eriehl at math dot jhu dot edu
Johns Hopkins University
Department of Mathematics
3400 N. Charles Street
Baltimore, MD 21218
312 Krieger Hall
I am an associate professor in the department of mathematics at Johns Hopkins University working on a variety of topics in category theory related to homotopy theory.
From 2015-2019, I was an assistant professor in the department of mathematics at Johns Hopkins University. From 2011-2015, I was a Benjamin Peirce and NSF postdoctoral fellow at Harvard University. In 2011, I completed my PhD at the University of Chicago under the direction of Peter May. I frequently collaborate with Dominic Verity at the Centre of Australian Category Theory. I am a host of the n-Category Café and founder of the Kan Extension Seminar, which recently completed its second iteration. I spent the spring of 2014 at MSRI as a member of the program on Algebraic Topology, the summer of 2015 at the Hausdorff Institute of Mathematics as a participant in the Homotopy theory, manifolds, and field theories program, and part of the winter of 2016 at the Max Planck Institute for Mathematics as part of the Program on Higher Structures in Geometry and Physics. I was in residence at the Center for Advanced Study at the Norweigian Academy of Science and Letters in May/June 2019 as part of the Homotopy Type theory and Univalent Foundations Program and am the lead organizer for an MSRI semester program Higher Categories and Categorification to take place in Spring 2020. Here is my CV.
I am the author of Categorical Homotopy Theory, published by Cambridge University Press in their New Mathematical Monographs series. (This material has been published by Cambridge University Press as Categorical Homotopy Theory by Emily Riehl. This version is free to view and download for personal use only. Not for re-distribution, re-sale or use in derivative works. © Emily Riehl 2014.) I am grateful to them for a special arrangement that also allows me to host a free PDF copy with the preceding disclaimer. More information can be found on the book website and in this blog post.
I am also the author of Category Theory in Context, published by Dover Publications in the Aurora: Modern Math Originals series. Once again, I am very grateful for a special arrangment with the publishers that allows me to host a free PDF copy. More information can be found on the book website and in this blog post.
Dominic Verity and I are in the process of writing a book developing the model-independent foundations of ∞-category theory, the most recent draft of which can be found here:
The 2018 MIT Talbot workhop was on the same topic and various additional resources can be found on the conference website.
I am a member of the editorial boards for Cahiers de Topologie et Géométrie Différentielle Catégoriques, Homology, Homotopy, and Applications, and the Journal of Homotopy and Related Structures. I organize the Johns Hopkins Category Theory Seminar. There is a rapidly expanding group of PhD students at Johns Hopkins who are interested in category theory, abstract homotopy theory, and homotopy type theory. If you are interested in joining us, more information about graduate admissions can be found here.
- Slides are available for a talk A formal theory for ∞-categories given at the 2019 International Category Theory Conference in Edinburgh.
- Slides are available for a talk The synthetic approach to ∞-category theory given at the Summer School on Higher Topos Theory and Univalent Foundations at Leeds.
- Slides are available for a talk The mathematics of social choice given as part of the 2019 National Math Festival.
- An expository book chapter Homotopical categories: from model categories to (∞,1)-categories, to appear in an as yet untitled volume in the MSRI Publications Series with Cambridge University Press, is now on the arXiv.
- Slides are available for three similar talks entitled “The complicial sets model of higher ∞-categories” given at the Perimeter Institute, Dan Freed's birthday conference, and the Higher Structures workshop at CIRM.
- Slides and video are available for a talk The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories given at the Vladimir Voevodsky Memorial Conference at IAS.
- Two new papers Recognizing quasi-categorical limits and colimits in homotopy coherent nerves and On the construction of limits and colimits in ∞-categories, both with Dominic Verity, are now available on the arXiv: 1808.09834, 1808.09835.
- Slides are available for a talk “Categorifying cardinal arithmetic” from the Category Theory for All invited paper session at MAA MathFest. The leisurely version gives you a chance to try to answer the questions for yourself before seeing the answer, while the condensed version is better for printing.
