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: Intuition for Kan Extensions

Ruby Khondaker (she/her) (Dec 08 2025 at 13:29):

I've been learning about Kan extensions recently, and they seem like an extremely cool concept - I can see why MacLane said that "all concepts are Kan extensions". I'd be very interested to hear how people like to think of Kan extensions (and their ubiquity in category theory) intuitively.

Paolo Perrone (Dec 08 2025 at 14:32):

I like looking at them as 'partial limits' (right Kan extensions) and 'partial colimits' (left Kan extensions).
As in, take a diagram, and instead of taking its limit, which is a single object:

• Cut the diagram into pieces ('slices')
• Take the limit of each piece
• By the universal property of limits, there are canonical arrows induced between these limits
• The resulting (usually smaller) diagram is a right Kan extension.

This can be made precise for pointwise Kan extensions.

Ruby Khondaker (she/her) (Dec 08 2025 at 14:32):

Ah yes, I recently watched the talk you gave on this! The monad of diagrams is a pretty neat concept :)

Paolo Perrone (Dec 08 2025 at 14:33):

(I've written about that here and here, but I'm by far not the only one, nor the first one to think this way, see the references.)

Nathanael Arkor (Dec 08 2025 at 14:49):

If you consider limits/colimits that are weighted by [[distributors]] (rather than just co/presheaves), then right and left extensions are precisely limits and colimits (weighted by representable distributors and corepresentable distributors respectively). I think this is the nicest way to view left/right extensions, as it means that all the facts you know about limits/colimits apply immediately to left/right extensions.

Nathanael Arkor (Dec 08 2025 at 14:51):

As far as I know, this idea is originally due to Street and Walters in the study of [[Yoneda structures]], but it's useful even for doing ordinary/enriched category theory.

Ruby Khondaker (she/her) (Dec 08 2025 at 14:53):

Hm, how does one change the weight to a distributor? As far as I can tell, it makes sense to take W-weighted limits _of_ distributors, but the weight in this case would still be a copresheaf.

Nathanael Arkor (Dec 08 2025 at 14:56):

If $p : Y \not\to Z$ is a distributor (here meaning contravariant in $Z$) and $f : Z \to X$ is a functor, then the $p$-weighted colimit of $f$ is a functor $p * f : Y \to X$ satisfying $X((p * f)(y), x) \cong [Z^{\text{op}}, \text{Set}](p(-, y), X(f{-}, x))$.

Nathanael Arkor (Dec 08 2025 at 14:57):

In particular, given a functor $j : Z \to Y$ and taking $p := Y(j, 1)$, we get the definition of a (pointwise) left extension.

Ruby Khondaker (she/her) (Dec 08 2025 at 14:58):

Ah I see what you mean now, makes sense - you want each object of Y to produce a presheaf on Z that you can take the weighted colimit with respect to, and that’s precisely what a distributor does for you!

So this is essentially the end/coend formula for right/left kan extensions, if I’ve understood correctly