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Stream: learning: questions

Topic: Limits and Stuff Structure Property

John Onstead (May 20 2025 at 11:23):

In the 1-category of 0-categories $\mathrm{Set}$, a topos and adhesive category, every monomorphism is a regular monomorphism and hence can be expressed as a 1-limit. If we view a monomorphism in $\mathrm{Set}$ as representing a property, then this shows a direct relation between the concept of a 1-limit and a property. In some sense, the elements of a limit can be thought of as elements of a set with extra property.

John Onstead (May 20 2025 at 11:25):

My very first instinct is to immediately try to generalize this correspondence. In the 2-category of 1-categories $\mathrm{Cat}$, which is a 2-topos, we have two kinds of "monomorphism": faithful and fully faithful functors, which represent structure and property respectively. Under the philosophy of n-stuff, this is just a straightforward generalization from the aforementioned $\mathrm{Set}$ case. The question then becomes: what are the two corresponding notion of higher regular monomorphism (IE, 2-regular monomorphisms) and higher adhesiveness (IE, 2-adhesive categories) for 2-limits? And is $\mathrm{Cat}$ such a higher adhesive category (in either or both kinds)?

John Onstead (May 20 2025 at 11:27):

Essentially, my question boils down to: is every subcategory (fully faithful functor) and concrete category (faithful functor) expressible as a 2-limit in $\mathrm{Cat}$ (up to equivalence of categories), in the same way as any subset is expressible as a 1-limit in $\mathrm{Set}$?
(Note: this article does state how to derive faithful and fully faithful functors as weak pullbacks, but this only gives such functors between core groupoids of categories, not the categories themselves)

John Baez (May 20 2025 at 12:22):

How would you express $\mathbb{N} \subset \mathbb{R}$ as a 1-limit in $\mathsf{Set}$? I'm mainly asking to make sure I know what you mean.

Nathan Corbyn (May 20 2025 at 12:43):

(deleted)

Nathan Corbyn (May 20 2025 at 12:48):

(deleted)

John Baez (May 20 2025 at 12:56):

I was asking @John Onstead what he meant, not "can anyone do this?" Of course it's fine to take a crack at it, but John seemed to be alluding to a general procedure for turning monics into 1-limits, so I just wanted to see how his procedure worked in a specific example.

(I did this partially because he would prefer to describe the procedure in general without ever mentioning an example. :smiling_imp:)

Nathanael Arkor (May 20 2025 at 13:36):

Have you read Bourke and Garner's article Two-dimensional regularity and exactness? It's very closely related to your question.

John Onstead (May 20 2025 at 19:12):

John Baez said:

How would you express $\mathbb{N} \subset \mathbb{R}$ as a 1-limit in $\mathsf{Set}$? I'm mainly asking to make sure I know what you mean.

There's a lot of ways to do this. The most obvious is to take the pullback of $f: \mathbb{R} \to 2$ on "True" where $f$ sends a real number to True if it is a natural number.
This is the direct decategorification of the system with core groupoids discussed in the nlab article.

John Onstead (May 20 2025 at 19:12):

But there are certainly other ways of expressing this limit. For instance, let $f: A \to B$ be an injective function, and let $g: B \to C$ be a function such that $g \circ f$ is constant at some element $e$. Then the pullback of $g$ along $e$ recovers this injective function up to isomorphism. Likewise in $\mathrm{Cat}$ this works for any functor up to being (essentially) injective on objects/faithful (so I guess this would answer my question when it comes to fully faithful functors)
(Edit: Actually I think $f$ must also be "maximal" in some sense among functors where $g \circ f$ is a constant functor, but this doesn't affect the proof since its still true that this is always possible given a smart enough choice of $g$)

John Onstead (May 20 2025 at 19:15):

Maybe here's a slightly more precise question when it comes to fully faithful functors, before we even get to faithful ones. The closest higher dimensional analogue of the equalizer is the equifier, which works on parallel 2-morphisms rather than parallel morphisms. Just as every equalizer is a monomorphism in a 1-category, every equifier is a fully faithful morphism in a 2-category. Then the question becomes: which categories are such that the opposite is true, that every fully faithful morphism is an equifier, and is Cat such a category?

John Onstead (May 20 2025 at 19:41):

Nathanael Arkor said:

Have you read Bourke and Garner's article Two-dimensional regularity and exactness? It's very closely related to your question.

I'm going through it currently. So far Propositions 22 ($F_{bo}$ congruences are effective in $\mathrm{Cat}$) and 31 ($F_{bof}$ congruences are effective in $\mathrm{Cat}$) are promising. In a 1-category, being effective implies being regular, and it seems something similar holds for the 2-category case.

John Onstead (May 20 2025 at 19:41):

I'm not exactly sure what is going on, but I have a guess. In $\mathrm{Set}$, there is a epi/mono factorization system that occurs through the regular image. The regular image can be given as a limit- the equalizer of the cokernal pair, and thus the resulting monomorphism is regular. Therefore, so long as any monomorphism can emerge through the epi/mono factorization system, it proves that any monomorphism is regular. I think this paper is saying something similar holds in $\mathrm{Cat}$, now where we have a factorization system for both faithful and fully faithful functors.

