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.

  • $\begingroup$ Have you looked at Emily Riehl's book yet? My gut instinct is that she would take this approach. $\endgroup$ – David White Sep 1 '15 at 12:22
  • $\begingroup$ 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. $\endgroup$ – Zhen Lin Sep 1 '15 at 16:11
  • $\begingroup$ @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. $\endgroup$ – James Waldron Sep 1 '15 at 21:31
  • $\begingroup$ @DavidWhite: Also, the categories there are 'enriched' rather that 'weakly enriched', as in the case of bicategories. $\endgroup$ – James Waldron Sep 1 '15 at 21:37
  • 2
    $\begingroup$ 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. $\endgroup$ – Zhen Lin Sep 1 '15 at 21:52

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.


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