Biased permutative equivariant categories

Abstract

For a finite group G, we introduce the complete suboperad Q G of the categorical G-Barratt-Eccles operad P G . We prove that P G is not finitely generated, but Q G is finitely generated and is a genuine E ∞ G-operad (i.e., it is N ∞ and includes all norms). For G cyclic of order 2 or 3, we determine presentations of the object operad of Q G and conclude with a discussion of algebras over Q G , which we call biased permutative equivariant categories.