- Slides are available for a talk A proof of the model-independence of (∞,1)-category theory delivered at Category Theory 2018 at the University of Azores.
- Video is available for a series of four lectures entitled The model-independent theory of (∞,1)-categories delivered at the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge as part of the workshop Higher structures in homotopy theory to open the semester program Homotopy Harnessing Higher Structures.
- Slides are available for a talk A model-independent theory of ∞-categories delivered at the Joint International Meeting of the American and Chinese Mathematical Societies in Shanghai.
- Video is available for a talk A proof of the model-independence of ∞-category theory delivered at the workshop ∞-Categories, ∞-Operads, and their Applications at the Casa Matématica Oaxaca.
- Slides and video are available for a talk The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories in the Homotopy Type Theory Electronic Seminar Talks.
- Slides are available for two talks at the Joint Mathematics Meetings, A synthetic theory of ∞-categories in homotopy type theory and On the directed univalence axiom; see also A synthetic theory of ∞-categories in homotopy type theory delivered at CT2017 on joint work with Michael Shulman.
Homotopy type theory
- The 2-category theory of quasi-category theory (with Dominic Verity), arXiv:1306.5144, Advances in Mathematics. Volume 280 (2015), 549-642.
The objective of this paper and its sequels 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 than those introduced by Joyal and developed by Jacob Lurie, but they are equivalent. The advantage of our 2-categorical approach is that it is essentially trivial to prove, e.g., that right adjoints preserve limits. Our definitions and proofs also apply in more general contexts, for instance in the biequivalent 2-category of complete Segal spaces.
- Homotopy coherent adjunctions and the formal theory of monads (with Dominic Verity), arXiv:1310.8279, Advances in Mathematics. Volume 286 (2016), 802-888; blog, lecture, video, video.
This is the second installment in a joint project to develop the formal category theory of quasi-categories. This paper introduces the free homotopy coherent adjunction, proving that any adjunction of quasi-categories (in the homotopy 2-category of quasi-categories) extends to a homotopy coherent adjunction and that such extensions are homotopically unique. Using the free homotopy coherent adjunction, we give a formal proof of the quasi-categorical monadicity theorem that is “all in the weights.”
- Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions (with Dominic Verity), arXiv:1401.6247,
Homology, Homotopy & Applications Volume 17 (2015), 1-33; lecture, video.
In the third paper of our series on the foundational category theory of quasi-categories, 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.
- Fibrations and Yoneda's lemma in an ∞-cosmos (with Dominic Verity), arXiv:1506.05500, Journal of Pure and Applied Algebra, Volume 221, Issue 3 (2017), 499-564; lecture.
The principle objectives in our fourth paper in this series are to develop the theory of cartesian and groupoidal cartesian fibrations and to prove the Yoneda lemma, in a form inspired by classical 2-categorical work of Ross Street. But another notable feature is that we finally describe the explicit axiomatic framework within which the majority of the results from the previous papers in this series apply. Namely, an ∞-cosmos is a quasi-categorically enriched category with specified classes of weak equivalences and isofibrations satisfying axioms that suggest an enriched category of fibrant objects. The best behaved models of (∞,1)-categories and even (∞,n)-categories define the objects of some ∞-cosmos, to which our foundational work applies.
- Kan extensions and the calculus
of modules for ∞-categories (with Dominic Verity), arXiv:1507.01460, Algebraic and Geometric Topology, Volume 17, Issue 1 (2017) 189–271; lecture.
Our fifth paper introduces “modules” (aka “profunctors” or “correspondences”) between ∞-categories and develops their calculus, with notions of unit modules and represented modules and operations of restriction and composition that are familiar for (bi)modules between rings. We prove that any ∞-cosmos has an associated virtual equipment of modules, a structure that has been recognized as a vehicle to describe the “complete formal category theory.” In particular, we use the virtual equipment to define and study pointwise Kan extensions between ∞-categories.
