The n-Category Café

Dedicated to Max Kelly, June 5 1930 to January 26 2007

This is not a proceedings of the 2004 conference “$n$-Categories: Foundations and Applications” that we organized and ran at the IMA during the two weeks June 7–18, 2004! We thank all the participants for helping make that a vibrant and inspiring occasion. There has been a great deal of work in higher category theory since then, but we still feel that it is not yet time to offer a volume devoted to the main topic of the conference. At that time, we felt that we were at the beginnings of a large new area of mathematics, but one with many different natural approaches in desperate need of integration into a cohesive field. We feel that way still.

So, instead of an introduction to higher category theory, we have decided to publish a series of papers that provide useful background for this subject. This volume is aimed towards the wider mathematical community, rather than those knowledgeable in category theory and especially higher category theory. We are particularly sensitive to the paucity of young Americans knowledgeable enough in the subject to be potential readers.

The focus of the conference was on comparing the many approaches to higher category theory. These approaches, as they existed at the time of the conference, have been summarized by Leinster [Lein1] and by Cheng and Lauda [ChLau]. The earliest was based on filtered simplicial sets. It is due to Street [Str1, Str3] and is being developed in detail by Verity [Verity1, Verity2, Verity3], with a contribution by Gurski [Gur]. Another approach, called “opetopic” since operads were used to describe the shapes of diagrams used in the theory, is due to Baez and Dolan [BD] and has been developed by Leinster [Lein2], Cheng [Ch1, Ch2, Ch3, Ch4], and Makkai and his collaborators [HMP,Makkai]. There is also a topologically motivated approach using operads due to Trimble [Trimble], which has been studied and generalized by Cheng and Gurski [Ch5, ChGur]. There is a quite different and more extensively developed operadic approach to weak $\infty$-categories due to Batanin [Bat1, Str2], with a variant due to Leinster [Lein3]. Penon [Penon] gave a related, very compact definition of $\infty$-category; a modified version correcting a small but critical flaw has been proposed by Cheng and Makkai [ChMakkai]. Another highly developed approach, based on $n$-fold simplicial sets and inspired by work of Segal in infinite loop space theory, is due to Tamsamani and Simpson [Simpson1, Simpson2, Simpson3, Simpson4, Tam1]. Yet another theory, due to Joyal [Berger1,Joyal], is based on a presheaf category called the category of “theta-sets”. Also, Lurie [Lurie3] has begun making extensive use of Barwick’s approach to $(\infty,n)$-categories, which are roughly weak $\infty$-categories where all morphisms above dimension $n$ are invertible.

The theme of the 2004 conference was comparisons among all these approaches to higher categories — or at least, those that existed at the time. While there are papers that tackle aspects of this immense unification project [Berger2, Ch3, Ch5, Simpson5], it is still quite unclear how a unified theory of higher categories will evolve. In proposing the conference, we wrote as follows: “It is not to be expected that a single all embracing definition that is equally suited for all purposes will emerge. It is not a question as to whether or not a good definition exists. Not one, but many, good definitions already do exist, although they have been worked out to varying degrees. There is growing general agreement on the basic desiderata of a good definition of $n$-category, but there does not yet exist an axiomatization, and there are grounds for believing that only a partial axiomatization may be in the cards.”

One can make analogies with many other areas where a number of interrelated definitions exist. In algebraic topology, there are various symmetric monoidal categories of spectra, and they are related by a web of Quillen equivalences of model categories. In this context, all theories are in some sense “the same”, but the applications require use of different models: many things that can be proven in one model cannot easily be proven in another [May]. In algebraic geometry, there are many different cohomology theories, definitely not all the same, but connected by various comparison functors. Motivic theory is in part a search for a universal source of such comparisons.

