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Stream: learning: questions

Topic: ✔ homotopy 2-limits as biadjunctions

Nico Beck (Jun 14 2023 at 11:51):

I have cross-posted my question on math-stackexchange (https://math.stackexchange.com/questions/4718696/equation-in-the-paper-homotopy-limits-for-2-categories :). I am trying to find a left biadjoint for the homotopy limit functor of a 2-category for a fixed weight (by homotopy limit I mean what the nLab calls weighted 2-limit). Say my 2-category $K$ has copowers or weak copowers. Is it legal to do the following?

$\textrm{Hom}(\mathscr J,\textrm{Cat})(W,K(A,D-)) \simeq \textrm{Hom}(\mathscr J,K)(W(-)\ast A,D)$

Here $\textrm{Hom}(\mathscr J,K)$ is the 2-category of pseudo-functors, pseudo-transformations and modifications. There is a similar formula in Gambino's paper, but I don't see why it is true :/ I thought maybe I there is some kind of end-formula for the diagram category, so that I can apply the copower universal property pointwise, but I am not sure. Is it true at all that pseudo-limits, homotopy limits and lax limits for a fixed weight are right 2-adjoints/biadjoints?

Nico Beck (Jun 21 2023 at 10:19):

I found this paper on lax ends and it solves my problem :) https://arxiv.org/abs/2210.01522

Notification Bot (Jun 21 2023 at 10:19):

Nico Beck has marked this topic as resolved.