Mike Shulman (May 20 2025 at 19:49):

John Onstead said:

But there are certainly other ways of expressing this limit. For instance, let $f: A \to B$ be an injective function, and let $g: B \to C$ be a function such that $g \circ f$ is constant at some element $e$. Then the pullback of $g$ along $e$ recovers this injective function up to isomorphism. Likewise in $\mathrm{Cat}$ this works for any functor up to being (essentially) injective on objects/faithful (so I guess this would answer my question when it comes to fully faithful functors)
(Edit: Actually I think $f$ must also be "maximal" in some sense among functors where $g \circ f$ is a constant functor, but this doesn't affect the proof since its still true that this is always possible given a smart enough choice of $g$)

It's true in Set that you can always make a smart enough choice of $g$. For instance, you can take $C=2$ and $g$ the characteristic function of $f$, as in your previous example. But I don't see any way to do this in Cat.

Mike Shulman (May 20 2025 at 19:53):

John Onstead said:

which categories are such that the opposite is true, that every fully faithful morphism is an equifier, and is Cat such a category?

It's not true in Cat. For example, every equifier is closed under retracts, but not every full subcategory is.

Mike Shulman (May 20 2025 at 19:55):

John Onstead said:

The regular image can be given as a limit- the equalizer of the cokernal pair, and thus the resulting monomorphism is regular.

I would call this the coregular image. The regular image, being the one that exists in any regular category, is the coequalizer of the kernel pair, and in that case the resulting epimorphism is regular. It's a fact that Set, and other 1-toposes, are not just regular but coregular. However, I don't think this generalizes to 2-toposes.

John Onstead (May 20 2025 at 22:04):

Mike Shulman said:

I would call this the coregular image. The regular image, being the one that exists in any regular category, is the coequalizer of the kernel pair, and in that case the resulting epimorphism is regular. It's a fact that Set, and other 1-toposes, are not just regular but coregular. However, I don't think this generalizes to 2-toposes.

Well, I think I found a factorization method that works, at least for fully faithful functors:

John Onstead (May 20 2025 at 22:04):

In the nlab page for full image, it gives a way to construct the full image of a functor $C \to D$ as a limit (a specific pullback in Cat), with the embedding of the full image into $D$ being one of the projections. The full image is the notion of image in the (bo/ff) factorization system, meaning any fully faithful functor that arises this way can be given as the aforementioned limit. Thereby proving any fully faithful functor emerges as a limit.

John Onstead (May 20 2025 at 22:07):

Nlab also says the (bo/ff) factorization system can be constructed via generalized kernels (IE, the construction in terms of coequalizers of kernel pairs), meaning that somehow the full image can be expressed as either a colimit or a limit! If I can show this also holds for the notion of "image" in the bijective on objects and full/faithful factorization, then I can extend the above result to faithful functors and answer my question fully.

Mike Shulman (May 20 2025 at 22:34):

John Onstead said:

In the nlab page for full image, it gives a way to construct the full image of a functor $C \to D$ as a limit (a specific pullback in Cat), with the embedding of the full image into $D$ being one of the projections. The full image is the notion of image in the (bo/ff) factorization system, meaning any fully faithful functor that arises this way can be given as the aforementioned limit. Thereby proving any fully faithful functor emerges as a limit.

That's true, but that only works in the 1-category Cat: that pullback is a 1-categorical pullback but not a 2-categorical pullback. Your original question asked about the 2-category Cat, so I was answering in that context.

John Onstead (May 21 2025 at 00:48):

Mike Shulman said:

Your original question asked about the 2-category Cat, so I was answering in that context.

I see. Well, it seems I have a clear objective: characterize $\mathrm{Im}_1$ (the full image, the central object of eso/ff factorization), and $\mathrm{Im}_2$ (the faithful image, the central object of eso and full/faithful factorization) as a 2-limit (not colimit) in the 2-category $\mathrm{Cat}$. I really tried thinking through this but as I only have experience with conical limits, and the answer very most likely is some sort of weighted limit, I'm not getting very far. Would you be able to provide a hint (at least what the weight should be, or the diagrams involved)?

Mike Shulman (May 21 2025 at 00:51):

I do not know of a way to do that.

John Onstead (May 21 2025 at 02:52):

Mike Shulman said:

I do not know of a way to do that.

Oh I see... well, I'll keep reading more of the article above as well as more about generalized kernels to see if I can find any more information!

John Onstead (May 21 2025 at 02:52):

As for my more general question of limits vs faithful/fully faithful functors, I already have another idea in mind other than factorizations. I'll work on formulating it a bit more before discussion. To be continued!

John Onstead (May 21 2025 at 11:42):

I want to discuss a motivating example for why I think there's a connection between limits and SSP (stuff structure property) in the first place. In $\mathrm{Cat}$, given two faithful functors into $\mathrm{Set}$, one can think of their pullback as forming the category of objects with both structures (for instance, with $\mathrm{Top}$ and $\mathrm{Grp}$, you get the category of topological groups). From this perspective it seems that limits are good at adding extra structure onto objects, making them relevant to SSP.

