# Kan extensions of pseudofunctors

Can anyone suggest a reference for (left) Kan extensions of pseudofunctors?

In particular, say we are given bicategories $\mathscr{A,B,C}$ and pseudo functors $\mathscr A \xrightarrow{G} \mathscr C$ and $\mathscr A \xrightarrow{F} \mathscr B$, where $\mathscr C$ has all pseudo colimits. Then is the pseudo left Kan extension $\mathrm{PsLan}_F(G)$ of $G$ along $F$ given pointwise by the pseudo analogue of the standard formula for left Kan extensions of functors between 1-categories?

(I'm assuming that $\mathscr A$ is appropriately small).

EDIT: I am mostly interested in the case where $\mathscr A$ is actually a 1-category, and the functors $F$ and $G$ are strict functors.

• Have you looked at Emily Riehl's book yet? My gut instinct is that she would take this approach. – David White Sep 1 '15 at 12:22
• The "standard" formula doesn't generalise well. You would be better off starting with the formula from enriched category theory in terms of weighted colimits. – Zhen Lin Sep 1 '15 at 16:11
• @DavidWhite: Yes, I have looked in Riehl's book. Kan extensions of enriched functors between enriched categories are covered there, but as far as I can tell everything is done for strict and not pseudo functors. – James Waldron Sep 1 '15 at 21:31
• It depends on what you mean. As Finn Lawler explained, you only need (pseudo)colimits of diagrams of shape $\mathcal{A}$ weighted by certain (pseudo)functors. Whether you can reduce to conical (pseudo)colimits or not depends on the weights – specifically, whether the weights themselves can be reduced to conical (pseudo)colimits of representables. – Zhen Lin Sep 1 '15 at 21:52
• Sorry for the delayed comment. I just saw your question now (I am sorry about that). But I had worked out the concept of pointwise pseudo-Kan extensions: actually, I gave a talk mentioning them at University of Aveiro (CT 2015). I have two papers mentioning pseudo-Kan extensions. One was published "On Biadjoint Triangles" (TAC) and the other one can be found in CMUC's preprints (16-30). On one hand, conical bilimits do not work in general. On the other hand, the "formula" for pointwise right pseudo-Kan extensions is pretty similar to the pointwise right Kan extensions (via weighted bilimits – Fernando Jun 15 '16 at 2:08

## 2 Answers

I haven't seen this written down anywhere, but I've worked it out myself in a paper I'm working on. The paper isn't quite ready, so I'll just sketch the idea here.

As Zhen says, you need to generalise the weighted-colimit definition and put $$(\mathrm{Lan}_F G) b = \mathcal{B}(F-,b) \star G$$ Then you can express its universal property in two ways: $$\mathrm{Ps}(\mathrm{Lan}_F G, E) \simeq \mathrm{Ps}\big(\mathcal{B}(F-, -), \mathcal{C}(G-, E-)\big) \simeq \mathrm{Ps}(G, EF)\,.$$ The first equivalence is the definition of a weighted colimit, and the second follows from some profunctor-y manipulations and the bicategorical Yoneda lemma. But now the composite of the two expresses precisely what you want a bicategorical left Kan extension to be.

Another reference is section 10 of https://arxiv.org/abs/1301.3191.