Research Papers:

1.  A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), 2043-2058.

(math.AT/0406507)

2. Three models for the homotopy theory of homotopy theories, Topology 46 (2007), 397-436. 

(math.AT/0504334)  

Longer thesis version (University of Notre Dame, 2005)

3. Rigidification of algebras over multi-sorted theories, Algebr. Geom. Topol. 6 (2006) 1925-1955.

(math.AT/0508152)

4. Simplicial monoids and Segal categories, Contemp. Math. 431 (2007) 59-83.

and correction

(math.AT/0508416; math.AT/0806.1767)

5. A characterization of fibrant Segal categories, Proc. Amer. Math. Soc. 135 (2007) 4031 4037.

(math.AT/0603400)

6. Adding inverses to diagrams encoding algebraic structures, Homology, Homotopy Appl. 10(2), 2008, 149–174.

(math.AT/0610291)

and erratum

7. Complete Segal spaces arising from simplicial categories, Trans. Amer. Math. Soc. 361 (2009), 525-546.

(math.AT/0704.1624) 

8. Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy Appl. 10(2), 2008, 175–193.

(math.AT/0710.2254)

and erratum 

9. Homotopy fiber products of homotopy theories, Israel Journal of Mathematics 185 (2011), 389-411.

(math.AT/0811.3175)

10. Derived Hall algebras for stable homotopy theories, Cah. Topol. Géom. Différ. Catég. 54 (2013), no. 1, 28–55.

(math.AT/0910.1861)

11. Homotopy limits of model categories and more general homotopy theories, Bull. Lond. Math. Soc. 44 (2012), no. 2, 311–322. 

(math.AT/1010.0717)

12. Reedy categories and the θ-construction (with C. Rezk), Math. Z. 274 (1), 2013, 499-514.

(math.AT/1110.1066)

13. Comparison of models for (∞,n)-categories, I (with C. Rezk), Geom. Topol. 17 (2013), no. 4, 2163–2202. 

(math.AT/1204.2013)

14. Group actions on Segal operads (with P. Hackney), Israel J. Math. 202 (2014), no. 1, 423–460. 

(math.AT/1207.3465)

15. Reedy categories which encode the notion of category actions (with P. Hackney), Fund. Math.  228 (2015), no. 3, 193–222.

(math.AT/1207.3467)

16. Homotopy colimits of model categories, An Alpine Expedition through Algebraic Topology, Contemp. Math., 617 (2014), 31-37.

(math.AT/1212.4541)

17. Group actions on Γ-spaces (with P. Hackney), Manifolds and K-Theory, Contemp. Math, 682 (2017) 39-50.

(math.AT/1212.4542)

18. Fixed points of p-toral groups acting on partition complexes (with R. Joachimi, K. Lesh, V. Stojanoska, and K. Wickelgren), Women in Topology: Collaborations in Homotopy Theory, Contemp. Math. 641 (2015), 83-96. 

(math.AT/1401.0491)

19. Comparison of models for (∞,1)-categories II (with C. Rezk), J. Topol. 13 (2020), 1554--1581. 

(math.AT/1406.4182)

20. Classification of problematic subgroups of U(n) (with R. Joachimi, K. Lesh, V. Stojanoska, and K. Wickelgren), Trans. Amer. Math. Soc.  371 (2019) 6739-6777.

(math.AT/1407.0062)

21. Equivalence of models for equivariant (∞,1)-categories, Glasg. Math. J., 59 (2017), no. 1, 237–253. 

(math.AT/1408.0038)

22. Equivariant complete Segal spaces (with S.G. Chadwick), Homology, Homotopy, and Appl. 17(2), 2015, 371-381. 

(math.AT/1502.06637)

23. 2-Segal sets and the Waldhausen construction (with A. Osorno, V. Ozornova, M. Rovelli, and C.I. Scheimbauer), Topology Appl.,235 (2018) 445--484.

(math.AT/1609.02853)

24. The edgewise subdivision criterion for 2-Segal objects (with A. Osorno, V. Ozornova, M. Rovelli, and C. Scheimbauer), Proc. Amer. Math. Soc. 148 (2020) 71-82.

(math.AT/1807.05069)

25. 2-Segal objects and the Waldhausen construction (with A. Osorno, V. Ozornova, M. Rovelli, and C.I. Scheimbauer), Alg. Geom. Topol. 21 (2021) 1267-1326.

(math.AT/1809.10924)

26. Comparison of Waldhausen constructions (with A. Osorno, V. Ozornova, M. Rovelli, and C.I. Scheimbauer), Ann. K-Theory, 6-1 (2021).