- The comprehension construction (with Dominic Verity), arXiv; blog.
Our sixth paper constructs an analogue of Lurie's “straightening” construction, defining a comprehension functor associated to any cartesian or cocartesian fibration between ∞-categories. The comprehension functor associated to a cocartesian fibration p:E→B and another ∞-category A is a homotopy coherent diagram from the quasi-category of functors from A to B to the quasi-category of cocartesian fibrations and cartesian functors over A. In the case of a particular classifying cocartesian fibration, the comprehension functor is analogous to the Lurie's “unstraightening” construction, but this level of generality permits other important applications. Applied to the codomain fibration in a slice ∞-cosmos over A, the comprehension functor specializes to define the Yoneda embedding for the ∞-category A.
- Recognizing quasi-categorical limits and colimits in homotopy coherent nerves (with Dominic Verity), arXiv.
- On the construction of limits and colimits in ∞-categories (with Dominic Verity), arXiv.
- Lecture notes from a mini-course ∞-category theory from scratch given at the Young Topologists' Meeting 2015, arXiv:1608.05314;
- Video from a mini-course entitled The model-independent theory of (∞,1)-categories delivered at the Isaac Newton Institutefor Mathematical Sciences at the University of Cambridge as part of the workshop Higher structures in homotopy theory to open the semester program Homotopy Harnessing Higher Structures;
- The latest draft of a rapidly evolving book in progress: Elements of ∞-Category Theory (with Dominic Verity; please notify me of any comments, corrections, or confusions).
New model structures
- A type theory for synthetic ∞-categories (with Michael Shulman); arXiv:1705.07442 , Higher Structures, Volume 1, Issue 1 (2017) 116-193; blog, video, video, video, slides, slides, slides, slides, slides, slides.
This paper proposes foundations for a synthetic theory of (∞,1)-categories within homotopy type theory motivated by Shulman's model of homotopy theory in the category of Reedy fibration bisimplicial sets. Among the Reedy fibrant bisimplicial sets, which we think of as “types”, one can identify the Segal spaces “types with composition” and complete Segal spaces “types with composition and univalence”, which we refer to as Rezk spaces. New internal characterizations of Segal and Rezk spaces as Reedy fibrant bisimplicial sets motivate our definition of Segal and Rezk types. To probe the internal categorical structure of these types, we propose a new three-layered type theory with shapes, whose contexts are extended by polytopes within directed cubes, which allows us to extend dependent type families along subobjects of simplices. As an enriched category of fibrant objects, the Rezk spaces form an ∞-cosmos, a universe that supports the sythentic development of their formal category theory, providing a library of categorical definitions that can be stated in the language of homotopy type theory. We define covariant fibrations, which are type families varying functorially over a Segal type, prove a “dependent Yoneda lemma”, and apply this to the study of coherent adjunctions between Segal and Rezk types.
Algebraic model structures
- Lifting accessible model structures (with Richard Garner and Magdalena Kędziorek), arXiv:1802.09889.
- A necessary and sufficient condition for induced model structures (with Kathryn Hess, Magdalena Kędziorek, and Brooke Shipley),
This paper introduces “accessible model categories”, a generalization of the familiar combinatorial model categories, which facilitate the transfer of model structures along adjunctions. A model category is accessible if its underlying category is locally presentable and if its weak factorization systems can be promoted to accessible algebraic weak factorization systems. An accessible model structure can be lifted along any left or right adjoint between locally presentable categories if and only if an appropriately-handed “acyclicity” condition is satisfied. The transferred model structure is again accessible and so this process can be iterated. New model structures on bialgebras are obtained via this process.
- Coalgebraic models for combinatorial model categories (with Michael Ching), arXiv:1403.5303, Homology, Homotopy & Applications Volume 16, Number 2 (2014) 171 – 184.