In the case of weak $n$-categories, it is unclear whether there is a useful sense in which all known theories are the same. We do not have a complete web of comparison maps relating different theories. Nor are we sure what it means for two theories to be “the same”, despite important insights by Grothendieck [Gro] and Makkai [Makkai2]. The terms in which comparisons should be made are not yet clear. Quillen model category theory should capture some comparisons, but it may be too coarse to give the complete story. A smaller related theme of the conference was that there should be a “baby” comparison project, for which model category theory would in fact be sufficient. Precisely, the idea was that there should be a web of Quillen equivalences among the various notions of $(\infty,1)$ category. These include topologically or simplicially enriched categories, Segal categories, complete Segal spaces, and quasi-categories. In the years since the conference, this comparison project has been largely completed by Bergner [Be1, Be2] and Joyal and Tierney [JT].

Another smaller related theme was the higher categorical modelling of $n$-types of topological spaces. It was a dream of Grothendieck [Gro] that weak $n$-groupoids should model $n$-types. The algebraic model of $n$-types due to Loday [Loday] is one implementation of this idea. More recently we are seeing attempts to implement Grothendieck’s idea in various approaches to $n$-categories [Bat2, Bat3, Tam2, Cisinski]. There is also a large body of work on simpler $n$-categorical structures, such as “strict” $n$-groupoids, that capture only part of the information in a homotopy $n$-type. The prime mover of this line of work is Brown [BHS].

As mentioned, the goal of this volume is merely to prepare the reader for more detailed study of these fast-moving topics. So, we begin with a light-hearted paper that treats Grothendieck’s dream as a starting-point for speculations on the relationship between $n$-categories and cohomology. It is based on notes that Michael Shulman took of John Baez’s 2007 Namboodiri Lectures at Chicago. Higher category theory has largely developed from a series of analogies with and potential applications to other subjects, including algebraic topology, algebraic geometry, mathematical physics, computer science, logic, and, of course, category theory. This paper illustrates this, and raises the challenge of formalizing the patterns that become visible thereby.

The second paper, by Julie Bergner, is a survey of $(\infty,1)$-categories. She begins by describing four approaches: simplicial categories, complete Segal spaces, Segal categories and quasi-categories. Then she describes the network of Quillen equivalences relating these: the “baby comparison project” mentioned above.

The third paper, by Simona Paoli, focuses on a number of algebraic ways of modelling $n$-types of topological spaces in terms of strict categorical structures. Her focus is on the role of “internal” structures in higher category theory, that is, structures that live in categories other than the category of sets. She surveys the known comparisons among such algebraic models for $n$-types. It is a part of the comparison project to relate various notions of weak $n$-groupoid to these strict algebraic models for topological $n$-types.

Logically, we might next delve into various approaches to $n$-categories and full-fledged $\infty$-categories. But this seems premature. So instead, the rest of the volume goes back to the beginnings of the subject. By now every well-educated young mathematician can be expected to be familiar with categories, as introduced by Eilenberg and Mac Lane in 1945. It has taken longer to understand that what they introduced was a $2$-category: Cat, with categories as objects, functors as morphisms, and natural transformations as 2-morphisms. Ehresmann [Ehresmann] introduced strict $n$-categories sometime in the 1960’s, and Eilenberg and Kelly discussed them in 1965 [EK], with Cat as a key example of a strict 2-category. Bénabou introduced the more general weak $2$-categories or “bicategories” the following year [Benabou]. But even today, the mathematics of 2-categories is considered somewhat recondite, even by many mathematicians who implicitly use these structures all the time. There is a great deal of basic $2$-category theory that can illuminate everyday mathematics. The second author rediscovered a chunk of this while writing a book on parametrized homotopy theory [MS], and he was chastened to see how little he knew of something that was so very basic to his own work.

For this reason, the next paper is the longest in this volume: a thorough introduction to the theory of $2$-categories, by Stephen Lack. This paper gives a solid grounding for anyone who wants some idea of what lies beyond mere categories and how to work with higher categorical notions. Anybody interested in higher category theory must learn something of the richness of $2$-categories.

Lawrence Breen’s paper, on gerbes and 2-gerbes, gives an idea of how naturally $2$-categorical algebra arises in the study of algebraic and differential geometry. His paper also illustrates the need for “enriched” higher category theory, in which one deals with hom objects that have more structure than is seen in merely set-based categories. Many more such applications could be cited.

