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.BANGS, BINEGAR, KIM, ORMSBY, OSORNO, TAMAS-PARRIS, AND XU a finite group G. This operad, P G , has the property that its classifying space is a genuine E ∞ G-operad, and thus, its algebras give rise to genuine equivariant infinite loop spaces. Because of this, Guillou and May define permutative G-categories as algebras over P G .Following [GMMO18], 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 2.15 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.Using the work of Rubin [Rub17, Rub18], we construct suboperads Q G of P G that are still E ∞ , yet are finitely generated. The key insight from Rubin, which is inspired by the work on N ∞ operads of Blumberg and Hill [BH15], is that the full suboperad generated by a collection of norms will be E ∞ , as long as one includes all the norms for orbits as generators.Finally, in Theorems 3.5 and 3.7 we give explicit presentations for the operads Q G in the cases where G = C 2 and G = C 3 . Although the statements of the proofs look very similar, the proofs that the relations given are sufficient are strikingly different. We use these results together with Rubin's coherence theorem for normed symmetric monoidal categories [Rub18] to give a biased definition of Q G -algebras.Organization. In Section 1, we recall necessary preliminary notions regarding permutations, operads in general, and the categorical G-Barratt-Eccles operad P G and its operad of objects P G . In Section 2, we prove that P G is not finitely generated for nontrivial G (Theorem 2.15). In Section 3, we introduce the finitely generated E ∞ G-operads Q G and determine presentations of the operad of objects when G = C 2 or C 3 . Finally, in Section 4, we define the notion of a biased permutative G-category for G = C 2 or C 3 and prove that these are in one-to-one correspondence with Q G -algebras.Acknowledgements. The authors express their deep gratitude to Jonathan Rubin, who very generously explained the results of [Rub18] in detail.

SYMMETRIC MONOIDALG-CATEGORIES AND THEIR STRICTIFICATION

“…We include it because it is often convenient and much of the relevant categorical literature focuses on it. 2 We are greatly indepted to Power and Lack for correspondence about this result.…”

“…Power discovered [27] and Lack elaborated [18] a remarkably simple way to strictify structures over a 2-monad. 2 Power's short paper defined the strictification St on pseudoalgebras, and Lack's short paper (on codescent objects) defined St on 1-cells and 2-cells. The result and its proof are truly beautiful category theory.…”

“…That is, it is desirable to determine explicit additional structure on a naive permutative or symmetric monoidal G-category that suffices to give it a genuine structure. As shown in [2] by the fourth author and her collaborators, this problem cannot be solved. More precisely, they show that if G is a nontrivial finite group, the operad P G is not finitely generated.…”

“…Using ideas from Rubin [30], one can produce a finitely generated suboperad Q G of P G that is equivariantly equivalent, in the sense that it is also an E ∞ G-operad. Bangs et al solve in [2] the problem of identifying biased specifications for algebras over Q G for G = C p when p = 2, 3.…”