EMILY RIEHLeriehl at math dot harvard dot edu
Harvard University
Science Center 320 
New  Research  Exposition  Talks  Teaching 
∞category theory from scratch:
Lecture notes from a minicourse given at the Young Topologists' Meeting 2015 can be found here: ∞category theory from scratch (last update 7/10). 


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.
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.
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.