John Onstead (May 21 2025 at 11:44):

But if topological groups are the pullback of spaces and groups, then what are groups or spaces the pullbacks of? This led me down an interesting path. First, I learned that magmas (sets equipped with any binary operation) are a set $S$ equipped with $m: S \times S \to S$, which can be thought of as a limit involving the arrow category. Specifically, it's the category of endofunctor algebras for $(-)^2$, or the equalizer involving the category $\mathrm{Arr}(\mathrm{Set}) \times \mathrm{Set}$ where we equalize the pairs $((-)^2 \circ \pi_S, \mathrm{dom} \circ \pi_A)$ and $(\mathrm{cod} \circ \pi_A, \pi_S)$. This can be thought of as adding the initial extra structure of the map $m$ onto a set $S$ in the same way as taking the above pullback adds group structure onto topological spaces.

John Onstead (May 21 2025 at 11:45):

However, now let's say I want the category of associative magmas. How would I alter the limit above to impose the associativity condition? Do I need some sort of inserter or equifier (or both)?

Mike Shulman (May 21 2025 at 15:00):

Topological groups are not the pullback of spaces and groups over Set. An object of that pullback would be a set with a topology and an unrelated group structure, whereas a topological group requires the group operations to be continuous.

Kevin Carlson (May 21 2025 at 17:24):

You're really going to want to know about inserters and equifiers (which are important weighted limits, but you don't really need the general weighted limit story to get to them) to get this story to go smoothly.

John Onstead (May 21 2025 at 19:25):

Kevin Carlson said:

You're really going to want to know about inserters and equifiers (which are important weighted limits, but you don't really need the general weighted limit story to get to them) to get this story to go smoothly.

Ok, well I reviewed more about them. It seems that, as per my observation above, equifiers are kind of like the 2-cell equivalent of equalizers for 1-cells.

John Onstead (May 21 2025 at 19:25):

Here's my attempt to perform my construction of associative magmas, please let me know if I did it right. Start with the category of magmas $\mathrm{Mag}$ (the inserter of the endofunctors $(-)^2$ and $\mathrm{id}$ on $\mathrm{Set}$). Now, identify the two functors $(-)^3 \circ U$ and $U$ (with $U$ the forgetful functor). We consider the parallel natural transformations, whose components, for every $A$, are $\alpha_A = m \circ m \times 1: A^3 \to A$ and $\beta_A = m \circ 1 \times m: A^3 \to A$. If we take the equifier of these two paths, it should make the relevant diagram commute, which should give the associativity condition.

John Onstead (May 21 2025 at 20:24):

Assuming the above is true, it means we can construct the category of associative magmas via equifiers, which makes me want to revisit a point made above:

Mike Shulman said:

It's not true in Cat. For example, every equifier is closed under retracts, but not every full subcategory is.

John Onstead (May 21 2025 at 20:24):

I looked up "closed under retracts" online and found very few results. Could you do the following. First, briefly tell me what it means to be closed under retracts, then give concrete examples of categories that are and are not closed under retracts.

Mike Shulman (May 21 2025 at 20:26):

A subcategory $C\subseteq D$ is closed under retracts if whenever an object $x\in D$ is a retract of an object $y\in D$, and $y\in C$, then also $x\in C$.

Mike Shulman (May 21 2025 at 20:43):

For example, the category of finite sets is closed under retracts in Set, since a retract of a finite set is finite. But the category of infinite sets is not closed under retracts in Set, since an infinite set can have a finite retract.

John Onstead (May 21 2025 at 23:21):

Mike Shulman said:

For example, the category of finite sets is closed under retracts in Set, since a retract of a finite set is finite. But the category of infinite sets is not closed under retracts in Set, since an infinite set can have a finite retract.

Ah, so the closure under retracts is a property of a subcategory, rather than an inherent property of the category in and of itself. That's interesting! In any case if I want to show any subcategory is equivalent to some limit construction, I'll have to somehow construct the category of infinite sets as a limit involving $\mathrm{Set}$. Sounds like a challenge!

John Onstead (May 21 2025 at 23:38):

I think I got it! Define the functor $(-) \cup *: \mathrm{Set} \to \mathrm{Set}$. This functor will change the set if it is finite by adding an element, but infinite sets (regardless of the specific set theory chosen- at least for many types of set theory) will not change under the action of this functor. Therefore, infinite sets form a subcategory of fixed points for this endofunctor, and can be exhibited as the equalizer of this endofunctor along the identity. So this shows it's possible to define a subcategory using limits even if it isn't closed under retracts.

James Deikun (May 21 2025 at 23:46):

I don't think this works. Even though they land at the same (an isomorphic) object, they're not really part of the "fixed point" because they have a non-trivial automorphism performed on them.

