nLab Emily Riehl

I am an Assistant Professor in the Department of Mathematics at Johns Hopkins University.

My webpage can be found here.

Selected writings

On rigidification of quasi-categories:

• Emily Riehl, On the structure of simplicial categories associated to quasi-categories, Math. Proc. Camb. Phil. Soc. 150 (2011) 489-504 [arXiv:0912.4809, doi:10.1017/S0305004111000053]

On topos levels in simplicial sets and cubical sets:

• C. Kennett, Emily Riehl, Michael Roy, M. Zaks, Levels in the toposes of simplicial sets and cubical sets, JPAA 215 5 (2011) 949-961 [arXiv:1003.5944, doi:j.jpaa.2010.07.002]

On Reedy model structures via weighted colimits:

• Emily Riehl, Dominic Verity, The theory and practice of Reedy categories, Theory and Applications of Categories 29 9 (2014) 256-301 [tac:29-09,arXiv:1304.6871]

An introductory category theory textbook for beginning graduate students or advanced undergraduates with an emphasis on applications of categorical concepts to a variety of areas of mathematics.

• Emily Riehl, Category Theory in Context (2017)

Textbooks on (simplicial) homotopy theory and (∞,1)-category theory with emphasis on tools from category theory and 2-category theory (via ∞-cosmoi and the homotopy 2-category of (∞,1)-categories):

• Emily Riehl, Categorical Homotopy Theory, Cambridge University Press (2014) [doi:10.1017/CBO9781107261457, pdf]

• Emily Riehl, Dominic Verity, Elements of ∞-Category Theory, Cambridge studies in advanced mathematics 194, Cambridge University Press (2022) [doi:10.1017/9781108936880, ISBN:978-1-108-83798-9, pdf]

On directed homotopy type theory via a form of simplicial type theory

• Emily Riehl, Mike Shulman, A type theory for synthetic ∞-categories, Higher Structures 1 1 (2017) [arxiv:1705.07442, published article]

Survey of homotopy theory from homotopical categories to (∞,1)-categories:

• Emily Riehl, Homotopical categories: from model categories to (∞,1)-categories, in: Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill (eds,) Stable categories and structured ring spectra, MSRI Book Series, Cambridge University Press (arXiv:1904.00886)

A new proof of the Strøm model structure using algebraic weak factorization systems:

• Tobias Barthel, Emily Riehl, On the construction of functorial factorizations for model categories, Algebr. Geom. Topol. 13 (2013) 1089-1124 (arXiv:1204.5427, doi:10.2140/agt.2013.13.1089)

On monads in computer science:

• Emily Riehl, A categorical view of computational effects, talk at C$\circ$mp$\circ$se$\colon\!\colon$Conference (2017) [pdf, pdf]

On left-transferred model structures and model structures on functors:

• Marzieh Bayeh, Kathryn Hess, Varvara Karpova, Magdalena Kedziorek, Emily Riehl, Brooke Shipley, Left-induced model structures and diagram categories, in: Women in Topology: Collaborations in Homotopy Theory, Contemporary Mathematics 641 American Mathematical Society (2015) [arXiv:1401.3651, ISBN:978-1-4704-2495-4]

• Kathryn Hess, Magdalena Kedziorek, Emily Riehl, Brooke Shipley, A necessary and sufficient condition for induced model structures, J. Topology 10 2 (2017) 324-369 [arXiv:1509.08154, doi:10.1112/topo.12011]

  (this contains an error which is corrected in Garner, Kedziorek & Riehl 2018)

• Richard Garner, Magdalena Kedziorek, Emily Riehl, Lifting accessible model structures, J. Topology 13 1 (2020) 59-76 [arXiv:1802.09889, doi:10.1112/topo.12123]

On (∞,1)-category theory via the homotopy 2-category of (∞,1)-categories (formal $(\infty,1)$-category theory):

• Emily Riehl, Dominic Verity, The 2-category theory of quasi-categories, Advances in Mathematics Volume 280, 6 August 2015, Pages 549-642 (arXiv:1306.5144, doi:10.1016/j.aim.2015.04.021)

• Emily Riehl, Dominic Verity, Infinity category theory from scratch, Higher Structures Vol 4, No 1 (2020) (arXiv:1608.05314, pdf)

• Emily Riehl, The formal theory of ∞-categories, talk at Categories and Companions Symposium June 8–12, 2021 (video)

On (∞,1)-functors and (∞,1)-monads:

• Emily Riehl, Dominic Verity, Homotopy coherent adjunctions and the formal theory of monads, Advances in Mathematics, Volume 286, 2 January 2016, Pages 802-888 (arXiv:1310.8279, doi:10.1016/j.aim.2015.09.011)

On the Yoneda lemma for (∞,1)-categories:

• Emily Riehl, Dominic Verity, Fibrations and Yoneda’s lemma in an $\infty$-cosmos, Journal of Pure and Applied Algebra Volume 221, Issue 3, March 2017, Pages 499-564 (arXiv:1506.05500, doi:10.1016/j.jpaa.2016.07.003)

On (discrete) Grothendieck fibrations and profunctors:

• Emily Riehl and Fosco Loregian, Categorical notions of fibration, Expositiones Mathematicae 38 4 (2020) 496-514 [arXiv:1806.06129, doi:10.1016/j.exmath.2019.02.004]

On the categorical semantics of univalent homotopy type theory in $\infty$-topoi (proof of Awodey's conjecture):

• Emily Riehl, On the $\infty$-topos semantics of homotopy type theory, lecture at Logic and higher structures CIRM (Feb. 2022) $[$pdf, pdf$]$

Exposition of (∞,1)-category theory with an eye towards homotopy type theory:

• Emily Riehl, $\infty$-Category theory for undergraduates, talk at CQTS (Dec. 2022) [web, video: YT]

On teaching (∞,1)-category theory to undergraduate students:

• Emily Riehl, Could $\infty$-category theory be taught to undergraduates?, Notices of the AMS (May 2023) [published pdf, arxiv:2302.07855]

category: people