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Daniel Plácido (Jul 20 2021 at 15:43):Is there a model structure on bicategories and strictly unitary lax functors between them? I wanted to understand if/how the Duskin nerve, which goes from this category to simplicial sets, is right Quillen. Lack's popular model structure won't fit into this picture, because it's a model structure on strict 2-functors.
Mike Shulman (Jul 20 2021 at 16:49):Does that category have limits and colimits? It seems doubtful to me, which would preclude the existence of a model structure in the ordinary sense.
Daniel Plácido (Jul 23 2021 at 13:47):I guess it does not. However Lurie outlines in kerodon that the Duskin nerve of strict 2-categories can be written as a map , so perhabs that is a right Quillen functor. Thanks!
Alexander Campbell (Jul 24 2021 at 00:22):Daniel Plácido said:
I guess it does not. However Lurie outlines in kerodon that the Duskin nerve of strict 2-categories can be written as a map , so perhabs that is a right Quillen functor. Thanks!
But with respect to which model structure on sSet?
Alexander Campbell (Jul 24 2021 at 00:27):The most natural thing to do is to take the codomain of the Duskin nerve functor to be the category of scaled simplicial sets, equipped with the model structure for ∞-bicategories. I think this is shown to be right Quillen in a paper of Gagna--Harpaz--Lanari.