I think, topological over $\mathsf{Set}$!

David Egolf (Nov 28 2023 at 19:48):

A few thoughts:

• The $\mathrm{Spa}(T)$ construction reminds me a lot of the category of elements. I wonder exactly how these constructions are related, and if there are other similar constructions.
• It's kind of fun to start with a functor $T: C \to \mathsf{Set}$ and see what $\mathrm{Spa}(T)$ is. For example, as a short puzzle, you might enjoy figuring out what $\mathrm{Spa}(1_{\mathsf{Set}})$ is, where $1_{\mathsf{Set}}: \mathsf{Set} \to \mathsf{Set}$ is the identity functor.
• Is there an nLab article on $\mathrm{Spa}(T)$? I couldn't find one in an initial search.

Nathaniel Virgo (Nov 29 2023 at 03:59):

I really enjoyed the blog post too. One question that occurs to me: are these categories (Top, Meas, etc.) "special" in some way, i.e. do they have universal properties of some kind, among the categories that can be constructed this way?

John Baez (Nov 29 2023 at 09:21):

If so, it must fall straight out of how a topology or measurable space structure on $X$ is defined as a sub-poset of $P(X)$ that has certain nice properties, e.g. closed under all sups and finite infs.

John Baez (Nov 29 2023 at 09:23):

But I don't see how this gives the category of topological spaces or measurable space a universal property. It just suggests this universal property will be a property of categories over $\mathsf{Set}$.

John Baez (Nov 29 2023 at 09:23):

It's a great question, Nathaniel. Someone must have studied this.... no?

Matteo Capucci (he/him) (Nov 29 2023 at 10:02):

It seem this address it, doesn't it?
image.png

Matteo Capucci (he/him) (Nov 29 2023 at 10:05):

David Egolf said:

A few thoughts:

• The $\mathrm{Spa}(T)$ construction reminds me a lot of the category of elements. I wonder exactly how these constructions are related, and if there are other similar constructions.

$\rm Spa$ is the pullback of the subobjects fibration of $\bf Set$ along $T$:
image.png
a fact which also immediately implies that $U:{\rm Spa}(T) \to \cal C$ is a fibration too!

Matteo Capucci (he/him) (Nov 29 2023 at 10:06):

It might remind you of the category of elements because that's also the pullback of a fibration over set, that of points:
image.png

Matteo Capucci (he/him) (Nov 29 2023 at 10:06):

Here $\bf Set_*$ is the category of pointed sets and the projection to $\bf Set$ forgets point

Matteo Capucci (he/him) (Nov 29 2023 at 10:07):

In fact it turns out every discrete fibration arise as such, a fact that earns to $\bf Set_* \to Set$ the name of discrete fibrations classifier (compare this with other classifiers, like subobject classifier)

Matteo Capucci (he/him) (Nov 29 2023 at 10:09):

This makes me suspect that Theorem VI.22.3 might be expressing a similar fact about the Spa construction, namely that $\bf Set^\subseteq \to Set$ is almost classifying topological categories

Matteo Capucci (he/him) (Nov 29 2023 at 10:10):

This isn't true up to that 'concretely reflective subcategory of', I wonder if there is a way to modify the notion of classifier to accomodate this extra bit

Matteo Capucci (he/him) (Nov 29 2023 at 10:11):

Chris Grossack (they/them) said:

It sounds like you want a topologically concrete category. I have an old blog post about them here, if you're interested in an overview

By the way Chris, another amazing post, kudos! I knew nothing about topological categories, what an interesting subject.

Nathaniel Virgo (Nov 29 2023 at 12:43):

Matteo Capucci (he/him) said:

It seem this address it, doesn't it?
image.png

I meant, among these categories does Top have a universal property, does Meas have one etc.?

Chris Grossack (they/them) (Nov 29 2023 at 15:41):

@Matteo Capucci (he/him) -- thanks for both the kind words and for making precise the connection to fibrations. I knew there should be a connection, but (especially back then) I didn't know how to make it precise. A lot of fibration stuff is still just on the boundary of what I'm comfortable with, so these examples were really helpful for me ^_^

Matteo Capucci (he/him) (Nov 29 2023 at 15:52):

Nathaniel Virgo said:

Matteo Capucci (he/him) said:

It seem this address it, doesn't it?
image.png

I meant, among these categories does Top have a universal property, does Meas have one etc.?

The best story I know about the 'universality' of topological spaces is the result characterizing them as being β-relational modules, where β is the codensity monad of the inclusion of finite sets in set, aka the ultrafilter monad. It'd be interesting to link this characterization to the subject of topological categories. And also to figure out if a version of this story still holds for measurable spaces!

Graham Manuell (Nov 29 2023 at 17:15):

There is a result that says the category of (T,V)-categories is always topological over Set. (See the Monoidal Topology book for details.) Topological spaces are the case T = the ultrafilter monad, V = {0,1}. I don't know how measurable spaces fit in, though.

Sam Staton (Nov 29 2023 at 21:16):

Just to add, this Spa(T) construction is really useful. Maybe the Spa notation is not so common: in "Quasitoposes, Quasiadhesive Categories and Artin Glueing" Sec 4, I think it is denoted C//T, which might be more common. Since it is a way of building interesting categories with nice properties, it is connected to the "logical relations" method in type theory / programming languages, e.g. via the pullback of fibrations angle that Matteo explained.

