Emily Riehl, a mathematician at Johns Hopkins University, won a huge prize from the university recently: the $250,000 President’s Frontier Award. Riehl works in category theory related to homotopy theory, such as $(\infty,1)$-categories. Her work has roots in earlier work of Quillen, Dwyer, Kan, Lurie, and others, but has significantly pushed the field forward.

Riehl was a Ph.D. student at the University of Chicago, with J. Peter May as advisor. She was then a B.P. Instructor at Harvard, before moving to Johns Hopkins. Riehl is a host of the n-Category Café, who proudly wrote about her winning the prize in this post. You can read more about Riehl in the announcement from Johns Hopkins, from her web page, her posts on the n-Category Café, or by looking up her work in MathSciNet. A few reviews from MathSciNet are copied below.

Congratulations, Emily Riehl!

## Some reviews of Riehl’s work.

**MR2781914** ** **

Riehl, Emily(1-CHI)

Algebraic model structures.

*New York J. Math.* 17 (2011), 173–231.

55U35 (18A32)

The author defines and develops the theory of algebraic model categories. The adjective algebraic means that the two functorial weak factorization systems formed by the pairs trivial cofibrations-fibrations and cofibrations-trivial fibrations are algebraic in the sense of R. Garner [Appl. Categ. Structures 17 (2009), no. 3, 247–285; MR2506256] and that there is a morphism of algebraic weak factorization systems called a comparison map from the former algebraic weak factorization system to the latter. The algebraic weak factorization system was originally called a natural weak factorization system by M. Grandis and W. Tholen [Arch. Math. (Brno) 42 (2006), no. 4, 397–408; MR2283020]. This means that the functorial factorizations come from a comonad and a monad. In this setting, cofibrations and fibrations are retracts of coalgebras for comonads and algebras for monads, a result which may make it easier to prove the cofibrancy or the fibrancy of a given map. It is proved that every cofibrantly generated model category underlies a cofibrantly generated algebraic model category. Various algebraic analogues of classic results are given: transfer along adjunctions of an algebraic model structure, characterization of algebraic Quillen adjunctions, algebraic generalization of the projective model structure. Note that a non-cofibrantly generated weak factorization system may be cofibrantly generated in the algebraic sense: the generating set must be then replaced by a non-discrete small category. And a non-cofibrantly generated model category may underlie an algebraic model structure which is cofibrantly generated.

Reviewed by Philippe Gaucher

**MR3221774**

Riehl, Emily(1-HRV)

Categorical homotopy theory.

New Mathematical Monographs, 24. *Cambridge University Press, Cambridge,* 2014. xviii+352 pp. ISBN: 978-1-107-04845-4

18G55 (18D20 55U35)

Categorical homotopy theory, like homological algebra and category theory itself, grew out of the need of algebraic topologists to generalize notions which arose in the study of topological spaces. It has since been applied to such areas as symplectic topology, algebraic geometry, and representation theory.

The first complete formulation of an abstract approach to homotopy theory was provided by D. G. Quillen in [Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer, Berlin, 1967; MR0223432]: it is based on the notion of a model category $\scr M$ (such as the category $\bf{Top}$ of topological spaces, or that of chain complexes) in which one can carry out the actual constructions needed to define the coarser invariants captured by the corresponding homotopy category ${\rm ho}\,\scr M$, in which maps are replaced by their homotopy classes.

The need to describe more refined invariants, which might depend on a choice of (higher) homotopies, led W. G. Dwyer and D. M. Kan to formulate an alternative approach, presented in terms of topologically (or simplicially) enriched categories [see Topology 19 (1980), no. 4, 427–440; MR0584566]. In particular, they showed that any model category can be endowed with such an enrichment. Variants of this approach have appeared in work of Rezk, Bergner, Joyal, Lurie, and others, all subsumed under the notion of an $(\infty,1)$-category.

This is the first book to attempt a comprehensive treatment of both approaches. It differs from earlier accounts of one or the other, such as M. A. Hovey’s [Model categories, Math. Surveys Monogr., 63, Amer. Math. Soc., Providence, RI, 1999; MR1650134], P. S. Hirschhorn’s [Model categories and their localizations, Math. Surveys Monogr., 99, Amer. Math. Soc., Providence, RI, 2003; MR1944041], and J. Lurie’s [Higher topos theory, Ann. of Math. Stud., 170, Princeton Univ. Press, Princeton, NJ, 2009; MR2522659], in that it emphasizes the categorical aspects of the theory, rather than trying to address the needs of the “working algebraic topologist”.

The first part of the book is devoted to the two related notions of derived functors and homotopy (co)limits—both of which can be defined in any category with a suitable notion of weak equivalences. After an exposition of Kan extensions, the author (following W. G. Dwyer et al. in [Homotopy limit functors on model categories and homotopical categories, Math. Surveys Monogr., 113, Amer. Math. Soc., Providence, RI, 2004; MR2102294]) constructs derived functors using deformations. Thus homotopy limits and colimits are provided in a simplicial model category (such as $\bf{Top}$) by the (co)bar construction.

