title: Monoidal structures in orthogonal calculus
Creators: name: Hendrian, Leon
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2023
Subjects: subject: 510 Mathematics

description: Der "Orthogonal Calculus", zuerst beschrieben von Michael Weiss in 1991, ist ein Funktorkalkül für Funktoren von euklidischen Vektorräumen nach Räumen. Viele der Funktoren auf die der Orthogonal Calculus angewendet wurde lassen sich mit einer lax symmetrisch monoidalen Struktur ausstatten, die bisher keine Beachtung fand. Beispielsweise lässt sich der Funktor BO(V) mit Abbildungen BO(V) x BO(W) -&gt; BO(V + W) ausstatten die eine lax symmetrisch monoidale Struktur bilden. Eines der Hauptresultate dieser Dissertation besagt, dass auch die Taylorapproximationen eines lax symmetrisch monoidalen Funktors selbst lax symmetrisch monoidal sind. Außerdem studieren wir die Ableitungsspektren lax symmetrisch monoidaler Funktoren und beweisen die Existenz von O(n)-äquivarianten Abbildungen der Form Theta(F) (x) Theta(F) -&gt; Theta(F) (x) DO(n). Hier bezeichnet DO(n) das dualisierende Spektrum der topologischen Gruppe O(n).
description: 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 BO(V) admits maps BO(V) x BO(W) -&gt; BO(V + W) which determine a lax symmetric monoidal structure. One of the main results of this thesis is that the Taylor approximations of a lax symmetric monoidal functor are themselves lax symmetric monoidal as well. We also study the derivative spectra of lax symmetric monoidal functors, and prove that they admit O(n)-equivariant structure maps of the form Theta(F) (x) Theta(F) -&gt; Theta(F) (x) DO(n) where DO(n) is the Klein-Spivak dualising spectrum of the topological group O(n). We also formulate Orthogonal Calculus in the infinity-categorical language before proving our results.
description: 1 Introduction ..... 1 1.1 Overview of this Thesis ..... 1 1.2 What is Orthogonal Calculus? ..... 2 1.2.1 Functor Calculus in General ..... 2 1.2.2 Orthogonal Calculus ..... 5 1.3 Monoidal Structures ..... 7 1.3.1 The Pointwise Monoidal Structure ..... 8 1.3.2 The Day Convolution Monoidal Structure ..... 9 2 Other Work on Orthogonal Calculus ..... 13 2.1 Formalisms for Orthogonal Calculus ..... 13 2.1.1 The Original Version (Weiss) ..... 13 2.1.2 Model Categories (Barnes-Oman) ..... 14 2.1.3 ∞-categorical Orthogonal Calculus ..... 15 2.2 Some Known Calculations in Orthogonal Calculus ..... 16 2.2.1 Weiss’s Original Article ..... 16 2.2.2 Arone’s Work on BO and BU ..... 17 2.2.3 Θ2btop (Krannich-Randal-Williams) ..... 18 2.2.4 Pontryagin Classes etc. (Reis-Weiss et al.) ..... 19 2.3 Analyticity Conditions ..... 19 2.3.1 Barnes-Eldred ..... 19 2.3.2 Taggart ..... 20 2.4 Various ..... 20 2.4.1 Unitary Calculus ..... 20 2.4.2 Comparison with Goodwillie Calculus ..... 21 3 ∞-Categories for Orthogonal Calculus ..... 23 3.1 Basic Definitions of Relevant Categories and Functors ..... 23 3.2 Polynomial Functors ..... 26 3.3 Homogeneous Functors and Derivative Spectra ..... 30 3.4 Some Topos-Theoretical Remarks ..... 34 4 Orthogonal Calculus and Monoidal Structures ..... 37 4.1 Setup / Preliminaries ..... 38 4.1.1 Symmetric Monoidal ∞-Categories ..... 38 4.1.2 Day Convolution and the Internal Hom Functor ..... 39 4.1.2.1 The Day Convolution Monoidal Structure ..... 39 4.1.2.2 The Internal Hom Functor ..... 42 4.1.3 Generalities on Monoidal Localisations ..... 45 4.2 A Monoidal Localisation Theorem for Taylor Approximations in Orthogonal Calculus ..... 48 5 Monoidality and Derivative Spectra ..... 53 5.1 Pointed and Unpointed Functors ..... 53 5.2 Monoidality of Homogeneous Layers ..... 55 5.3 Classification of Homogeneous Layers and the Internal Hom ..... 56 5.3.1 Calculation of ΘnF∗(A, B) for A, B n-homogeneous ..... 58 5.3.2 Calculation of Θn(A ⊛ B) for A, B n-homogeneous ..... 65 5.4 Final Result on Derivative Spectra of Monoidal Functors ..... 67 6 Applications and Open Questions ..... 71 6.1 Examples of Monoidal Functors ..... 71 6.2 A Potential Alternative Description of the Multiplication On Θ1F ..... 74 6.3 Further Open Questions ..... 75 Bibliography ..... 79 Index ..... 83