SYMMETRIC MONOIDALG-CATEGORIES AND THEIR STRICTIFICATION

Abstract

We give an operadic definition of a genuine symmetric monoidal G-category, and we prove that its classifying space is a genuine E∞ G-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power, and Lack, to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal G-categories to genuine permutative G-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations.When G is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal G-categories as input to an equivariant infinite loop space machine that gives genuine Ω-G-spectra as output.

Smashing localizations in equivariant stable homotopy

“…Therefore we have 12) ẼF (1234) (12) ẼF (1234) is the universal (12) D 8 space ẼF (1342) , and hence i (12) D 8 V 4…”

“…We end this section with an example illustrating the necessity of the normality conditions in Corollary 3. 20 24), ( 12)( 34), ( 14)( 23), ( 1234), ( 1432), ( 13), (24 12) D 8 = {e, (13)( 24), ( 12) (34), ( 14)( 23), ( 1342), ( 1243), ( 14), ( 23)} D 8 ∩ (12) D 8 = {e, (13)( 24), ( 12) (34), (14…”

“…An account of this construction is to appear in [31]. This construction appears in the context of the symmetric monoidal G-categories of Guillou, May, Merling, and Osorno (see 3.7 of [12]), the normed symmetric monoidal categories of Rubin (see 3.7 of [39]), and the symmetric monoidal Mackey functors of Hill-Hopkins (see 2.6 [16]). By use of N , we may therefore send an idempotent e in Mod C (A) to an idempotent N (e) in L A .…”

Biased permutative equivariant categories

“…Following [GMMO20], one may ask if there is a biased definition of permutative G-categories, as there is for permutative categories. One of the main results of this paper, Theorem 2.13 is that in the strictest sense, the answer is no for nontrivial groups G. Indeed, we prove that the object part of P G is not finitely generated, meaning that one needs to specify infinitely many operations to give an algebra over it.…”

2-Segal sets and the Waldhausen construction

“…The appendix of[17] gives sufficient conditions for d * to preserve finite limits and hence pullbacks.…”