Weighted limits and Kan extension in Dist

Question

(noting $\otimes$ for composition in distributors, $\phi_f : A \nrightarrow B = B(-,f=)$ and $\phi^f : B \nrightarrow A = B(f-,=)$ the embeddings of a functor $f:A\to B$ in $Dist$, and $Dist(A,B) = [B^{op}\times A, V]$ for the category of distributors from $A \nrightarrow B$)

I am looking at 2 universal properties:

• The UP of the weighted limit of a functor $D : A \to M $ and weights $J : A \nrightarrow B$ is, naturally for $m :1 \to M$ and $b : 1 \to B$

$$ [1,M] (m,(Lim^JD)b) \simeq [A,V] (J(b,-), M(m,D -)) $$

which can be rewritten

$$ Dist(B,M)(\phi_m\otimes\phi^b, \phi_{Lim^JD}) \simeq Dist(A,M) (\phi_m\otimes \phi^b \otimes J, \phi_D ) $$

• The UP for the right kan extension in Dist $R : M^{op} \times B \to V$ of $M(-, D=) : M^{op}\times A \to V$ along $J:B^{op} \times A \to V$ is, for $H : M^{op}\times B \to V$

$$Dist(B, M)(H, Ran_{J}\phi_D) \simeq Dist(A,M)(H\otimes J, \phi_D)$$

When special cased to $H$ of the form $\phi_m \otimes \phi^b$, this gives the previous UP. So some $l : B \to M $ representing the right kan extension as $\phi_l \simeq Ran_{J}\phi_D: B \nrightarrow M$ is required to verify more than the UP for $l : B \to M $ to be a weighted limit.

Is there a clever way to highlight the difference between a weighted limit and a representation of a right kan extension in $Dist$, which would shed lights on the difference of nature between the two ?

(Maybe some additional property which says when a weighted limit also a representation.. ? Or some case where one exist and not the other ?)

Answer 1

I'm not sure I've followed your question exactly, so let me rephrase it to how I've understood it, and you can tell me if I've misunderstood. I shall use different letters to make sure I'm not accidentally using them inconsistently with your conventions.

For a functor $s \colon X \to Z$, I will denote by $s_* \colon X \nrightarrow Z$ the corepresentable distributor $Z({-}_2, s{-}_1)$.

The question essentially appears to be this:

What is the relationship between right (Kan) extensions in the bicategory of distributors, and weighted limits in the 2-category of small categories?

As you say, a right extension in $\mathrm{Dist}$ is defined by the following universal property: $$\mathrm{Dist}(Y, Z)(h, \mathrm{Ran}_g f) \cong \mathrm{Dist}(X, Z)(h \otimes g, f)$$

Now suppose we have a distributor (the weight) $\Psi \colon X \nrightarrow Y$ and a functor $s \colon X \to Z$ (the diagram). The $\Psi$-weighted limit of $s$ is a functor $\{ \Psi, s \} \colon Y \to Z$, defined to satisfy $(\{ \Psi, s \})_* \cong \mathrm{Ran}_\Psi (s_*)$. The weighted limit thus satisfies the following universal property: $$\mathrm{Dist}(Y, Z)(h, (\{ \Psi, s \})_*) \cong \mathrm{Dist}(X, Z)(h \otimes \Psi, s_*)$$ In other words, a weighted limit is precisely a corepresentable right extension along a corepresentable distributor.

One reference I find helpful for this topic is Wood's Abstract pro arrows I, though this perspective also appears in other literature by Street, Betti, and others.