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Combinatorial homotopy theory

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Lecture notes

Version of 04/10/2012.

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Advanced course given in Autumn 2012. The notes cover the first two lectures. They will eventually be expanded to include the remaining lectures, and the numerous typos corrected.

The lectures not yet written up outlined a construction of a model structure upon cubical sets with connections, and the construction of a Quillen equivalence between this model structure and the Serre model structure upon spaces.

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Abstract

An idea of profound importance in homotopy theory is that we can model homotopy types by gadgets which are of a very different character to topological spaces. First among these are presheaf models, the most well known being simplicial sets. One can think of a presheaf model of a homotopy type as akin to a CW-complex, with better categorical properties.

The course will present this classical idea with a twist — we will work in the main not with simplicial sets but with cubical sets. I will develop the homotopy theory of cubical sets from scratch, building up to a proof that it is 'equivalent' to that of topological spaces, in a sense which we will explore along the way.

The prerequisites are few. Familiarity and comfort with categories would be helpful, as would a little rudimentary algebraic topology (homotopy groups, for example).

Last updated: 10:29 (GMT+1), 03/12/2012.