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Stream: learning: questions

Topic: Are weighted (co)limits the more “fundamental” concept?

Ruby Khondaker (she/her) (Aug 15 2025 at 17:15):

Since reading Paolo’s wonderful “How to Represent Non-Representable Functors” paper, as well as appreciating the value of ends/coends, I’m wondering whether weighted (co)limits are more fundamental than ordinary “conical” (co)limits?

When working in Set the distinction isn’t as important, since they can be converted into each other - though admittedly I do like how things like powers, tensors and kernel pairs seem to have a “nicer” weighted description than an unweighted one. Ends and coends as Hom-weighted limits and colimits are very useful too, as well as the characterisation of pointwise Kan extensions through them, and stuff like co-yoneda as well. I’ve also heard that they’re the correct formulation in the enriched setting, as well as for formal category theory more broadly.

In the (admittedly introductory) category theory resources I’ve learned from, ordinary limits and colimits are emphasised over their weighted cousins, if the latter are even mentioned. Is there a reason for that beyond pedagogical tradition? Because otherwise it feels like, naively, weighted limits and colimits are the “natural” notion, with conical limits and colimits a special case…

Mike Shulman (Aug 15 2025 at 17:53):

I think it's pedagogical simplicity as well as tradition. Learning the abstract concept of limit is hard enough at first for the ordinary version.

Ruby Khondaker (she/her) (Aug 15 2025 at 17:58):

Interesting, perhaps I’ve underestimated how hard limits are on a first pass. What do people typically find difficult about them? When I was learning category theory, I found them hard but not harder than the surrounding material; indeed, yoneda was probably the thing I found hardest out of my introductory course!

Maybe I’ve just been spoiled by (co)end calculus…

John Baez (Aug 15 2025 at 17:59):

Weighted limits and colimits are more natural, and this becomes very apparent in enriched category theory - but as Mike pointed out, math is hard enough to learn without having to learn the ultimate "best" concepts right away, since they are usually more abstract and general.

Weighted colimits are a categorified version of integration. When you first learn about integrals, the little $dx$ in

$\displaystyle{ \int_a^b f(x) \, dx }$

seems like a stupid thing you put in just because the teacher tells you to: you think you're trying to integrate a function.

But in fact you're trying to integrate a function against a measure. The interval $[a,b]$ comes with a fairly god-given measure $d x$, but a general measure space $X$ does not, and then the integral

$\displaystyle{ \int_a^b f \, d\mu }$

cares just as much about the measure $\mu$ as the function $f$. Measures and functions have opposite variance, and they mate to produce numbers.

When we categorify, the measure becomes the weight, and the function becomes the functor that we're trying to take the weighted colimit of. The weight and the functor have opposite variances, and they mate to produce an object: the colimit.

When we're first learning about colimits we encounter situations where there's a canonical weight, so the weight recedes into the background, but in general we need to pay attention to the weight.

Mike Shulman (Aug 15 2025 at 18:06):

One thing that I've found that some people find difficult about even ordinary limits is representing an arbitrary "diagram drawn on the page" by "a functor out of a small category". If that's already challenging, then having another set-valued functor around telling you "how many arrows to draw to each target" would be extra difficult.

Ruby Khondaker (she/her) (Aug 15 2025 at 18:14):

Ah, that’s fascinating! And come to think of it, I remember struggling with that myself when I learned category theory. How do students manage to get over that conceptual hurdle?

John Baez (Aug 15 2025 at 18:21):

They just wake up one morning and it's so obvious they forget how they figured it out. :stuck_out_tongue_wink:

John Baez (Aug 15 2025 at 18:24):

Seriously, I had to think about the 'free category on a graph' and how we can use a graph to present a category, and how sometimes when we draw a diagram we say "this is a commuting diagram" and sometimes we don't, and what that choice means in terms of the 'free category on a graph' stuff.... there's a fair amount to work through here.

Mike Shulman (Aug 15 2025 at 18:33):

It's probably different for everyone. After I figure something out I usually have a hard time remembering what it was like to not understand it.

Ruby Khondaker (she/her) (Aug 15 2025 at 18:54):

That's very interesting indeed! I wonder how one might best introduce them to the concept of thinking of diagrams as functors (and conversely, functors as diagrams)..

John Baez (Aug 15 2025 at 19:05):

I just draw a diagram "in a category" $\mathsf{C}$ and then the abstract diagram "as a category" $\mathsf{D}$, and then a functor mapping $\mathsf{D}$ to $\mathsf{C}$. Like for example the walking pullback $\mathsf{D}$ getting mapped into some specific pullback diagram in $\mathsf{C}$.

John Baez (Aug 15 2025 at 19:06):

The category $\mathsf{C}$ is, as usual, a big blob with a bunch of dots and arrows in it....

I believe that people benefit a lot from pictures, even obvious pictures - i.e., the mental pictures I automatically get as soon as someone describes something. Because pictures make things more concrete.

Mike Shulman (Aug 15 2025 at 19:30):

And because a picture that's obvious or automatic to one person may not yet be obvious to another.

John Baez (Aug 15 2025 at 20:26):

Right!

