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Stream: learning: questions

Topic: Left adjoints preserve left Kan extensions

David Egolf (he/him) (Apr 04 2026 at 15:52):

I was recently delighted to learn that not only do left adjoints preserve colimits, they in fact preserve all left Kan extensions! (I learned this from Riehl's book, "Categorical Homotopy Theory", on page 7).

If we have a functor $F:J \to C$ which has a colimit, then its left Kan extension along $P:J \to 1$ (where $1$ is a category with a single object and a single morphism) is in fact a colimit for $F$. So, if I understand correctly, "left adjoints preserve colimits" is a nice special case of "left adjoints preserve left Kan extensions".

I am curious about other examples! What are some left Kan extensions (not corresponding to taking colimits), where it is interesting or useful to note that they are preserved by left adjoints?

Oscar Cunningham (Apr 04 2026 at 16:24):

A left adjoint is the Kan extension of identity along the right adjoint. So left adjoints preserve left adjoints.

David Egolf (he/him) (Apr 04 2026 at 17:48):

Thanks for pointing out that there is a close connection between adjoints and Kan extensions! That's quite interesting to know. However, the left adjoint corresponds to a right Kan extension, though, doesn't it?

Referencing "Categories for the Working Mathematician" (page 248):

David Egolf (he/him) (Apr 04 2026 at 17:52):

However, I think right adjoints are closely related to certain left Kan extensions! Referencing Riehl's book again (on page 9):

David Egolf (he/him) (Apr 04 2026 at 18:08):

At any rate, the kind of setup we have changes when we apply a left adjoint $L$ to the situation involving a left Kan extension associated to a right adjoint. In particular, after application of $L$, we aren't contemplating a left Kan extension of an identity functor anymore:

Here, $L:C \to X$ is a left adjoint, and $\mathrm{Lan}_F(1_C)$ is a left Kan extension of $1_C$ along $F$. When we apply $L$ to get a new left Kan extension (using the fact that left adjoints preserve left Kan extensions), we notice that the resulting left Kan extension is a left Kan extension of $L$ along $F$ (not of an identity functor along $F$).

David Egolf (he/him) (Apr 04 2026 at 18:13):

So I would assume this new left Kan extension doesn't describe a right adjoint anymore.

Peva Blanchard (Apr 05 2026 at 21:29):

David Egolf (he/him) said:

So, if I understand correctly, "left adjoints preserve colimits" is a nice special case of "left adjoints preserve left Kan extensions".

Actually, I think that, in some situations, the two statements are equivalent. If I remember correctly, a left Kan extension can be defined pointwise as a colimit, provided $C$ is, e.g., cocomplete.

David Egolf (he/him) (Apr 06 2026 at 22:36):

Thanks for your comment!

I agree that (under certain conditions) the functor part of a left Kan extension can indeed be computed at each object as a colimit. It's not clear to me that preserving a left Kan extension then (under these conditions) corresponds to preserving some colimit, though.

Morgan Rogers (he/him) (Apr 07 2026 at 17:47):

That is an apparently stronger claim, but it turns out that the entire functor can be derived from the presentation in terms of colimits at the objects. That's a technical but potentially enlightening exercise.
It's also worth saying that there aren't so many Kan extensions we are able to construct outside of the nice pointwise cases, so you'll have to find someone experienced in the topic to give some examples.

David Egolf (he/him) (Apr 07 2026 at 18:44):

Here's the formula given in "Categorical Homotopy Theory", by Riehl:

Riehl also gives this:

David Egolf (he/him) (Apr 07 2026 at 18:44):

Riehl goes on to say that each formula encodes a "weighted colimit" of $F$, which is a new term to me!

David Egolf (he/him) (Apr 07 2026 at 18:46):

So, in this case, it sounds like one can think of a left Kan extension as a collection of colimits (one for each object it maps from). It would take me some more effort to understand what "left adjoints preserve left Kan extensions" tells us in this context.

David Egolf (he/him) (Apr 07 2026 at 18:49):

One might guess something like "left adjoints preserve certain collections of related colimits".

