Monoidal Structures in Orthogonal Calculus

TL;DR

This paper explores the integration of monoidal structures into Orthogonal Calculus, demonstrating that Taylor approximations preserve monoidal properties and analyzing derivative spectra with equivariant structures using advanced categorical methods.

Contribution

It introduces the study of lax symmetric monoidal functors within Orthogonal Calculus, proving their Taylor approximations are also monoidal and analyzing their derivative spectra with equivariant maps.

Findings

Taylor approximations of lax symmetric monoidal functors are monoidal

Derivative spectra admit $O(n)$-equivariant structure maps

Framework formulated using $alategorical$ methods

Abstract

Orthogonal Calculus, first developed by Weiss in 1991, provides a calculus of functors for functors from real inner product spaces to spaces. Many of the functors to which Orthogonal Calculus has been applied since carry an additional lax symmetric monoidal structure which has so far been ignored. For instance, the functor $V \mapsto \text{BO}(V)$ admits maps $$\text{BO}(V) \times \text{BO}(W) \to \text{BO}(V \oplus W)$$ which determine a lax symmetric monoidal structure. Our first main result, Corollary 4$.$2$.$0$.$2, states that the Taylor approximations of a lax symmetric monoidal functor are themselves lax symmetric monoidal. We also study the derivative spectra of lax symmetric monoidal functors, and prove in Corollary 5$.$4$.$0$.$1 that they admit $O(n)$-equivariant structure maps of the form $$\Theta^nF \otimes \Theta^nF \to D_{O(n)} \otimes \Theta^nF$$ where $D_{O(n)} \simeq S^{\text{Ad}_n}$ is the Klein-Spivak dualising spectrum of the topological group $O(n)$. As our proof methods are largely abstract and $\infty$-categorical, we also formulate Orthogonal Calculus in that language before proving our results.