RECENT TEACHING

In the fall of 2012, I taught Math 131: Topology I. The course website is here.

In the spring of 2012, I taught Math 266x: categorical homotopy theory. More information can be found on the course website. Lecture notes written at the time have been assembled into a book that will appear in the New Mathematical Monographs series published by Cambridge University Press. Cambridge has graciously allowed me to host a free PDF copy in perpetuity, which can be found here.

RESEARCH

• Reedy categories

  An expository preprint, joint with Dominic Verity, also on the arXiv, re-develops Reedy category theory using (unenriched) weighted limits and colimits. We conclude with a few sections applying this general theory to deduce formulae for homotopy limits and colimits of diagrams indexed by Reedy categories.

• Parametrised mates

  Joint work with Eugenia Cheng and Nick Gurski studies a new categorical correspondence that arose naturally in the course of my work on monoidal algebraic model categories. Certain natural transformations involving multivariable adjoint functors admit parametrised mates. The central theorem in our preprint Multivariable adjunctions and mates, also on the arXiv, describes the multifunctoriality of the parametrised mates correspondence. In practice, this allows one to characterize which commutative diagrams transform into others. I have blogged about this on the n-Category Café.

• Homotopical resolutions

  In joint work with Andrew Blumberg appearing in a preprint Homotopical resolutions associated to deformable adjunctions, also on the arXiv, we define a new derived bar and cobar construction associated Quillen adjunctions between cofibrantly generated model categories, giving a homotopical model of the (co)completion of the associated (co)monad. Our main observation is that others' work with similar resolutions generalizes to situations in which objects are not assumed to be (co)fibrant.

• Functorial factorizations and h-model structures

  This is joint work with Tobias Barthel appearing in a paper On the construction of functorial factorizations for model categories, published in Algebraic & Geometric Topology and also appearing on the arXiv. We establish Hurewicz-type model structures on any locally bounded topologically bicomplete category by constructing the missing functorial factorizations. We expect our methods to apply more generally to the construction of functorial factorizations appropriate to other non-cofibrantly generated model structures. Notes from a talk I gave on this topic at the Midwest Topology Seminar at Northwestern were live-TeXed by Gabriel C. Drummond-Cole.

• Algebraic model structures

  An extension of Quillen's model categories is introduced in the paper Algebraic model structures published in the New York Journal of Mathematics. This work is also on the arXiv, where it was originally entitled "Natural weak factorization systems in model structures." I've blogged about this topic on the n-Category Café. Here are slides from talks I've given on this topic to more topological and more categorical audiences.

• Monoidal algebraic model structures

  The extension of this theory to monoidal categories, enabling a further extension to enriched categories, is the subject of the preprint Monoidal algebraic model structures, also available on the arXiv. I've blogged about this on the n-Category Café. This work formed part II of my PhD thesis.

• Simplicial categories associated to quasi-categories

  A paper On the structure of simplicial categories associated to quasi-categories has been published by the Mathematical Proceedings of the Cambridge Philosophical Society (copyright Cambridge University Press). It uses recent work of Dugger and Spivak to prove a few facts about the hom-spaces of the simplicial categories associated to simplicial sets by means of the left adjoint to the homotopy coherent nerve. This work is also on the arXiv. I have also blogged about this topic on the n-Category Café.

• Levels in the toposes of simplicial sets and cubical sets

  The paper Levels in the toposes of simplicial and cubical sets is joint work with Carolyn Kennett, Michael Roy, and Michael Zaks. It has been published in the Journal of Pure and Applied Algebra and is also available on the arXiv.

• A comparison of norm maps

  I contributed to the appendix of this preprint, available on the arXiv, by Anna Marie Bohmann. Her work compares the norm maps of Greenlees-May and Hill-Hopkins-Ravenel. In the appendix, we present a unifying categorical framework for the indexed tensor products employed by both constructions.

My "topic" proposal: A model structure for quasi-categories

My Part III essay: Model categories and weak factorisation systems

My undergraduate senior thesis: Lubin-Tate formal groups and local class field theory

MISCELLANEOUS EXPOSITION

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

A formalist's introduction to simplicial sets, intended to establish a firm foundation for understanding the categorical and topological applications: A leisurely introduction to simplicial sets.

Lecture notes for talks given by Mike Shulman in the fall of 2008 introducing weighted limits, with some preliminary ideas about homs and tensors of bimodules expanding into their full gorey detail. The level is appropriate for someone whose knowledge of enriched category theory is more-or-less contained in the first three pages of Max Kelly's Basic concepts of enriched category theory: Weighted limits and colimits.

A short note, originally written for my advisor, proves the equivalence between an alternative (and my preferred) definition of a model structure on a category and the usual axioms: A concise definition of a model category.

Notes describing the appropriate topologies for spaces constructed as products, subspaces, quotients, or by gluing, written to accompany a series of lectures in an undergraduate point-set topology course taught at Harvard in the fall of 2012: On the construction of new topological spaces from existing ones.

Comments are always welcome.

NOTES FROM TALKS

The following notes were written — usually quite hastily and with sparce, if any, editing — to accompany talks I've given in the University of Chicago Topology Proseminar.

• Two-sided discrete fibrations in 2-categories and bicategories

• Homotopy (limits and) colimits

• Factorization systems

TALKS FOR A GENERAL AUDIENCE

• The stable marriage problem. Here are slides from a PechaKucha presentation on the same topic, geared toward members of the Harvard math department.

• The algebra and geometry of ∞-categories

Finally, here is my CV.