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Tom Bachmann will speak on

Motivic Colimits

Abstract: Given infinity categories I and C and a functor X: I -> C, there may or may not exist a so-called homotopy colimit of the I-indexed diagram X. This construction generalises for example homotopy pushout and homotopy orbits; it is clearly of fundamental importance in homotopy theory. In equivariant homotopy theory there is a generalisation of this concept into so-called G-homotopy colimits, where the group G acts on the diagram (I, X) [Dotte-Moi, Homotopy theory of G-diagrams and equivariant excision]. This suggests that in the case where C is a "motivic" homotopy category, one might look for a theory of colimits indexed on "motivic diagrams", in some sense.

In this talk we provide a definition of "motivic homotopy colimits" indexed on (more or less) motivic infinity *groupoids*. We explain some of the basic properties of this construction, and then mostly focus on examples to illustrate its utility.