Operads in unstable global homotopy theory

Abstract

We study operads in unstable global homotopy theory, which is the homotopy theory of spaces with compatible actions by all compact Lie groups. We show that the theory of these operads works remarkably well, as for example it is possible to give a model structure for the category of algebras over any such operad. We define global E 1 -operads, a good generalization of E 1 -operads to the global setting, and we give a rectification result for algebras over them. 55P91 1. Introduction 3293 2. Background 3299 3. G-orthogonal spaces 3307 4. Main results for operads in .Spc; / 3320 5. Global E 1 -operads 3334Appendix. More about G-orthogonal spaces 3338References 33542 Background OperadsLet Ꮿ be a cocomplete symmetric monoidal category, where the tensor product preserves all small colimits in both variables. We follow the exposition of [9] to define operads

Genuine versus naïve symmetric monoidal $G$-categories

“…Recently, Barrero [1] studied a notion of global E 1 -algebras in Schwede's orthogonal space model of global homotopy theory [38]. Our approach presented here differs from his treatment in basically two ways.…”

“…Applying our equivalence G -ParSumCat 1 ' Alg E † E G .G -Cat/ 1 from Theorem 5.29 now produces a genuine permutative G-category P G .R/ from this, which we can feed into the Guillou-May machinery. Theorem 6.7.…”

“…underlying . By [12, Corollary 2.5 (iii)] it therefore suffices to construct the equivalence filling (A.2) after postcomposition with the forgetful functor to G -Cat 1 underlying , where both paths through the diagram can simply be identified with the forgetful functor (see [20, proof of Lemma 6.12] for the left-hand composite).…”