The second part is devoted to enriched homotopy theory, with an emphasis on weighted limits and colimits (in which one decorates a diagram $F\:I\to\scr C$ by a weight $W\:I\to{\bf Set}$). This notion is useful mainly in the enriched context: for a simplicial model category, homotopy (co)limits are weighted by the nerve functor into simplicial sets. In fact, one even has a notion of a weighted homotopy (co)limit. This allows one to enrich the homotopy category ${\rm ho}\,\scr M$ of a (simplicial) model category $\scr M$ over ${\rm ho}\,\bf{Top}$ (a result due to M. Shulman [cf. “Homotopy limits and colimits and enriched homotopy theory”, preprint, arXiv:math/0610194]), which can be thought of as a homotopy version of Dwyer-Kan localization.

The third part of the book describes the classical notion of a model category, with an emphasis on the weak factorization systems of the author’s thesis [New York J. Math. 17 (2011), 173–231; MR2781914], enhanced by R. Garner’s version of the small object argument in [Appl. Categ. Structures 17 (2009), no. 3, 247–285; MR2506256]. It also recapitulates the author’s work with D. Verity on Reedy categories in [Theory Appl. Categ. 29 (2014), 256–301; MR3217884].

The last part deals with $(\infty,1)$-categories, in their quasi-category version, beginning with a survey of Joyal’s (as yet unpublished) monograph on the subject. The treatment of this vast subject is necessarily somewhat sporadic: the topics covered are the topological enrichment of quasi-categories, the treatment of (homotopy) isomorphisms, and some 2-categorical aspects.

In summary, the book provides an interesting slant on the emerging subject of abstract homotopy theory, with an emphasis on categorical tools which may not be familiar to many practitioners in the field.

Reviewed by David A. Blanc

**MR3350229**

Riehl, Emily(1-HRV); Verity, Dominic(5-MCQR-CT)

The 2-category theory of quasi-categories.

*Adv. Math.* 280 (2015), 549–642.

18G55 (18A05 18D20 18G30 55U10 55U35)

Quasi-categories are simplicial sets satisfying the inner horn-filling condition. They were introduced by J. M. Boardman and R. M. Vogt under the name “weak Kan complexes” [Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer, Berlin, 1973; MR0420609], and they provide a convenient model for $(\infty,1)$-categories (i.e. categories weakly enriched in $\infty$-groupoids). In particular, they include ordinary categories (via the nerve functor), and it is natural to try to extend the definitions and theorems of ordinary category theory into the quasi-categorical context. There has been significant work in this direction, mostly by A. Joyal [J. Pure Appl. Algebra 175 (2002), no. 1-3, 207–222; MR1935979; “The theory of quasi-categories and its applications. Vol. II”, Quadern 45, CRM Barcelona, 2008] and J. Lurie [Higher topos theory, Ann. of Math. Stud., 170, Princeton Univ. Press, Princeton, NJ, 2009; MR2522659].

This work is a new contribution in this direction. More precisely, the paper develops a formal category theory of quasi-categories using 2-category theory. The starting point is a (strict) 2-category of quasi-categories $\underline{qCat}_2$ defined as a quotient of the simplicially enriched category of quasi-categories $\underline{qCat}_\infty$. The underlying category of both enriched categories is the usual category of quasi-categories and simplicial maps. By translating simplicial universal properties into 2-categorical ones, it is shown that $\underline{qCat}_2$ is cartesian closed, and that equivalences in $\underline{qCat}_2$ are precisely the (weak) equivalences of quasi-categories introduced by Joyal, proving that this 2-category appropriately captures the homotopy theory of quasi-categories. It is also shown that $\underline{qCat}_2$ admits several weak 2-limits of a sufficiently strict variety with which to develop formal category theory. In particular, it is shown that $\underline{qCat}_2$ admits weak cotensors by categories freely generated by a graph (including, in particular, the walking arrow), weak 2-pullbacks and weak comma objects. These are used to encode the universal properties associated to limits, colimits, adjunctions, and so forth.

The work provides a self-contained account and to some outsiders hoping to understand the foundations of quasi-category theory it may be more approachable than the previous ones.

Reviewed by Josep Elgueta

**MR3415698**

Riehl, Emily(1-JHOP); Verity, Dominic(5-MCQR-CT)

Homotopy coherent adjunctions and the formal theory of monads.

*Adv. Math.* 286 (2016), 802–888.

18G55 (18C15 55U10 55U35 55U40)

The impact of categories on the mathematics of structure reaches well beyond the wildest expectations of researchers in the 1950s. Needless to say, there are specific problems which may not be helped at all by category theory.

