Version of 04/10/2012.
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.
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).
These notes became an ingredient in, and/or inspiration for, Thierry Coquand, Marc Bezem, and Simon Huber arriving at the first of the 'cubical models of dependent type theory', which has since become an entire research field. Thierry got in touch with me at the time over email, and we corresponded a little; I believe that Marc and I corresponded once too, although I do not remember the exact topic, or precisely when it was. Thierry kindly refers to the role my notes played in this talk. The original paper of Bezem, Coquand, and Huber is this one. An early draft, which I have (I hope!) somewhere, contained an appendix discussing a question raised in a footnote to my notes.
These developments in homotopy type theory led to wider interest in use of cubical sets and related ideas in homotopy theory and adjacent fields. At the time I wrote the notes, I felt like the only person who had an idea that cubical sets (in their various incarnations) have both conceptual and technical advantages that could be useful in approaching several significant open problems; and like the only person who was actively working on pursuing this vision. In particular, though my work built upon that of several others, especially Grandis, Kamps, Porter, Brown, Cisinski, and Maltsiniotis (the latter two building upon Grothendieck's ideas, which I was also closely familiar with in their original sources), it did not feel that a bridge had been built towards modern combinatorial homotopy theory as founded upon simplicial sets; even Cisinski (who I met in Oxford in the late 2000s and corresponded with several times) who — there was slightly earlier work of Jardine too, but Cisinski went further and used very different techniques — placed the homotopy theory of cubical sets in a model categorical setting and (importantly) characterised cubical Kan fibrations, did not have this point of view as far as I know.
Even a presentation of cubical Kan complexes as in my notes, though there is nothing difficult in it (it indeed has origins to a certain degree in Kan's very first paper on combinatorial homotopy theory, predating simplicial sets!), had not, as far as I know, been given before.
I have no idea how Thierry found my notes, or why he initially was interested in them, but I am glad he did and was! I worked on a number of hard, foundational problems for several years around this time in which I made intensive use of cubical sets (in numerous flavours), but never published anything; I am happy that these notes, even though they almost do not touch at all on what I actually was doing at the time and did over the following few years, have at least had some influence! I was struck at the time by Thierry — the leading type theorist in the world — having the humility, and unconceited intellectual curiosity, to engage with the work of an unknown.
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