Monoidal categories and $\Delta$, $\Theta_2$ and $\Gamma$

Specialeforsvar ved Peter James Dawson

Titel: Monoidal categories and $\Delta$, $\Theta_2$ and $\Gamma$

Abstract: We investigate Segal's infinite loop space machine in order to develop a double loop space machine in the same vein. We make Segal's constructions explicitly functorial, fully exploring the relationships between symmetric monoidal categories, $\Gamma$-spaces and spectra, where $\Gamma$ is Segal's category of finite sets. Our double loop space machine relates braided monoidal categories to $\Theta_2$-spaces and to double loop spaces, where $\Theta_2$ is the categorical wreath product of $\Delta$ with itself (although we define it directly). The second half of this machine relies heavily on the work of Berger, whereas, to our knowledge, the first half is new. For completeness, we also present a single loop space machine, relating monoidal categories in general to $\Delta$-spaces (otherwise known as simplicial spaces) and to single loop spaces. We briefly explore appropriate model structures on categories of $\Delta$-spaces, $\Theta_2$-spaces and $\Gamma$-spaces, and present a proof of the Barratt-Priddy-Quillen theorem, following Segal. In the course of our exposition, we make use of unbiased definitions of the various types of monoidal category, and we show that these are equivalent to the classical definitions 

Vejleder:  Dustin Clausen
Censor:    Iver M. Ottosen, Aalborg Universitet