Investigations began in the 1960s to show how 2-categories could be used to express commonalities in the study of many variants of the notion of category. In those days, we had in mind categories with specified structure, enriched categories, fibred categories, and so on. New examples are still arising. In their previous paper [Adv. Math. 280 (2015), 549–642; MR3350229], the authors showed how penetrating 2-category theory is in studying quasi-categories (simplicial sets for which all inner horns have fillers). Some new 2-category theory is invented for that purpose. For example, their notion of weak 2-limit lies between the strict notion of weighted limit for 2-categories and the bicategorical notion, involving as it does the concept of smothering functor. Also, what it means for a morphism of quasi-categories to have an adjoint is purely 2-categorical.

The present paper delves more deeply into adjunctions between quasi-categories and the theory of monads. A cofibrant simplicial category they call the free homotopy coherent adjunction is introduced and described by means of a well-founded graphical calculus. Any adjunction of quasi-categories is shown to extend to a homotopy coherent adjunction; and these extensions are shown homotopically unique (the relevant spaces of extensions are contractible Kan complexes).

The reviewer [in Category Seminar (Proc. Sem., Sydney, 1972/1973), 134–180. Lecture Notes in Math., 420, Springer, Berlin, 1974 (p. 167); MR0354813] described the weight required to obtain the Kleisli (and so, dually, the Eilenberg-Moore) construction of a monad as a colimit (limit). In the present paper, the appropriate weights are found to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad. The authors show that each vertex in the quasi-category of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras.

The paper concludes with the quasicategorical version of the Beck monadicity theorem. Indeed, this paper makes clear that a mild variant of Beck’s argument can be expressed totally in terms of the weights themselves and is independent of the quasi-categorical context.

Reviewed by R. H. Street

**MR3917428 **

Riehl, Emily(1-JHOP); Verity, Dominic(5-MCQR-CT)

The comprehension construction.

*High. Struct.* 2 (2018), no. 1, 116–190.

18G55 (55U35)

The article under review is a continuation of the programme of “synthetic theory of $\infty$-categories” initiated by the authors in [E. Riehl and D. Verity, Adv. Math. 280 (2015), 549–642; MR3350229]. The fundamental framework is that of an $\infty$-cosmos, i.e., essentially a category of fibrant objects in the sense of Brown, enriched in quasi-categories. In this context, “$\infty$-category” refers to an object of an abstract $\infty$-cosmos and a great deal of the theory of $(\infty, 1)$-categories (or even higher structures) can be developed at this level of generality. This includes notions such as quasi-categories, complete Segal spaces or complicial sets.

In the present article, the authors define cartesian and cocartesian fibrations, develop their basic theory and establish comprehension of cocartesian fibratinos, which generalizes a number of well-known constructions, such as unstraightening, of J. Lurie [Higher topos theory, Ann. of Math. Stud., 170, Princeton Univ. Press, Princeton, NJ, 2009; MR2522659].

A cocartesian fibration in an $\infty$-cosmos $\mathcal{K}$ is an isofibration $p \colon E \to B$ (here “isofibration” refers simply to a fibration in the category of fibrant objects underlying $\mathcal{K}$) such that the canonical functor $E \to p \downarrow B$ admits a left adjoint in the slice $\infty$-cosmos $\mathcal{K}_{/B}$. The notions of the comma object $p \downarrow B$ and an adjoint used here are defined in terms of the 2-categorical structure of $\mathcal{K}$ which arises from its enrichment.

This definition captures the idea that the fibers of $p$ vary (covariantly) functorially over the $\infty$-category $B$. That idea is made precise in the main theorem which states that for any $\infty$-category $A$, there is a functor (the comprehension functor) $\mathsf{Fun}_{\mathcal{K}}(A, B) \to \mathsf{coCart(K)}_{/A}$. Here, $\mathsf{Fun}_{\mathcal{K}}(A, B)$ is the quasi-category of maps from $A$ to $B$ and $\mathsf{coCart(K)}_{/A}$ is the quasi-category of cocartesian fibrations over $K$. (The latter is the homotopy coherent nerve of the Kan complex enriched category whose objects are cocartesian fibrations over $A$ and whose morphisms are cocartesian functors between them.) The comprehension functor sends a morphism $a \colon A \to B$ to the pullback $E_a$ of $p$ along $a$ and its action on higher morphisms encodes the functoriality of such pullbacks.

The main technical ingredient of the proof is the theory of simplicial computads (i.e., cofibrant simplicial categories) the basics of which are described in great detail. As an application, the authors explain how the Yoneda embedding of an $\infty$-category $A$ can be constructed in terms of comprehension of the cocartesian functor $A^{\mathbf{2}} \to A \times A$ in the slice $\mathcal{K}_{/A}$ and prove a very general version of the higher categorical Yoneda lemma.

Reviewed by Karol Szumiło