  We define a notion of colimit for diagrams in a motivic category indexed by a
presheaf of spaces (e.g. an \'etale classifying space), and we study basic
properties of this construction. As a case study, we construct the motivic
analogs of the classical extended and generalized powers, which refine the
categorical versions of these constructions as special cases. We also offer
more computationally tractable models of these constructions using equivariant
motivic homotopy theory. This is the first in a series of papers on power
operations in motivic stable homotopy theory.