(math.AT/1901.03606)

27. Enriched functor categories for functor calculus (with L. Bandklayder, R. Griffiths, B. Johnson, and R. Santhanam), Topology Appl. 316 (2022), Paper No. 108099, 33 pp.

(math.AT/2003.00071)

28. An explicit comparison between 2-complicial sets and θ2-spaces (with V. Ozornova and M. Rovelli), Alg. Geom. Topol. 24 (2024) 3827–3873.

(math.AT/21041.3292)

29. Discreteness and completeness for Θn-models for (∞, n)-categories, Tunis. J. Math. 6 (2024) no. 1, 49-96.

(math.AT/2209.08156) 

Note: The version here corrects an error in the original published version.

(Fun fact: As part of the choir for the album recording of Karl Jenkins' Let's Go, I read a paragraph of this paper during the "babbling" section.)

30. Equivariant trees and partition complexes (with P. Bonventre, M. Calle, D. Chan, and M. Sarazola), Theory Appl. Categ. 45 (2026), Paper No. 15, 501–536.

(math.AT/2302.08949)

31. Cofibrantly generated model structures for functor calculus (with L. Bandklayder, R. Griffiths, B. Johnson, and R. Santhanam), Alg. Geom. Topol. 25-7 (2025), 3931-3973.

(math.AT/2304.12954)

32. 2-Segal sets from cuts of rooted trees (with O. Borghi, P. Dey, I. Gálvez-Carrillo, and T. Hoekstra-Mendoza), Topology Appl. 376 (2025), Paper No. 109447, 29 pp.

(math.AT/2501.10518)

Book:

The Homotopy Theory of (∞,1)-Categories, Cambridge University Press, 2018.

Errata

Edited Volume:

Higher Segal spaces and applications (with J. Kock and M. Sarazola), Contemp. Math., 838 American Mathematical Society, Providence, RI, 2026, x+331 pp.

Expository Papers:

1. A survey of (∞,1)-categories, in J. Baez and J. P. May, Towards Higher Categories, IMA Volumes in Mathematics and Its Applications, Springer, 2010, 69-83.

(math.AT/0610239)

2. Models for (∞, n)-categories and the cobordism hypothesis, in H. Sati and U. Schreiber, ed., Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Proc. Sympos. Pure Math. 83 (2011) 17-30.

(math.AT/1011.0110)

3. Cluster categories for topologists (with M. Robertson), Stacks and Categories in Geometry, Topology and Algebra, Contemp. Math. 643 (2015) 25-35.

(math.AT/1308.2560)

4. A survey of models for (∞,n)-categories, in Handbook of Homotopy Theory (Haynes Miller, ed,), CRC Press, 2020, 263-295.

(math.AT/1810.10052)

5. Modeling homotopy theories, Notices Amer. Math. Soc. 66 (2019) 1423-1432.

6. Homotopy limits of model categories, revisited, in Equivariant Topology and Derived Algebra (S. Balchin et al, editors), London Mathematical Society Lecture Note Series 474 (2021).

(math.AT/2411.18546)

7. Simplicial sets in topology, category theory, and beyond, Matematica 1 (2022), no. 4, 886–912.

(math.AT/2411.18561)

8. Cyclic Segal spaces (with W. Stern), Contemp. Math., 838 American Mathematical Society, Providence, RI, 2026, 211–253.

(math.AT/2409.11945)

9. Combinatorial examples and applications of 2-Segal sets, to appear in Boletín de Matemáticas  (math.AT/2411.18544)

Papers written for or with undergraduates:

1. Groupoid cardinality and Egyptian fractions (with C. Walker), College Math. J. 46 (2015) 122-129.

(older, more formal version)

2. Action graphs and Catalan numbers (with G. Alvarez and R. Lopez), J. Integer Seq. 18 (2015), Article 15.7.2.

(math.CO/1503.00044)

3. Action graphs, planar rooted forests, and self-convolutions of the Catalan numbers (with C. Harper, R. Keller, and M. Rosi-Marshall)

(math.CO/1807.03005)

Non-technical papers:

1. Writing papers in the mathematical sciences, Academe: First Forays into Academic Writing, Vol. 2, No. 1 (January 2020).

2. What do I do when my paper or grant proposal is rejected?, Notices Amer. Math. Soc. 67 No. 3 (2020) 364—365.

Submitted preprints:

1. A comparison of definitions of equivariant trees (with M.E. Calle, D. Chan, A.M. Osorno, and M. Sarazola)

(math.AT/2512.14956).

Unpublished Notes:

1. Workshop on the homotopy theory of homotopy theories

2. Examples and applications of 2-Segal spaces