# Traces, homotopy theory, and motivic Galois groups

Gallauer Alves de Souza, Martin. Traces, homotopy theory, and motivic Galois groups. 2015, University of Zurich, Faculty of Science.

## Abstract

This thesis consists of two independent parts. In the first part we ask how traces in monoidal categories behave under homotopical operations. In order to investigate this question we define traces in closedmonoidal derivators and establish some of their properties.
In the stable setting we derive an explicit formula for the trace of the homotopy colimit over finite categories in which every endomorphism is invertible. In the second part, we study motives of algebraic varieties over a subfield of the complex numbers, as defined by Nori on the one hand and by Voevodsky, Levine, and Hanamura on the other. Ayoub attached to the latter theory a motivic Galois group using the Betti realization, based on a weak Tannakian formalism. Our main theorem states that Nori’s and Ayoub’s motivic Galois groups are isomorphic. In the process of proving this result we construct well-behaved functors relating the two theories which are of independent interest.

## Abstract

This thesis consists of two independent parts. In the first part we ask how traces in monoidal categories behave under homotopical operations. In order to investigate this question we define traces in closedmonoidal derivators and establish some of their properties.
In the stable setting we derive an explicit formula for the trace of the homotopy colimit over finite categories in which every endomorphism is invertible. In the second part, we study motives of algebraic varieties over a subfield of the complex numbers, as defined by Nori on the one hand and by Voevodsky, Levine, and Hanamura on the other. Ayoub attached to the latter theory a motivic Galois group using the Betti realization, based on a weak Tannakian formalism. Our main theorem states that Nori’s and Ayoub’s motivic Galois groups are isomorphic. In the process of proving this result we construct well-behaved functors relating the two theories which are of independent interest.