Mike Shulman (May 22 2025 at 00:12):

Yeah, this is a 1- vs 2-category thing. For almost no sets will $X \sqcup \{\ast\}$ be literally equal to $X$, so you can't take the equalizer in the 1-category Cat (or the strict equalizer in the strict 2-category Cat). The next most obvious thing you could try is the "iso-inserter" or "pseudo-equalizer", but its objects will be sets $X$ equipped with the structure of an isomorphism $X \cong X \sqcup \{\ast\}$, and its morphisms will be functions that commute with those isomorphisms. So every object of that category will be infinite, and every infinite set will be represented in that category, but not every function between infinite sets will appear therein.

John Onstead (May 22 2025 at 01:31):

James Deikun said:

I don't think this works. Even though they land at the same (an isomorphic) object, they're not really part of the "fixed point" because they have a non-trivial automorphism performed on them.

Mike Shulman said:

Yeah, this is a 1- vs 2-category thing. For almost no sets will $X \sqcup \{\ast\}$ be literally equal to $X$, so you can't take the equalizer in the 1-category Cat (or the strict equalizer in the strict 2-category Cat).

Ah, I see. I guess you could always try using the skeleton category. If you let $\mathrm{Sk}: \mathrm{Set} \to \mathrm{Sk}(\mathrm{Set})$, then maybe there's better luck in equalizing $\mathrm{Sk}$ and $\mathrm{Sk} \circ (X \cup *)$. So even though $X \cup *$ is not equal to $X$, as long as it is isomorphic, it should be equal when moving to the skeleton. But of course this is a very 1-categorical operation, I don't know if it'd work too well in the 2-categorical setting (since the skeleton functor is an equivalence in that setting)

Mike Shulman (May 22 2025 at 04:18):

In 99.9999% of cases, skeletons do not solve any problems in category theory.

In this case, although using the skeleton would give you the right objects, it will still give you the wrong morphisms.

Mike Shulman (May 22 2025 at 04:31):

In that, it's similar to the iso-inserter. But unlike the iso-inserter, the morphisms you do get are not even sensible: they depend on your chosen equivalence $\rm Set \to Sk(Set)$, which in turn comes from the axiom of choice so you have no information about it.

John Onstead (May 22 2025 at 05:08):

Mike Shulman said:

In 99.9999% of cases, skeletons do not solve any problems in category theory.

I'll keep that in mind!

John Onstead (May 22 2025 at 05:08):

It's just a bit disappointing that I have a concrete picture of the construction I want to do- find the full subcategory on all objects, up to isomorphism, that remain invariant under an endofunctor $E$- but it's hard to express this as a precise construct. There might still be a way to do this using universal constructions- maybe something like "the universal $P: FP \to C$ such that for any other functor $F: D \to C$ satisfying the property that $E \circ F \cong F$, there is a unique functor $H: D \to FP$ making the diagram commute $F \cong P \circ H$". But I'm not sure if this would qualify as a limit or can be rewritten as one.

John Baez (May 22 2025 at 06:45):

I think the problem is that you're wanting to treat "invariant under some isomorphism" as a property of an object, rather than treating a particular choice of isomorphism as an extra structure on the object. Category theory tends to push us away from this sort of property, because it involves an existential quantifier over isomorphisms, which makes it more complex and hard to work with.

Morgan Rogers (he/him) (May 22 2025 at 09:02):

Note that there is a bo-ff factorization that recovers the full subcategory that you're looking for, but the intermediate thing isn't simply expressible as a limit.

John Onstead (May 22 2025 at 09:07):

John Baez said:

I think the problem is that you're wanting to treat "invariant under some isomorphism" as a property of an object, rather than treating a particular choice of isomorphism as an extra structure on the object. Category theory tends to push us away from this sort of property, because it involves an existential quantifier over isomorphisms, which makes it more complex and hard to work with.

I see, that must be where the difficulty lies. I'll certainly have to think more on this!

John Onstead (May 22 2025 at 09:15):

Morgan Rogers (he/him) said:

Note that there is a bo-ff factorization that recovers the full subcategory that you're looking for, but the intermediate thing isn't simply expressible as a limit.

Interesting. I think I can see how something like this might work: we can take the iso-inserter as suggested above, which will give a functor into $\mathrm{Set}$ that's essentially injective on objects yet not full. We can factor this using bo/ff, in which case the full image should be the full subcategory on infinite sets. I was also thinking about finding the discrete category of objects of $\mathrm{FinSet}$ and taking its "complement" with respect to the objects of $\mathrm{Set}$ to get the discrete category of infinite sets. Then the inclusion of this into the full $\mathrm{Set}$ can be factored via bo/ff. But there's probably a lot of set theoretic size issues with trying to take complements of things that are too big to be sets.

John Onstead (May 22 2025 at 11:08):

Often times, if a certain limit is hard to construct, the best advice is to usually move to a setting where they work a little nicer. So perhaps I should work smarter not harder, and instead of trying to show any faithful/fully faithful functor emerges as a limit in $\mathrm{Cat}$, I should find some similar structure where such a thing might be far easier to show or prove. For 1-categories, we usually move to something like $\mathrm{Set}$ or a presheaf category. But in some sense, $\mathrm{Cat}$ is already the notion of $\mathrm{Set}$ for 2-categories, thus making it difficult to figure out a setting where 2-limits might be even better behaved. Still, there's many options to consider for broader settings beyond the usual 2-category of categories.