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. We prove that this model structure exists provided that the original combinatorial model category is also simplicial. This leads to a Quillen equivalent model category in which all objects are cofibrant.
- Left-induced model structures and diagram categories (with Marzieh Bayeh, Kathryn Hess, Varvara Karpova, Magdalena Kędziorek, and Brooke Shipley), arXiv:1401.3651, Contemporary Mathematics, Volume 641 (2015) 49 - 81.
Model structures that are fibrantly generated or equipped with a Postnikov presentation (distinct notions whose precise relationship is described here) 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. In the combinatorial context, a theorem of Makkai and Rosicky enables the construction of suitable functorial factorizations, which define a model structure in the presence of the dual of the usual acyclicity condition. An application is given in Coalgebraic models for combinatorial model categories.
- Six model structures for DG-modules over DGAs: Model category theory in homological action (with Tobias Barthel and Peter May), arXiv:1310.1159, New York Journal of Mathematics Volume 20 (2014) 1077-1159; blog.
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 the construction of functorial factorizations for model categories. Part II discusses cofibrant approximations and applications to homological algebra.
- On the construction of functorial factorizations for model categories (with Tobias Barthel), arXiv:1204.5427, Algebraic & Geometric Topology Volume 13, Issue 2 (2013) 1089–1124; lecture.
This paper establishs Hurewicz-type model structures on any locally bounded topologically bicomplete category by constructing the missing functorial factorizations. Its methods apply more generally to the construction of functorial factorizations appropriate to other non-cofibrantly generated model structures; see Six model structures for DG-modules over DGAs: Model category theory in homological action
- Algebraic model structures, arXiv:0910.2733, New York Journal of Mathematics Volume 17 (2011) 173-231; blog.
Most (perhaps all?) Quillen model categories admit an algebraic model structure, with superior categorical properties provided by a well-chosen pair of functorial factorizations. This paper, part I of my PhD thesis, begins the development of this “algebraic” approach to abstract homotopy theory, with chosen (sometimes natural) solutions to lifting problems. Any algebraic model category has a fibrant replacement monad and a cofibrant replacement comonad, an observation that enabled the paper Homotopical resolutions associated to deformable adjunctions
- Monoidal algebraic model structures, arXiv:1109.2883, Journal of Pure and Applied Algebra, Volume 217, Issue 6 (2013) 1069–1104; blog.
Part II of my PhD thesis investigates the interaction between a monoidal or enriched category structure and the algebraic weak factorization systems of an algebraic model category. A main theorem is that chosen solutions to lifting problems are respected by pushout-products and pullback-homs if and only if the pushout-products of the generating (trivial) cofibrations are cellular, i.e., are relative cell complexes, not mere retracts thereof.
- The theory and practice of Reedy categories (with Dominic Verity), arXiv:1304.6871, Theory & Applications of Categories Volume 29, Number 9 (2014) 256-301.
This paper redevelops Reedy category theory using (unenriched) weighted limits and colimits and applies this general theory to deduce formulae for homotopy limits and colimits of diagrams indexed by Reedy categories.
- Inductive presentations of generalized 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 presence of an algebraic weak factorization system. This project is still evolving and comments are extremely welcome.
- Directional derivatives and higher order chain rules for abelian functor calculus (with KristineBauer, Brenda Johnson, Christina Osborne, and Amelia Tebbe) arXiv:1610.01930.
Abelian functor calculus studies horrible functors between abelian categories (that are non-additive and may fail to preserve zero). We describe the appropriate categorical context for chain rules, in which the components of the Taylor tower are composable and the linear approximations are functorial. This category is a quotient of the Kleisli bicategory for the chain complex pseudomonad, where pointwise chain homotopy equivalent functors are identified. We show that the Johnson-McCarthy directional derivative equips this category with the structure of a cartesian differential category and deduce various higher order chain rules expressed in terms of higher directional derivatives.