Stephen Lack is an Australian, and it is noteworthy that the premier world center for category theory has long been Sydney. We have dedicated this volume to Max Kelly, the founder of the Australian school of category theory, who died in 2007. Kelly visited Chicago in 1970-71, just before becoming Chair of the Department of Mathematics at Sydney. That was long before e-mail, and Max was considering how best to build up a department that would necessarily suffer from a significant degree of isolation. He decided to focus in large part on the development of his own subject, and he succeeded admirably. The final paper in this volume, by Kelly’s student Ross Street, gives a fascinating mathematical and personal account of the development of higher category theory in Australia.

We had very much hoped to include a survey by André Joyal of his important work on quasi-categories. As shown in the work of Lurie [Lurie1, Lurie2], these give probably the most tractable model of $(\infty,1)$-categories. Joyal’s work showing that one “can do category theory” in quasi-categories is an essential precursor to Lurie’s work and is unquestionably one of the most important recent developments in higher category theory. However, Joyal’s survey is not yet complete: it has grown to hefty proportions and is still growing. So, it will appear separately.

References

[BD1] J. Baez and J. Dolan, Higher-dimensional algebra III: $n$-categories and the algebra of opetopes, Adv. Math. 135 (1998), 145–206. Also available as arXiv:q-alg/9702014.

[Bat1] M. Batanin, Monoidal globular categories as natural environment for the theory of weak $n$-categories, Adv. Math. 136 (1998), 39–103.

[Bat2] M. Batanin, The Eckmann–Hilton argument, higher operads and $E_n$-spaces. Available as arXiv:math/0207281.

[Bat3] M. Batanin, The combinatorics of iterated loop spaces. Available as arXiv:math/0301221.

[Benabou] J. Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1–77.

[Berger1] C. Berger, Iterated wreath product of the simplex category and iterated loop spaces, Adv. Math. 213 (2007), 230–270. Also available as arXiv:math/0512575.

[Berger2] C. Berger, A cellular nerve for higher categories, Adv. Math. 169 (2002), 118–175. Also available at http://math.ucr.edu.fr/~cberger/.

[Be1] J. Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007), 2043-2058. Also available as arXiv:math/0406507.

[Be2] J. Bergner, Three models of the homotopy theory of homotopy theory, Topology 46 (2006), 1925–1955. Also available as arXiv:math/0504334.

[BHS] R. Brown, P. Higgins and R. Sivera, Nonabelian Algebraic Topology: Higher Homotopy Groupoids of Filtered Spaces Available at http://www.bangor.ac.uk/~mas010/nonab-a-t.html.

[Ch1] E. Cheng, The category of opetopes and the category of opetopic sets, Th. Appl. Cat. 11 (2003), 353–374. Also available as arXiv:math/0304284.

[Ch2] E. Cheng, Weak $n$-categories: opetopic and multitopic foundations, Jour. Pure Appl. Alg. 186 (2004), 109–137. Also available as arXiv:math/0304277.

[Ch3] E. Cheng, Weak $n$-categories: comparing opetopic foundations, Jour. Pure Appl. Alg. 186 (2004), 219–231. Also available as arXiv:math/0304279.

[Ch4] E. Cheng, Opetopic bicategories: comparison with the classical theory, available as arXiv:math/0304285.

[Ch5] E. Cheng, Comparing operadic theories of $n$-category, available as arXiv:0809.2070.

[ChGur] E. Cheng and N. Gurski, Toward an $n$-category of cobordisms, Th. Appl. Cat. 18 (2007), 274–302. Available at http://www.tac.mta.ca/tac/volumes/18/10/18-10abs.html.

[ChLau] E. Cheng and A. Lauda, Higher-Dimensional Categories: an Illustrated Guidebook. Available at http://www.dpmms.cam.ac.uk/$\sim$elgc2/guidebook/.

[ChMakkai] E. Cheng and M. Makkai, A note on the Penon definition of $n$-category, to appear in Cah. Top. Géeom. Diff.

[Cisinski] D.-C. Cisinski, Batanin higher groupoids and homotopy types, in Categories in Algebra, Geometry and Mathematical Physics, eds. A. Davydov et al, Contemp. Math. 431, AMS, Providence, Rhode Island, 2007, pp. 171–186. Also available as arXiv:math/0604442.