John Onstead (May 22 2025 at 11:10):

To start, from many perspectives, the 2-category $\mathrm{Cat}$ isn't "the full picture", with that instead being the double category (and equipment) $\mathrm{Prof}$ of categories, functors, and profunctors. My question is: do we gain any new insights or abilities by moving to this double categorical setting over the usual 2-categorical setting when it comes to limits? For instance, are there things double limits in $\mathrm{Prof}$ can do that 2-limits in $\mathrm{Cat}$ cannot, or do they somehow end up being equivalent?

Nathanael Arkor (May 22 2025 at 12:40):

There are many interesting limits (and especially colimits) in the double category of distributors. For instance, the [[tabulator]] of a distributor is its [[category of elements]].

Nathanael Arkor (May 22 2025 at 12:41):

On the side of colimits, the [[full image]] of a functor, and the [[collage]] of a distributor are examples of colimits that are not 2-categorical colimits.

John Onstead (May 22 2025 at 19:08):

Nathanael Arkor said:

There are many interesting limits (and especially colimits) in the double category of distributors. For instance, the [[tabulator]] of a distributor is its [[category of elements]].

I'm wondering why it's not possible in general to represent the tabulator as a 2-limit then. For instance, the category of elements of a $\mathrm{Set}$ functor can be rewritten as the comma category $(*/F)$, and comma categories are a type of 2-limit. So why doesn't this work in general for profunctors?

John Onstead (May 22 2025 at 19:52):

I looked into it a bit more, but this does appear to be the case. But if any double limit in $\mathrm{Prof}$ can be constructed by products, equalizers, and tabulators, and each of those can be constructed by 2-limits, then it means there's an equivalence between double limits in $\mathrm{Prof}$ and 2-limits in $\mathrm{Cat}$. If so then it means moving to $\mathrm{Prof}$ won't solve any of my problems. It's also fascinating in that this represents another clear "symmetry breaking" between limits and colimits, in that double limits are equivalent to 2-limits but, as shown above, double colimits are not equivalent to 2-colimits! Isn't that weird?

Kevin Carlson (May 22 2025 at 20:44):

What, exactly, are you claiming "appear[s] to be the case"?

John Onstead (May 22 2025 at 21:16):

Kevin Carlson said:

What, exactly, are you claiming "appear[s] to be the case"?

That there's an equivalence between double limits in $\mathrm{Prof}$ and 2-limits in $\mathrm{Cat}$. (In the sense that anything you can construct with one you can also construct with the other)

Kevin Carlson (May 22 2025 at 21:24):

Ah, I assumed you meant something more general. Sure, that's true in $\mathrm{Cat}.$

John Onstead (May 22 2025 at 22:33):

So that was first attempt at extending $\mathrm{Cat}$ to get better limits... here's my second. What if instead of considering the usual tensor product, we instead considered the funny tensor product? That is, what if you took the equifier of parallel unnatural transformations? My main question is: are unnatural equifiers still fully faithful, or not necessarily? Second, of the unnatural equifiers that are still fully faithful, might it be possible some of them are not closed under retracts?

John Onstead (May 23 2025 at 07:05):

Alright, well there's no direct reference for equifiers of unnatural transformations in the literature, so guess I'm on my own! Now, given two transformations $\alpha, \beta: F \to G$, the objects of the equifier are any object $A$ of the domain such that $\alpha_A = \beta_A$. This works regardless of if these transformations are natural or not, meaning that the equifier will always be (essentially) injective on objects. However, to answer the questions, we need to know how it acts on morphisms.

John Onstead (May 23 2025 at 07:05):

I can't find anything on the internet that tells me what properties the morphisms in the equifier category need to have. If we want to make two transformations equal, we want, for every morphism $f: A \to B$ in the domain, the resulting squares for $\alpha$ and $\beta$ to be identical. But a transformation (natural or otherwise) is defined by its components, so if all components are equal, then the transformations are equal as a whole. But the components being equal is a condition on the objects, not any morphisms. So, maybe this means any morphism would work?

John Onstead (May 23 2025 at 07:07):

Also, any equifier is a fully faithful morphism in the 2-category it is taken in, meaning $\mathrm{Hom}(Z, f)$ is fully faithful for all $Z$. So unless a functor can be a fully faithful morphism in the 2-category of categories with unnatural transformations without being an actual fully faithful functor (which I doubt), then this should give further support to equifiers of unnatural transformations being fully faithful. It still doesn't answer my question about the retracts, though, and I have no idea where to begin on that one.
(As long as $*$ is a good enough separator in either setting, then we only need to consider morphisms from it. But since $\mathrm{Hom}(*, C) \cong C$ in both the usual category of categories and in the category of categories with unnatural transformations, then this should imply that anything that is fully faithful in one setting will be in the other)

John Onstead (May 23 2025 at 07:10):

Also, as a fun aside, I found the following math exchange answer that basically did the same construction I did with magmas and associative magmas above. Strange I didn't run into it when doing research on that! The answer even says "Once you have all this, you can use finite 2-categorical limits and the "internal logic" to construct all the usual concrete categories out of the object "set"". Of course it didn't make explicit what it meant by "usual". Obviously that would include all algebraic categories (since they are all given by monad algebras and categories of monad algebras are given as limits) but it's not clear how far beyond that class of concrete category "usual" entails.

