Emily Riehleriehl at math dot jhu dot edu
Johns Hopkins University
312 Krieger Hall 
New  Research  Exposition  Talks  Teaching 
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 redistribution, resale 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.
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.
From 20112015, 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 Macquarie University in Sydney. I am a host of the nCategory Café and founder of the Kan Extension Seminar, now in 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. Here is my CV.


The objective of this paper and its sequels is to redevelop the foundational category theory of quasicategories using a strict 2category defined by André Joyal. Our definitions, particularly of limits and colimits in a quasicategory, have a different form than those introduced by Joyal and developed by Jacob Lurie, but they are equivalent. The advantage of our 2categorical 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 2category of complete Segal spaces.
This is the second installment in a joint project to develop the formal category theory of quasicategories. This paper introduces the free homotopy coherent adjunction, proving that any adjunction of quasicategories (in the homotopy 2category of quasicategories) 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 quasicategorical monadicity theorem that is “all in the weights.”
In the third paper of our series on the foundational category theory of quasicategories, we prove that weighted limits of diagrams of quasicategories 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 BousfieldKanstyle homotopy limits, quasicategories of algebras, and more.
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 2categorical 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 quasicategorically 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.
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.
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 quasicategory of functors from A to B to the quasicategory 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.
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 threelayered 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.
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 appropriatelyhanded “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.
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, leftinduced 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.
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 “leftinduced” 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.
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.
This paper establishs Hurewicztype 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 noncofibrantly generated model structures; see Six model structures for DGmodules over DGAs: Model category theory in homological action
Most (perhaps all?) Quillen model categories admit an algebraic model structure, with superior categorical properties provided by a wellchosen 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
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 pushoutproducts and pullbackhoms if and only if the pushoutproducts of the generating (trivial) cofibrations are cellular, i.e., are relative cell complexes, not mere retracts thereof.
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.
This preprint, with a very hastily written introduction and background sections, describes workinprogress 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.
Abelian functor calculus studies horrible functors between abelian categories (that are nonadditive 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 JohnsonMcCarthy 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.
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.
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.
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 homspaces produced by this construction are 3coskeletal. We also show that the cofibrant replacements of discrete simplicial categories produced in this manner are isomorphic to the DwyerKan simplicial resolutions.
This paper answers a question posed by Bill Lawvere: when does nskeletal imply kcoskeletal?
This paper compares the norm maps of GreenleesMay and HillHopkinsRavenel. The appendix presents a unifying categorical framework for the indexed tensor products employed by both constructions.


