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Tim Hosgood (Oct 26 2021 at 12:15):this Thursday (the 28th) at 17:00 UTC (back to our usual time)
Tim Hosgood (Oct 26 2021 at 12:15):Dorette Pronk: Doubly Lax Colimit of Double Categories with Applications
Thus far, lax and oplax pseudo colimits of double categories have been considered in two flavours [2]: horizontally lax and vertically lax, based on the notions of horizontal and vertical transformations (respectively) between double functors. Also, the diagrams of double categories have typically been indexed by a 2-category.
In this work we introduce diagrams indexed by a double category; in order to make sense of this we will map into a version of the quintets of the category of double categories, because this category itself is only enirched in double categories and is often taken as a 2-category. Between the new indexing functors we introduce a new notion of transformation, namely doubly lax transformation. We then introduce a double categorical version of the Grothendieck construction and show that it has a universal property as doubly lax colimit of the diagram; i.e., a colimit that is lax with respect to the new transformations.
As applications we obtain:
— a universal property as lax colimit for the Grothendieck construction for bicategories described in [1];
— a universal property for the elements construction for double categories;
— a notion of fibration for double categories, different from the internal one described by Street and others;
— a double categorical generalization of the classical tom Dieck fundamental groupoid for a space with an action by a topological group.
This is joint work with Marzieh Bayeh (University of Ottawa) and Martin Szyld (Dalhousie University).
Tim Hosgood (Oct 26 2021 at 12:16):Zoom: https://topos-institute.zoom.us/j/5344862882?pwd=Znh3UlUrek41T3RLQXJVRVNkM3Ewdz09
YouTube: https://youtu.be/0OesTY87uao
Tim Hosgood (Oct 28 2021 at 16:54):starting in 5!
Tim Hosgood (Oct 28 2021 at 16:54):(minutes, that is)