Alex Kreitzberg (Aug 15 2025 at 20:35):

(Thanks for recommending Paolo's essay, so far it looks very helpful)

Jonas Frey (Aug 15 2025 at 22:18):

Contrary to some people, I like the question of what mathematical concepts are more "fundamental" than others, but I typically equate fundamental with "prior" or "preceding" in the historical and didactical hierarchy of how concepts have emerged and are best explained (which I think is at a first approximation often the same). From this perspective, conical (co)limits are clearly more fundamental than weighted (co)limits.

One interesting perspective is also given by $\infty$-categories: In Joyal's theory of quasicategories, cones and conical limits play a central role and are much easier conceptually than weighted limits, since the latter require the $\infty$-category of homotopy-types and the notions of (co)presheaf and hom-functor, which require a lot of work to define. These complications are somehow less visible for $1$-categories.

Ruby Khondaker (she/her) (Aug 15 2025 at 22:29):

Interesting - what makes the Hom-functor harder to define for $\infty$-categories?

Jonas Frey (Aug 15 2025 at 22:31):

A related point that I haven't seen spelled out anywhere: limits are more fundamental than colimits!

Specifically, the fundamental concept is that of limit in the "background" category of sets or types. For example, a commutative square

$$\begin{CD} A @>{f}>> B \\ @V{g}VV @VV{h}V \\ C @>>{i}> D \end{CD}$$

is a pushout in $\mathcal{C}$ iff

$$\begin{CD} \hom(A,X) @<{\hom(f,X)}<< \hom(B,X) \\ @A{\hom(g,X)}AA @A{\hom(h,X)}AA \\ \hom(C,X) @<<{\hom(i,X)}< \hom(D,X) \end{CD}$$

is a pullback in the background for all $X$.

Jonas Frey (Aug 15 2025 at 22:46):

Ruby Khondaker (she/her) said:

Interesting - what makes the Hom-functor harder to define for $\infty$-categories?

I think basically that the quasicategory of spaces/types (the codomain of hom-functors) is defined as the homotopy coherent nerve of a simplicial category, and you have to go through this indirection, which makes everything very non-explicit.

But other people might have other takes on this, and of course you might prefer a model-independent synthetic approach where you postulate hom-functors.

Jonas Frey (Aug 15 2025 at 23:10):

I now realize that the two points that I made are somewhat contradictory: the claim that limits are more fundamental requires hom-functors, and is not observable technically in the conical presentation of co/limits in quasicategories.

Mike Shulman (Aug 15 2025 at 23:55):

Actually all you need in order to define weighted limits is a notion of profunctor. You don't have to be able to reinterpret those as functors into some ambient enriching category. In particular, therefore, you can define weighted limits for $V$-enriched categories even when $V$ doesn't have enough internal-homs for there to be a $V$-category $V$. And Riehl and Verity, I believe, defined a calculus of profunctors (= virtual equipment) for quasicategories without going through a quasicategory of quasicategories.

Jonas Frey (Aug 16 2025 at 00:01):

That's true (and now I remember the treatment of weighted co/limits in equipments in your paper "Enriched indexed categories"), but if that makes it more fundamental really depends on what you mean by "fundamental".

Mike Shulman (Aug 16 2025 at 00:24):

A sentence that would be true for any value of "fundamental"! (-:

Morgan Rogers (he/him) (Aug 18 2025 at 17:18):

Ruby Khondaker (she/her) said:

That's very interesting indeed! I wonder how one might best introduce them to the concept of thinking of diagrams as functors (and conversely, functors as diagrams)..

I taught an intro to CT course last week in which I approached this the other way around: after introducing functors I used functors out of the walking object and the walking arrow as examples and hinted that we could use functors to pick out a general "situation" in a category. Then "diagram" was later defined as just another name for a functor. I pointed out that we could index by a directed graph instead of a category, but never needed the extra generality. On the other hand, I never gave a very formal definition of what it meant for a diagram to commute.

Ruby Khondaker (she/her) (Aug 18 2025 at 18:05):

@Morgan Rogers (he/him) thanks for sharing! I quite like this approach - it seems to lean into Lawvere's "philosophy of generalised elements", right? Namely how one can reinterpret a morphism $A \to B$ as an "A-shaped element of B", or "A-shaped figure in B". If you apply that to Cat, then you get the notion of a diagram!

Morgan Rogers (he/him) (Aug 18 2025 at 18:35):

More generally I like the principle of having several words for a given structure that reflect an attitude towards or perspective on that structure. Besides functor (which is fairly balanced) and diagram (which focuses on the codomain) we also have "presheaf" (which up to variance focuses on the indexing by the domain, although the codomain is usually fixed as Set in this case).

Ruby Khondaker (she/her) (Aug 18 2025 at 18:36):

Oh for sure! I would add "bundle" to that list, in terms of viewing a morphism $E \to B$ (so an object of the slice category over B) as a "bundle over B".

Joe Moeller (Aug 18 2025 at 19:19):

I learned most of my basic category theory in a class taught by @John Baez, so I'm adding on to what he's said. What I remember is him drawing diagrams for several weeks including talking about co/products, pullbacks, equalizers, then general co/limits, and only after all that mentioning that you can think of a diagram as a functor and cones as natural transformations. At that point it was obvious.