- Homotopical resolutions associated to
deformable adjunctions (with Andrew Blumberg), arXiv:1208.2844, Algebraic & Geometric Topology Volume 14, Issue 5 (2014) 3021–3048.
We introduce 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.
- Multivariable adjunctions and mates (with Eugenia
Cheng and Nick Gurski), arXiv:1208.4520, Journal of K-Theory Volume 13, Issue 02 (2014) 337-396; blog.
This paper studies a new categorical correspondence that arose in the course of my work on monoidal algebraic model categories. Certain natural transformations involving multivariable adjoint functors admit parametrised mates. The central theorem describes the multifunctoriality of the parametrised mates correspondence. In practice, this allows one to characterize which commutative diagrams transpose into others.
- On the structure of simplicial
categories associated to quasi-categories, arXiv:0912.4809, Mathematical
Proceedings of the Cambridge Philosophical Society, Volume 150, Issue 03 (2011) 489-504; blog.
The word on the street is that the left adjoint to the homotopy coherent nerve is impossible to understand. This paper applies work of Dugger and Spivak to prove that the hom-spaces produced by this construction are 3-coskeletal. We also show that the cofibrant replacements of discrete simplicial categories produced in this manner are isomorphic to the Dwyer-Kan simplicial resolutions.
- Levels in the toposes of simplicial and cubical sets (with Carolyn Kennett, Michael Roy,
and Michael Zaks), arXiv:1003.5944, Journal of Pure and Applied Algebra, Volume 215, Issue 5 (2011) 949–961.
This paper answers a question posed by Bill Lawvere: when does n-skeletal imply k-coskeletal?
- A comparison of norm maps (by Anna Marie Bohmann, with a joint appendix), arXiv:1201.6277, Proceedings of the AMS, Volume 142, Number 4 (2014) 1413–1423.
This paper compares the norm maps of Greenlees-May and Hill-Hopkins-Ravenel. The appendix presents a unifying categorical framework for the indexed tensor products employed by both constructions.
- A Sharp Bound for the Degree of Proper Monomial Mappings Between Balls (with John D'Angelo and Šimon Kos), Journal of Geometric Analysis Volume 13, Issue 4 (2003) 581-593.
- On the intersections of polynomials and the Cayley–Bacharach theorem (with E. Graham Evans Jr.), Journal of Pure and Applied Algebra Volume 183, Issues 1–3 (2003) 293–298.
Miscellaneous mathematical notes:
n-Category Café posts:
- Homotopical categories: from model categories to (∞,1)-categories, a book chapter to appear in an as yet untitled volume in the MSRI Publications Series with Cambridge University Press, also on the arXiv.
- Homotopy coherent structures, lecture notes written to accompany a mini-course as part of the Floer Homology and Homotopy Theory Summer School at UCLA.
- Complicial sets, an overture, lecture notes written to accompany a mini-course at the Higher Structures workshop at the MATRIX institute.
- Higher category theory, lecture notes from the Thursday Seminar held at Harvard in Spring 2013.
- The algebra and geometry of ∞-categories, written for the Friends of Harvard Mathematics.
- A survey of categorical concepts, written for a graduate topics course.
- On the construction of new topological spaces from existing ones, written for an undergraduate point-set topology class.
- A concise definition of a model category, written for Peter May.
- Weighted limits and colimits, a slightly expanded version of a talk given by Mike Shulman
- A leisurely introduction to simplicial sets, written for fun.
Research lecture notes:
- ∞-category theory from scratch, from the 2015 Young Topologists' Meeting.
- Toward the formal theory of (∞,n)-categories, from the 2014 Topologie workshop at Oberwolfach.
- The formal theory of adjunctions, monads, algebras, and descent, from Reimagining the Foundations of Algebraic Topology at MSRI, written by David White.
- Limits of quasi-categories with (co)limits, from Connections for Women: Algebraic Topology at MSRI.