John Onstead (May 23 2025 at 08:35):

Ah, I think I got it! I just had to understand the proof that equifiers were closed under retracts. So start with a retract diagram $X \to_r Y \to_s X$ in $C$ such that $s \circ r = \mathrm{id}_X$. Naturality of $\alpha$ forces $G(r) \circ \alpha_X = \alpha_Y \circ F(r)$. Now, if we do the postcomposition by $G(s)$, we have $G(s) \circ G(r) = G(s \circ r) = G(\mathrm{id}_X) = \mathrm{id}_{G(X)}$. So the equation becomes $\alpha_X = G(s) \circ \alpha_Y \circ F(r)$. By similar logic for $\beta$ we have $\beta_X = G(s) \circ \beta_Y \circ F(r)$.

John Onstead (May 23 2025 at 08:36):

Now, we want to equify these natural transformations. Let's assume that $Y$ is in the equifier. This means that $\alpha_Y = \beta_Y$. But this in turn implies $G(s) \circ \alpha_Y \circ F(r) = G(s) \circ \beta_Y \circ F(r)$. Which, by the above equation, immediately implies $\alpha_X = \beta_X$. But this directly implies that $X$ is in the equifier too. Thus, the equifier is closed under retracts!

John Onstead (May 23 2025 at 08:38):

And there's the catch- the proof that the equifier is closed under retracts depends entirely on the naturality of the transformations. Therefore, if the transformations are unnatural, it's no longer necessarily the case that the equifier is closed under retracts!!!!

Nathanael Arkor (May 23 2025 at 08:41):

John Onstead said:

I'm wondering why it's not possible in general to represent the tabulator as a 2-limit then. For instance, the category of elements of a $\mathrm{Set}$ functor can be rewritten as the comma category $(*/F)$, and comma categories are a type of 2-limit. So why doesn't this work in general for profunctors?

I looked into it a bit more, but this does appear to be the case.

How are you capturing the tabulator as a comma category? Note that the tabulator is not just the category of elements of the associated presheaf on $B^\circ \times A$.

Note too that, if your $\text{Cat}$ is the 2-category of small categories, then $\text{Set}$ isn't an object of the 2-category, and so you can't express the comma category you describe.

Nathanael Arkor (May 23 2025 at 08:46):

John Onstead said:

Kevin Carlson said:

What, exactly, are you claiming "appear[s] to be the case"?

That there's an equivalence between double limits in $\mathrm{Prof}$ and 2-limits in $\mathrm{Cat}$. (In the sense that anything you can construct with one you can also construct with the other)

What's the precise statement you have in mind? Every object in $\text{Cat}$ can be expressed as a trivial limit, so some care is required for this not to be tautological.

John Onstead (May 23 2025 at 09:03):

Nathanael Arkor said:

What's the precise statement you have in mind? Every object in $\text{Cat}$ can be expressed as a trivial limit, so some care is required for this not to be tautological.

I suppose what I meant was that, given any diagram and weight $F, W$ in the 2-category of categories, there is a canonical way to turn it into a double diagram in the double category of categories such that the weighted limit of the diagram and weight gives the same thing as the double limit of the double diagram. And vice versa.

John Onstead (May 23 2025 at 09:07):

Nathanael Arkor said:

How are you capturing the tabulator as a comma category? Note that the tabulator is not just the category of elements of the associated presheaf on $B^\circ \times A$.

How would the tabulator of a profunctor $F: B^{op} \times A \to \mathrm{Set}$ be different from the comma category $(*/F)$?

Nathanael Arkor (May 23 2025 at 09:18):

John Onstead said:

I suppose what I meant was that, given any diagram and weight $F, W$ in the 2-category of categories, there is a canonical way to turn it into a double diagram in the double category of categories such that the weighted limit of the diagram and weight gives the same thing as the double limit of the double diagram. And vice versa.

But this is trivially true: just pick the diagram $1 \to \text{Cat}$ picking out the limit of the original weight/diagram.

Nathanael Arkor (May 23 2025 at 09:19):

John Onstead said:

How would the tabulator of a profunctor $F: B^{op} \times A \to \mathrm{Set}$ be different from the comma category $(*/F)$?

The morphisms are different: it's the difference between the first two bullet points on [[graph of a functor]].

John Onstead (May 23 2025 at 10:22):

Nathanael Arkor said:

The morphisms are different: it's the difference between the first two bullet points on [[graph of a functor]].

Ah, that makes sense!

John Onstead (May 23 2025 at 17:54):

I suppose I should finish what I started, and answer- can any fully faithful functor be given as an equifier of unnatural transformations, now that we showed these equifiers don't have to be closed under retracts? I think the answer is yes! First, a fully faithful functor can be determined by a "proposition on objects" (or isomorphism classes if we're respecting equivalence). So, for every object, if we assign "true" or "false", then we've defined a fully faithful functor, and all such functors are defined in this way. What we need to do is to define an unnatural transformation to encode exactly the information of a proposition.