David Egolf (he/him) (Apr 07 2026 at 19:00):

If I worked it out correctly, I think we should have: $L(\mathrm{colim}(F \circ U:K/d \to C \to \mathcal{E})) \cong \mathrm{colim}(L \circ (F \circ U))$ for each object $d \in \mathcal{D}$, where $L$ is some left adjoint functor mapping from $\mathcal{E}$. So, indeed $L$ is preserving a colimit for each object of $\mathcal{D}$.

David Egolf (he/him) (Apr 07 2026 at 19:09):

The colimits involved in a left Kan extension of $F$ are related in a certain way, given that they assemble to form a left Kan extension of $F$. And the fact that $L$ preserves left Kan extensions means that it sends this collection of colimits to a collection of colimits which assemble to form a left Kan extension of $L \circ F$. One might wonder if there are applications in which the preservation of a relationship like this between colimits is useful.

John Baez (Apr 07 2026 at 23:23):

Just two tiny points:

1. When we do colimits in enriched categories, it's almost essential to use "weighted" colimits.

2. Colimits resemble integrals, and weighted colimits resemble them even more. When we start out learning integrals we write $\int f(x) dx$ but focus our attention on the $f(x)$: the $d x$ feels like an annoying thing the teacher makes us write down. Later we learn measure theory and write $\int f(x) d\mu(x)$, and we learn that the function $f$ and measure $d\mu$ are equally essential: integration is a way of pairing a function and a measure and getting a number. It turns out we had been blind to the measure's importance because the real line comes with a standard choice of measure. Similarly, when we do weighted colimits we learn that the "weight" is just as important as the diagram we are taking the colimit of. In the Set-enriched case we were blinded to its importance because there was a standard choice.

David Egolf (he/him) (Apr 09 2026 at 02:28):

That is interesting! It's intriguing to learn of that parallel.

Inspired by your post, I did some reading and attempted to better understand weighted colimits, but it felt difficult!

David Egolf (he/him) (Apr 09 2026 at 02:31):

Attempting to make a connection with left Kan extensions:

Let $F:C \to E$ (where $E$ is cocomplete) be a functor and $y:C \to [C^{\mathrm{op}}, \mathsf{Set}]$ be the Yoneda embedding. Then the left Kan extension of $F$ along $y$ is given on objects as:

$\mathrm{Lan}_yF:W \mapsto \int^{c \in C}[C^{\mathrm{op}}, \mathsf{Set}](C(-,c), W) \cdot F(c)$

David Egolf (he/him) (Apr 09 2026 at 02:34):

If it is safe to replace $[C^{\mathrm{op}}, \mathsf{Set}](C(-,c), W)$ with $W(c)$ using the Yoneda lemma, we get:

$\mathrm{Lan}_yF:W \mapsto \int^{c \in C} W(c) \cdot F(c)$

David Egolf (he/him) (Apr 09 2026 at 02:36):

But we also have that the weighted colimit of $F$ with respect to $W$ is given as $\mathrm{colim}^W F \cong \int^{c \in C} W(c) \cdot F(c)$ ("Categorical Homotopy Theory" p. 84)!

David Egolf (he/him) (Apr 09 2026 at 02:38):

If the above is true, it seems like the left Kan extension of a functor $F$ along the Yoneda embedding takes weighted colimits of $F$! That is, it sends a "weighting" functor $W$ to the weighted colimit of $F$ with respect to $W$.

David Egolf (he/him) (Apr 09 2026 at 03:38):

I think this can help me begin to understand why the weighting is so important. As a special case of the above, we can consider $F:\Delta \to \mathsf{Top}$ which sends a "combinatorial" simplex to the standard topological realization of that simplex. Then, the weighted colimit of $F$ with respect to a simplicial set $X$ is the "geometric realization" of $X$!

John Baez (Apr 09 2026 at 04:45):

That's a very good example. In a crude sense, we are "adding up" the standard topological realizations of simplices to get the geometric realization of X, and X serves as a weight telling us how many copies of each simplex we need. But X does more: it tells us how they're glued together.