nLab Charles Rezk (changes)

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Charles Waldo Rezk is a mathematician at the University of Illinois Urbana–Champaign.

He got his PhD degree in 1996 at MIT, advised by Michael J. Hopkins.

His PhD students include Nathaniel Stapleton and Nima Rasekh.

• website

Selected writings

Proof of the canonical model structure on Cat:

• Charles Rezk, A model category for categories, 1996 (dvi, pdf)

On the homotopy theory of homotopy theories (the (infinity,1)-category of (infinity,1)-categories) in terms of complete Segal spaces:

• Charles Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001), 973-1007 (arXiv:math/9811037, doi:10.1090/S0002-9947-00-02653-2)

Introducing model toposes/(∞,1)-toposes:

• Charles Rezk, Toposes and homotopy toposes, 2010 (pdf, pdf)

Introducing quasi-elliptic cohomology:

• Charles Rezk, Quasi-Elliptic Cohomology (2014) [pdf]

  (not publicly available until upload here, in 2023)

Review of higher topos theory:

• Charles Rezk, Lectures on Higher Topos Theory, Leeds 2019 (pdf, pdf)

On complete Segal spaces:

• Charles Rezk, A model for the homotopy theory of homotopy theory , Trans. Amer. Math. Soc., 353(3), 973-1007 (pdf)

On power operations:

• Charles Rezk, Lectures on power operations (dvi) (2006)

• Charles Rezk, Power operations in Morava E-theory – a survey (2009) (pdf)

• Charles Rezk, Isogenies, power operations, and homotopy theory, article (pdf) and talk at ICM 2014 (pdf)

Expressing logarithmic cohomology operations in terms of the Bousfield-Kuhn functor and power operations:

• Charles Rezk, The units of a ring spectrum and a logarithmic cohomology operation, J. Amer. Math. Soc. 19 (2006), 969-1014 (arXiv:math/0407022)

On ∞-groups of units, Thom spectra and twisted generalized cohomology:

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra, orientations, and Thom spectra via rigid infinite loop space theory, Journal of Topology, Volume7, Issue 4, December 2014 (arXiv:1403.4320, arXiv:10.1112/jtopol/jtu009)

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology, Journal of Topology Volume 7, Issue 3 2014 Pages 869–893 (arXiv:1403.4325, doi:10.1112/jtopol/jtt035)

On Theta-spaces:

• Charles Rezk, A cartesian presentation of weak $n$-categories Geom. Topol. 14 (2010), no. 1, 521–571 (arXiv:0901.3602)

  Correction to “A cartesian presentation of weak $n$-categories” Geom. Topol. 14 (2010), no. 4, 2301–2304. MR 2740648 (pdf)

• Charles Rezk, Cartesian presentations of weak $n$-categories – An introduction to $\Theta_n$-spaces (2009) (pdf)

On the string orientation of tmf:

• Matthew Ando, Michael Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (pdf)

On slices of global equivariant homotopy theory as a cohesive (∞,1)-topoi over $G$-equivariant homotopy theory (cohesion of global- over G-equivariant homotopy theory):

• Charles Rezk, Global Homotopy Theory and Cohesion, 2014 (pdf, pdf)

On sufficient conditions for geometric realization of simplicial topological spaces to commute with homotopy pullback

• Charles Rezk, When are homotopy colimits compatible with homotopy base change?, 2014 (pdf, pdf)

On elliptic cohomology:

• Charles Rezk, Elliptic cohomology and elliptic curves, Felix Klein Lectures, Bonn (2015) $[$web, pdf, pdf, video rec: part 1, 2, 3, 4$]$

On classifying spaces for equivariant principal bundles with 1-truncated compact Lie structure group:

• Charles Rezk, Classifying spaces for 1-truncated compact Lie groups, Algebr. Geom. Topol. 18 (2018) 525-546 (arXiv:1608.02999)

On quasi-categories:

• Charles Rezk, Stuff about quasicategories, Lecture Notes for course at University of Illinois at Urbana-Champaign, 2016, version May 2017 (pdf, pdf)

• Charles Rezk, Introduction to quasicategories (2022) [[pdf](https://faculty.math.illinois.edu/~rezk/quasicats.pdf), pdf]

On compactly generated topological spaces:

• Charles Rezk, Compactly Generated Spaces, 2018 (pdf, pdf)

On accessible $(\infty,1)$-categories:

• Charles Rezk, Generalizing accessible ∞-categories, 2021 (pdf).

On spectral algebraic geometry:

• Charles Rezk, Spectral algebraic geometry (pdf) in: Andrew J. Blumberg, Teena Gerhardt, Michael A. Hill (eds.), Stable categories and structured ring spectra, MSRI Book Series, Cambridge University Press

Related $n$Lab entries

in higher algebra/stable homotopy theory

• power operation

• logarithmic cohomology operation

• string orientation of tmf

• twisted cohomology, twisted Umkehr map

in higher category theory:

• (∞,n)-category

• Theta space/(n,r)-category

in higher topos theory:

• (∞,1)-topos

• higher topos theory

• global equivariant homotopy theory