- Made-to-order weak factorization systems, a less-abridged version of an extended conference abstract published by the Centre de Recerca Matemàtica following their Conference on Type Theory, Homotopy Theory, and Univalent Foundations.
- Quasi-categories as (∞,1)-categories, from a talk given in the Thursday Seminar at Harvard.
- Lifting properties and the small object argument, from the Midwest Topology Seminar at Northwestern, written by Gabriel C. Drummond-Cole.
- A formal theory for ∞-categories from the 2019 International Category Theory Conference in Edinburgh.
- The synthetic approach to ∞-category theory from the Summer School on Higher Topos Theory and Univalent Foundations at Leeds.
- The mathematics of social choice given as part of the 2019 National Math Festival.
- The complicial sets model of higher ∞-categories from Structures supérieures at CIRM.
- The complicial sets model of higher ∞-categories from Between Topology and Quantum Field Theory: A conference in celebration of Dan Freed's 60th birthday.
- The complicial sets model of higher ∞-categories, a colloquium talk given at the Perimeter Institute.
- The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories and also video from the Vladimir Voevodsky Memorial Conference at IAS.
- Categorifying cardinal arithmetic from the Category Theory for All invited paper session at MAA MathFest; condensed version (that gives away the answers to the questions posed).
- A proof of the model-independence of (∞,1)-category theory from the 2018 International Category Theory Conference.
- The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories and also video from the Homotopy Type Theory Electronic Seminar Talks.
- A synthetic theory of ∞-categories in homotopy type theory and On the directed univalence axiom, from the 2018 Joint Mathematics Meetings.
- A synthetic theory of ∞-categories in homotopy type theory, from Octoberfest at Carnegie Mellon University.
- A synthetic theory of ∞-categories in homotopy type theory, from CT2017 at the University of British Columbia.
- A categorical view of computational effects, a keynote talk delivered at the Compose Conference.
- Functoriality in algebra and topology, from a colloquium talk given to pure and applied mathematicians at Macquarie University.
- The formal theory of adjunctions, monads, algebras, and descent, from Reimagining the Foundations of Algebraic Topology at MSRI.
- Quasi-category theory you can use, from the Graduate Student Topology and Geometry Conference at UT Austin.
- Homotopy coherent adjunctions, from the AMS Special Session on Homotopy Theory at JMM 2014.
- The formal theory of homotopy coherent monads, from the Samuel Eilenberg Centenary Conference.
- Algebraic model structures, from the AWM Anniversary Conference at ICERM.
- Cellularity, composition, and morphisms of weak factorization systems, from CT 2011 at the University of British Columbia.
- Algebraic model structures, from the CMS Summer Meeting at the University of Alberta.
- Algebraic model structures, from CT2010 at the University of Genova.
University of Chicago Topology Proseminar lecture notes: (written hastily with little editing)
- The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories given at the Vladimir Voevodsky Memorial Conference at IAS; slides.
- A proof of the model-independence of ∞-category theory delivered at the workshop ∞-Categories, ∞-Operads, and their Applications at the Casa Matématica Oaxaca.
- The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories, with slides from the Homotopy Type Theory Electronic Seminar Talks.
- Foundations of (∞,2)-category theory, a talk at the Women in Topology workshop at MSRI outlining a research program.
- A categorical view of computational effects, a keynote lecture delivered at the Compose Conference devoted to the practice and craft of functional programming; slides.
- Towards a synthetic theory of (∞,1)-categories given at the 2016 Workshop on Homotopy Type Theory and Univalent Foundations of Mathematics at the Fields Institute.
- A mini course “∞-category theory from scratch” given at the 2015 Young Topologists' Meeting:
- A universal approach to universal algebra, a colloquium-style talk given at the Center for Geometry and Physics in Pohang, Korea.
- The formal theory of adjunctions, monads, algebras, and descent, from the workshop Reimagining the Foundations of Algebraic Topology at MSRI.
- Limits of quasi-categories with (co)limits, from the Connections for Women workshop at MSRI.