eriehl at math.harvard.edu
Science Center 320
In the spring of 2014, I will lead an online graduate reading course in category theory modeled after the Kan seminar at MIT. More information can be found on the course website.
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.
Any combinatorial model category admits a cofibrantly replacement comonad that is moreover accessible (i.e., preserves sufficiently large filtered colimits). It follows that the category of coalgebras, here the “algebraic cofibrant objects”, is locally presentable (in particular, complete and cocomplete) and thus a good candidate for a model structure, left-induced from the original model category. In joint work with Michael Ching, we show that this model structure exists, at least when the original combinatorial model category is also simplicial: see this paper, also on the arXiv. This leads to a Quillen equivalent model category in which all objects are cofibrant.
A fourth joint paper with Dominic Verity is now on the arXiv. This is the third paper in our series on the foundational category theory of quasi-categories. It has the rather unwieldy title Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions. We prove that weighted limits of diagrams of quasi-categories admitting and functors preserving limits or colimits of a fixed shape again admit such limits or colimits, provided the weights are projective cofibrant simplicial functors. Examples include Bousfield-Kan-style homotopy limits, quasi-categories of algebras, and more. Lectures notes from a short expository talk on this topic may be found here.
Joint work with Marzieh Bayeh, Kathryn Hess, Varvara Karpova, Magdalena Kędziorek, and Brooke Shipley completed at the Women in Topology workshop at BIRS lead to the following paper, Left-induced model structures and diagram categories, also on the arXiv. It considers model structures that are fibrantly generated or equipped with a Postnikov presentation (distinct notions whose precise relationship is described here). Such structures are convenient in contexts where one might wish to lift a model structure along a left adjoint; these are the “left-induced” model structures of the title, which we also consider more generally.
The second installment in my joint project with Dominic Verity to develop the formal category theory of quasi-categories is now on the arXiv. This paper, Homotopy coherent adjunctions and the formal theory of monads, introduces the free homotopy coherent adjunction, proving that any adjunction of quasi-categories extends to a homotopy coherent adjunction and that such extensions are homotopically unique. See also these slides. Using the free homotopy coherent adjunction, we give a formal proof of the quasi-categorical monadicity theorem that is “all in the weights.” An expository account can be found in this blog post or sketched on these slides.
Update: I gave a talk entitled “The formal theory of adjunctions, monads, algebras, descent” at the workshop Reimagining the Foundations of Algebraic Topology at MSRI. Here is a video and here are my slides.
In a joint paper with Tobias Barthel and Peter May entitled Six model structures for DG-modules over DGAs: Model category theory in homological action, now on the arXiv, we establish six model structures on the category of differential graded modules over a differential graded algebra over a commutative ring. New ideas are required to construct the functorial factorizations in two separate cases: One is an enriched algebraic small object argument suitable for a weak factorization system satisfying an enriched lifting property. The other is analogous to the construction in the paper “On functorial factorizations.” Part II discusses cofibrant approximations and applications to homological algebra.
The first installment of a joint project with Dominic Verity is available on the arXiv. The objective of the paper The 2-category theory of quasi-category theory is to redevelop the foundational category theory of quasi-categories using a strict 2-category defined by André Joyal. Our definitions, particularly of limits and colimits in a quasi-category, have a different form of those of Joyal and Jacob Lurie, but they are equivalent. Slides for an introductory talk on quasi-categories, which loosely touches on aspects of this project, can be found here.
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.
This preprint, with a very hastily written introduction and background sections, describes work-in-progress on the “algebraic” approach to generalized Reedy category theory. In this context, there are equivariance conditions needed to inductively define diagrams and natural transformations which are easily satisfied in the presense of an algebraic weak factorization system. This project is still evolving and comments are extremely welcome.
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 paper 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. This paper has been published in the Journal of K-Theory.
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.
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.
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.
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.
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.
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.
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
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.
A less-abridged version of an “extended conference abstract” to be published by the Centre de Recerca Matemàtica following their Conference on Type Theory, Homotopy Theory, and Univalent Foundations: Made-to-order weak factorization systems.
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 gave as a PhD student in the University of Chicago Topology Proseminar.
TALKS FOR A GENERAL AUDIENCE
Finally, here is my CV.