Operads in Unstable Global Homotopy Theory

Abstract

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 N∞$N_\infty$‐operads

“…The commutative monoids with respect to this box product are called ultra-commutative monoids, and they possess a rich structure, which includes transfer morphisms for all finite index closed subgroups of compact Lie groups. Ultra-commutative monoids are equivalent to algebras over global 𝐸 ∞ -operads, which were introduced in [3]. This equivalence is a consequence of [3,theorem II].…”

“…We use orthogonal spaces and global equivalences between them as a model for unstable global homotopy theory (see [19, chapter 1]), and we define global 𝑁 ∞ -operads to be operads in orthogonal spaces that satisfy a certain condition similar to that of an equivariant 𝑁 ∞ -operad. In [3], we constructed a model structure on the category of algebras over any operad in orthogonal spaces, we defined the equivalences between such operads, and we proved that these are precisely the morphisms of global operads that induce Quillen equivalences between the respective categories of algebras. The results of [3] justify the definitions of global 𝑁 ∞ -operad and equivalence between them that we give in this paper.…”

“…In [3], we constructed a model structure on the category of algebras over any operad in orthogonal spaces, we defined the equivalences between such operads, and we proved that these are precisely the morphisms of global operads that induce Quillen equivalences between the respective categories of algebras. The results of [3] justify the definitions of global 𝑁 ∞ -operad and equivalence between them that we give in this paper.…”

“…There is the greatest global transfer system, and the associated global 𝑁 ∞ -operads are the global 𝐸 ∞ -operads of [3]. In the unstable setting algebras over these operads are equivalent to the previously mentioned ultra-commutative monoids.…”

Parameterized stability and the universal property of global spectra

“…Just like non-equivariant and G-equivariant homotopy theory, global homotopy theory comes in various different flavours: unstable global homotopy theory studies global spaces [GH07] while stable global homotopy theory is concerned with socalled global spectra [Sch18]; in-between, one can also consider a variety of algebraic structures on global spaces [Bar21], with the most prominent example being ultracommutative monoids or the equivalent notion of special global Γ-spaces [Len20]. The goal of this article is to understand the relationship between these different variants.…”