I am currently visiting Dominic Verity at Macquarie University in Sydney. I will be in Australia until the end of March, 2010. Please come say hello if you are in the area.

Research

I recently posted this paper to the arXiv: Natural weak factorization systems in model structures

It starts from the following definition: a model structure on a complete and cocomplete category M consists of three classes of maps C, F, and W such that W has the 2 of 3 property and (C &cap W,F) and (C,F &cap W) are weak factorization systems (wfs). I won't define a wfs here, but those who are familiar with model categories should at least be able to guess some of their properties; look here for a quick explanation of this perspective. My paper considers the role that natural weak factorization systems (nwfs), which can be thought of as an algebraization of wfs, could play in a model structure on a category. The advantage of using nwfs in place of wfs is that the former have some nicer categorical properties. In 2007, Richard Garner came up with a small object argument, which applies in the cases where Quillen's can be used, giving rise to many examples. What I've done is give a definition of a natural model structure, which uses nwfs in place of wfs, and also includes a means of comparing the trivial cofibration - fibration factorization of a map with the cofibration - trivial fibration one. One result is a natural map RQX &rarr QRX between the fibrant - cofibrant replacements of an object.

Using Garner's small object argument, you can replace the wfs by nwfs in any cofibrantly generated model structure and get a natural model structure, so there are plenty of examples. I've generalized a theorem due to Kan that allows you to pass model structures across an adjunction and described the sort of Quillen adjunction you get in this case. (Usually the left adjoint preserves cofibrations; here what you can say is much stronger.) I show that a natural model structure on a category M gives rise to a "projective" natural model structure on any diagram category M^A, as in the classical theory.

Toward the end, there is a nice result that says that in the cofibrantly generated case when the cofibrations are monomorphisms, the arrows comparing the two factorizations mentioned above are themselves cofibrations. The proof illustrates a general technique that one can use for nwfs but not for wfs to detect which sort of arrows constructed as colimits are cofibrations. (Of course, not all pushouts of cofibrations are cofibrations but some are, and the algebraic structure of nwfs allows one to recognize when this will be true for a large class of examples.) There are a few other results which are of more interest to category theorists, but an explanation of these would require more details about the specifics of the definition of a nwfs, so I will save it for the paper.

Exposition

A document to accompany an n-Category Café post: Associativity data in an (∞,1)-category

My topic proposal: A model structure for Quasi-categories

My Part III essay: Model Categories and Weak Factorisation Sytems

My undergraduate senior thesis: Lubin-Tate Formal Groups and Local Class Field Theory

Notes

The following notes were written - usually quite hastily - to accompany talks I've given in the University of Chicago's Topology Proseminar.

Homotopy (Limits and) Colimits

Factorization Systems

CV available upon request.