There are nlab pages on Artin gluing and Freyd covers but maybe the Spa / C//T case deserves its own page and doesn't yet have one.

Chris Grossack (they/them) (Nov 30 2023 at 01:42):

That's super cool!

Mike Shulman (Nov 30 2023 at 02:20):

Isn't Spa(T) the [[Grothendieck construction]] of the composite $C \xrightarrow{T} \mathrm{Set} \xrightarrow{P} \mathrm{Cat}$ where $P$ is the covariant powerset functor, regarded as landing in posets and hence categories?

Mike Shulman (Nov 30 2023 at 02:22):

Or equivalently of $C^{\mathrm{op}} \xrightarrow{T^{\mathrm{op}}} \mathrm{Set}^{\mathrm{op}} \xrightarrow{P'} \mathrm{Cat}$ where $P'$ is the contravariant powerset functor.

Mike Shulman (Nov 30 2023 at 02:35):

Regarding classifiers, I believe a topological concrete category with small fibers is the same as a fibration whose fibers are small complete lattices and whose restriction functors preserve meets, i.e. an "indexed inf-lattice". And it's also the same as an opfibration whose fibers are small complete lattices and whose co-restriction functors preserve joins. So the "fiber-small topological-category classifier" should be $\mathbf{CjSLat}_* \to \mathbf{CjSLat}$, where $\mathbf{CjSLat}$ is the category of "complete join-semilattices", i.e. complete lattices and join-preserving maps.

John Baez (Nov 30 2023 at 08:40):

I started worrying about opposite categories as soon as David Egolf said:

We start with any functor $T: C \to \mathsf{Set}$. Then we form $\mathrm{Spa}(T)$, a "functor-structured category" as follows:

• each object is a pair $(c, \alpha)$, where $c \in C$ and $\alpha \subseteq T(c)$
• each morphism from $(c, \alpha)$ to $(c', \alpha')$ is a morphism $f: c \to c'$ in $C$ such that $T(f)(\alpha) \subseteq \alpha'$.

and then people started talking about topological spaces, since of course we say a map of topological spaces is continuous if the inverse image of an open set is open, while here it looks like we're saying the image of any set in $\alpha$ is in $\alpha'$. Probably Chris clarified this stuff in their original post - but looking at just the above, I got nervous.

John Baez (Nov 30 2023 at 08:40):

So I felt a lot better when Mike Shulman said:

Isn't Spa(T) the [[Grothendieck construction]] of the composite $C \xrightarrow{T} \mathrm{Set} \xrightarrow{P} \mathrm{Cat}$ where $P$ is the covariant powerset functor, regarded as landing in posets and hence categories? Or equivalently of $C^{\mathrm{op}} \xrightarrow{T^{\mathrm{op}}} \mathrm{Set}^{\mathrm{op}} \xrightarrow{P'} \mathrm{Cat}$ where $P'$ is the contravariant powerset functor.

John Baez (Nov 30 2023 at 08:44):

This helps clarify that $\mathrm{Spa}(T)$ is giving us a fibration over $C^{\textrm{op}}$, i.e. an opfibration over $C$.

John Baez (Nov 30 2023 at 08:45):

(I hope I'm not getting something backwards, which I have an amazing ability to do!)

John Baez (Nov 30 2023 at 08:52):

I'm even more relieved, of course, to hear that $\mathrm{Spa}$ is "just" a special case of the Grothendieck construction, not some fundamentally new thing.

(Here I'm using "just" in the way that category theorists use it, which is not as dismissive as it sounds: it's good when an $X$ is "just" a $Y$, if you like $Y$.)

Mike Shulman (Nov 30 2023 at 08:52):

I meant the Grothendieck construction of $P\circ T$ as a covariant functor, giving us an opfibration over $C$.

John Baez (Nov 30 2023 at 08:53):

Right. I think I said we've got an opfibration over $C$, so I'm happy.

Mike Shulman (Nov 30 2023 at 08:54):

Okay, when you said "giving us a fibration over $C^{\mathrm{op}}$" I thought you were thinking of considering $P\circ T$ as a contravariant functor defined on $C^{\mathrm{op}}$, building its Grothendieck construction as such a thing to get a fibration over $C^{\mathrm{op}}$, and then taking its opposite to get an opfibration over $C$, which is different.

Mike Shulman (Nov 30 2023 at 08:56):

(I included that kind of "just" in my own message at first, but then I took it out so as not to sound dismissive. But maybe you're right and we should hold the line and keep using it.)

Mike Shulman (Nov 30 2023 at 08:58):

Regarding topological spaces, I don't know exactly how they get represented as a reflective subcategory of some Spa(T), but my guess would be that the subsets $\alpha\subseteq T(c)$ are not themselves the topologies. There are definitely other ways to represent topological spaces that are covariant, e.g. using double powersets, and maybe one of those is what's happening here.

Matteo Capucci (he/him) (Dec 04 2023 at 08:33):

Uhm I see so I think my pullback characterization above isn't quite right... though we do have $Tf(\alpha) \subseteq \alpha' \iff \alpha \subseteq Tf^{-1}(\alpha')$