Diagram spaces and multiplicative structures

Abstract

This thesis explores diagrammatic E_n structures as models for E_n spaces.

Paper A: Braided injections and double loop spaces. (Christian Schlichtkrull and Mirjam Solberg.) We consider a framework for representing double loop spaces (and more generally E_2 spaces) as commutative monoids. There are analogous commutative rectifications of braided monoidal structures and we use this framework to define iterated double deloopings. We also consider commutative rectifications of E_\infty spaces and symmetric monoidal categories and we relate this to the category of symmetric spectra.

Paper B: Weak braided monoidal categories and their homotopy colimits. (Mirjam Solberg.) We show that the homotopy colimit construction for diagrams of categories with an operad action, recently introduced by Fiedorowicz, Stelzer and Vogt, has the desired homotopy type for diagrams of weak braided monoidal categories. This provides a more flexible way to realize E_2 spaces categorically.

Paper C: Operads and algebras in n-fold monoidal categories. (Mirjam Solberg.) We develop the concept of n-fold monoidal operads and algebras over n-fold monoidal operads in n-fold monoidal categories. We give examples of n-fold monoidal operads whose algebras generalize the concepts of monoids, commutative monoids and n-fold monoidal structures, to the setting of an n-fold monoidal category.

Paper D: Higher monoidal injections and diagrammatic E_n structures. (Christian Schlichtkrull and Mirjam Solberg.) We use the framework of n-fold monoidal categories to examine E_n structures in a diagrammatic setting. A major objective is to introduce the category I_n of n-fold monoidal injections as a counterpart to the symmetric monoidal category of finite sets and injective functions. This then leads to an n-fold monoidal version of the classical James construction. We also discuss applications to n-fold commutative strictification of E_n structures.