John Onstead (May 23 2025 at 17:54):

Here might be a good starting example. Let $C$ be a category with a terminal object and coproducts. Then define the endofunctor $F = (-) \cup *$. Now we have parallel functors $F, \mathrm{id}_C: C \to C$. Define a natural transformation $\alpha_X = X \to X \cup *$ the canonical morphism into this coproduct. Then, define the unnatural transformation component $\beta_X$ to be the canonical morphism into the coproduct if the proposition is "true", and define it to be the "else" morphism $X \to * \to X \cup *$ when the proposition is "false".

John Onstead (May 23 2025 at 17:54):

Now, if you take the equifier of $\alpha$ and $\beta$, any object where the proposition is "true" satisfies the equifier condition and any that do not, fail the equifier condition. Thus constructing precisely the subcategory of all objects satisfying the proposition. Even if $C$ doesn't have coproducts or a terminal object, I'm sure there's substitutions you can make. For instance, just use the Yoneda embedding instead of the identity. There's probably also a better way to formulate propositions as unnatural transformations, but this is what I could come up with. In any case, we now have a complete proof that any fully faithful functor can indeed be constructed as an equifier (with the caveat that this limit occurs in the category of categories with unnatural transformations, not the usual category of categories)

Mike Shulman (May 23 2025 at 18:00):

I haven't tried to follow everything you're saying about unnatural transformations, but just one warning: Cat with unnatural transformations is not a 2-category. Composition of unnatural transformations fails to satisfy the interchange law (that being precisely the naturality square of one of the transformations). You can still talk about "weighted limits" therein like equifiers, by using the fact that it's symmetric monoidal closed and hence enriched over itself, but you can't apply any existing 2-category theory directly.

John Onstead (May 23 2025 at 20:03):

Mike Shulman said:

I haven't tried to follow everything you're saying about unnatural transformations,

Sorry about that, I sometimes write too much, especially when I'm excited by something!

Mike Shulman said:

Cat with unnatural transformations is not a 2-category. Composition of unnatural transformations fails to satisfy the interchange law (that being precisely the naturality square of one of the transformations). You can still talk about "weighted limits" therein like equifiers, by using the fact that it's symmetric monoidal closed and hence enriched over itself, but you can't apply any existing 2-category theory directly.

Ah, well enriched category theory seems to be good enough for what I'm doing.

John Onstead (May 23 2025 at 20:15):

I've also been working on a way to generalize this result to categories of higher categories (hopefully the enriched perspective is sufficient for this too...) The first ingredient is to define an object $G$ of the $n$-category of $(n-1)$ categories to be the category with two objects and two nontrivial $(n-1)$ cells between them. So for 1-categories, $G$ is the set of 2 elements, for 2 categories it's the walking graph, for 3-categories it's the walking graph with two 2-cells from one of the arrows to the other, etc. Now, we assume we have an $n$-category that is copowered over its notion of $\mathrm{Cat}$ (IE, a 1-category copowered over $\mathrm{Set}$, etc.) Again, if not, we can always make use of the Yoneda embedding

John Onstead (May 23 2025 at 20:16):

Now, given an object $A$ of an n-category, take the copower $A \otimes G$. For 1-categories, this is just the coproduct $A \cup A$, with two canonical morphisms $A \to A \cup A$, which we can call "right" and "left". We can define a natural transformation $\alpha: \mathrm{id} \to (-) \otimes G$ whose components are all the "right" morphisms. Then, we can define an unnatural transformation whose components $\beta_A$ are "right" when a proposition is True for an object, and "left" when it is False. The equifier (2-cell equalizer) then gives the full subcategory on objects with "true".

John Onstead (May 23 2025 at 20:16):

But we can now generalize to 2-categories. If we take the copower of an object $A$ with the walking graph, we once again get two canonical morphisms which we can call "dom" and "cod", and then two canonical 2-cells from "dom" to "cod" we can call "right" and "left". We can define two natural transformations $\alpha,\beta: \mathrm{id} \to (-) \otimes G$ whose components are "dom" and "cod", and a modification between them whose components are all "right". Once again, we can define an "unnatural modification" whose component is "right" when a proposition is "True" for an object, and "left" when it is "False". We can then take the 3-cell equalizer of those two modifications to get the full 2-subcategory. And so on for all higher categories of categories!

John Onstead (May 23 2025 at 20:54):

Oh also I just had another idea that might allow us to ditch unnatural transformations and remain within the usual category of categories after all! So any natural transformation between two functors $F,G: C \to D$ can be equivalently given as a functor $C \to \mathrm{Arr}(D)$. Maybe somehow the equifier of the natural transformations becomes the equalizer of these corresponding functors! Now, an unnatural transformation can also be viewed as a functor, this time $C \to \mathrm{Sq}(D)$ where the codomain is the category of arrows and squares between them (the category $[2, D]$ under the funny tensor product's hom). So if the above statement about equifiers becoming equalizers is true, then an unnatural equifier can also be given as an equalizer purely in $\mathrm{Cat}$. So this achieves the original mission with no caveats!

