In this paper 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∞-operads, a good generalization of E∞-operads to the global setting, and we give a rectification result for algebras over them. Contents 1. Introduction 1 1.1. Structure of this paper 3 1.2. Notation and conventions 3 1.3. Acknowledgements 4 2. Background 4 2.1. Operads 4 2.2. Unstable global homotopy theory 6 2.3. Examples of operads in unstable global homotopy theory 6 3. G-orthogonal spaces 8 4. Main Results for Operads in (Spc, ⊠) 15 4.1. Lifting the positive global model structure to Alg (O) 15 4.2. Characterizing which morphisms of operads induce Quillen equivalences 18 5. Global E ∞ -operads 22 Appendix A. More about G-orthogonal spaces 24 A.1. G-level model structure 24 A.2. G-h-cofibrations and G-global equivalences 27 A.3. G-global model structure 29 References 32
Global transfer systems are equivalent to global
N
∞
N_\infty
-operads, which parametrize different levels of commutativity in globally equivariant homotopy theory, where objects have compatible actions by all compact Lie groups. In this paper we explicitly describe and completely classify global transfer systems for the family of all abelian compact Lie groups.
Global transfer systems are equivalent to global N∞-operads, which parametrize different levels of commutativity in globally equivariant homotopy theory, where objects have compatible actions by all compact Lie groups. In this paper we explicitly describe and completely classify global transfer systems for the family of all abelian compact Lie groups.
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