John Onstead (May 27 2025 at 11:54):

Hi, I hope all of you are well! We certainly made a lot of progress last week, but we're not quite done yet. I've been thinking about a new class of limit I think can help resolve my problem: "diagrammatic limits". These were inspired by the construction of the category of associative magmas above using limits, and are essentially an abstraction of that very same construction process. A diagrammatic limit category is essentially a category of labelled diagrams- the diagram $F(A) \to B \to G(C)$ would be an example (as opposed to the less general plain diagram like $A \to B \to C$, whose categories are given by the usual functor categories). For instance, we can think of associative magmas as a labelled diagram in $\mathrm{Set}$ where one of the labels is given by the functor $(-)^2 = (-) \times (-)$.

John Onstead (May 27 2025 at 11:55):

A diagrammatic limit starts by the construction of a product category of form $D \times E \times H... \times \mathrm{Arr}(C)^n$ where there is a category $D, E, H, ...$ for every functor $F: D \to C$, etc. into $C$ we wish to use to label diagrams, and a copy of the arrow category for every morphism in the diagram (so $n$ is the number of morphisms in our desired diagram). We then take an equalizer of functors from this product into $C$ with the equalizer satisfying the condition that every pair of equalized functors contains at least one functor of form $\mathrm{cod}$ or $\mathrm{dom}$ in the pair. Equalized pairs of form $(F, \mathrm{dom/cod})$ do the labelling while those of form $(\mathrm{dom/cod}, \mathrm{dom/cod})$ stitch together the arrows of the diagram. Lastly, we can take an equifier to add commutativity/equations to the diagram.

John Onstead (May 27 2025 at 11:57):

It seems many categories can be given by such diagrammatic limits- to start, any functor category (categories of unlabeled diagrams). Second, any sketchable category, since we can label our diagrams with limits and colimits via functors of form $\mathrm{col/lim}: [J, C] \to C$. The category of metric spaces also counts, since specifying a specific object $\mathbb{R}$ in the diagram is the same as labelling with a functor $\mathbb{R}: * \to \mathrm{Set}$. Obviously, any category of endofunctor algebras, but also I believe any category of monad algebras can arise this way. Even categories not usually amenable to categorical logic, like $\mathrm{Top}$ and any category of $(T, V)$ algebras can be given this way, since they are categories of monad algebras. It almost seems like any category of standard mathematical objects arises this way!

John Onstead (May 27 2025 at 11:58):

I have a lot of questions about this class of limit, but I need to formulate them into being more precise. In the meantime do give your thoughts on this class of limit if you have them. Also, let me know if it been studied in any formal context before, if there's any literature about the properties of these limits, etc.

John Onstead (May 27 2025 at 20:44):

Here's my first question, it's pretty basic and actually doesn't even require reading the above: When is the evaluation functor $ev_A: [J, C] \to C$ a faithful functor? Is it when $A$ is terminal in $J$, or is there some other criteria?

John Baez (May 28 2025 at 08:16):

I don't think $A$ being terminal in $J$ implies $ev_A: [J,C] \to C$ is faithful. Doesn't $J = \bullet \to \bullet$, $C = \mathsf{Set}$ provide a counterexample? Or am I confused? I haven't had much coffee yet this morning.

Kevin Carlson (May 28 2025 at 16:36):

I agree that doesn’t work. I believe evaluation is only faithful for all $C$ when $J$ is a connected groupoid. You could also make $C$ be a preorder, and then every functor into $C$ is faithful.

John Onstead (May 28 2025 at 18:56):

John Baez said:

I don't think $A$ being terminal in $J$ implies $ev_A: [J,C] \to C$ is faithful. Doesn't $J = \bullet \to \bullet$, $C = \mathsf{Set}$ provide a counterexample? Or am I confused? I haven't had much coffee yet this morning.

Ah, I should have tried thinking of some example. Yes, $J = \bullet \to \bullet$, $C = \mathsf{Set}$ isn't faithful for the same reason the "set of vertices" functor from directed graphs isn't faithful (since you are forgetting the "extra stuff" of the edges).

Kevin Carlson said:

I agree that doesn’t work. I believe evaluation is only faithful for all $C$ when $J$ is a connected groupoid. You could also make $C$ be a preorder, and then every functor into $C$ is faithful.

That's interesting. I guess this makes sense, since in the 2-category $\mathrm{Cat}$ at least, every connected groupoid is equivalent to a groupoid with a single object (IE, the delooping of a group). Then it makes sense the evaluation would be faithful, since it picks out only one bit of "stuff" via this single object.

John Onstead (May 28 2025 at 19:02):

So here's my next question. Labelled graphs $F(A) \to A$ for an endofunctor $A$ on $C$ are faithful over $C$ via $\mathrm{cod}$, as the category of endofunctor algebras for $F$. Yet, the category of the underlying unlabelled graphs $B \to A$, as shown above, is not faithful via $ev_A$, setting up an interesting question. When is a category of labelled graphs faithful over the category in question? Since it seems to not depend on if the underlying unlabelled graphs are faithful or not.