[Ehresmann] C. Ehresmann, Catégories et Structures, Dunod, Paris, 1965.

[EK] S. Eilenberg and G. M. Kelly, Closed categories, in Proceedings of the Conference on Categorical Algebra, eds. S. Eilenberg et al, Springer, New York, 1966.

[Gro] A. Grothendieck, Pursuing Stacks, letter to D. Quillen, 1983. To be published, eds. G. Maltsiniotis, M. Künzer and B. Toen, Documents Mathématiques, Soc. Math. France, Paris, France.

[Gur] M. Gurski, Nerves of bicategories as stratified simplicial sets. To appear in Jour. Pure Appl. Alg..

[HMP] C. Hermida, M. Makkai, and J. Power: On weak higher-dimensional categories I, II. Jour. Pure Appl. Alg. 157 (2001), 221–277.

[Joyal] A. Joyal, Disks, duality and $\theta$-categories, preprint, 1997.

[JT] A. Joyal and M. Tierney, Quasi-categories vs Segal spaces, available as arXiv:math/0607820.

[Lein1] T. Leinster, A survey of definitions of $n$-category, Th. Appl. Cat. 10 (2002), 1–70. Also available as arXiv:math/0107188.

[Lein2] T. Leinster, Structures in higher-dimensional category theory. Available as arXiv:math/0109021.

[Lein3] T. Leinster, Higher Operads, Higher Categories, Cambridge U. Press, Cambridge, 2003. Also available as arXiv:math/0305049.

[Loday] J. L. Loday, Spaces with finitely many non-trivial homotopy groups, Jour. Pure Appl. Alg. 24 (1982), 179–202.

[Lurie1] J. Lurie, Higher Topos Theory, available as arXiv:math/0608040.

[Lurie2] J. Lurie, Stable infinity categories, available as arXiv:math/0608228.

[Lurie3] J. Lurie, On the classification of topological field theories, available as arXiv:0905.0465.

[Makkai] M. Makkai, The multitopic $\omega$-category of all multitopic $\omega$-categories. Available at http://www.math.mcgill.ca/makkai.

[Makkai2] M. Makkai, On comparing definitions of weak $n$-category”. Available at http://www.math.mcgill.ca/makkai.

[May] J. P. May… something on model categories.

[MS] J. P. May and J. Siggurdson, Parametrized Homotopy Theory, AMS, Providence, Rhode Island, 2006.

[Penon] J. Penon, Approche polygraphique des $\infty$-categories non strictes, Cah. Top. Géom. Diff. 40 (1999), 31–80.

[Simpson1] C. Simpson, A closed model structure for $n$-categories, internal Hom, $n$-stacks and generalized Seifert–Van Kampen. Available as alg-geom/9704006.

[Simpson2] C. Simpson, Limits in $n$-categories, available as arXiv:alg-geom/9708010.

[Simpson3] C. Simpson, Calculating maps between $n$-categories, available as arXiv:math/0009107.

[Simpson4] C. Simpson, On the Breen–Baez–Dolan stabilization hypothesis for Tamsamani’s weak $n$-categories, available as arXiv:math/9810058.

[Simpson5] C. Simpson, Some properties of the theory of $n$-categories, available as arXiv:math/0110273.

[Str1] R. Street, The algebra of oriented simplexes, Jour. Pure Appl. Alg. 49 (1987), 283–335.

[Str2] R. Street, The role of Michael Batanin’s monoidal globular categories, in Higher Category Theory, Contemp. Math. 230, AMS, Providence, Rhode Island, 1998, pp. 99–116. Also available at http://www.math.mq.edu.au/~street.

[Str3] R. Street, Weak omega-categories, in Diagrammatic Morphisms and Applications, Contemp. Math. 318, AMS, Providence, RI, 2003, pp. 207–213. Also available at http://www.math.mq.edu.au/~street.

[Tam1] Z. Tamsamani, Sur des notions de $n$-catégorie et $n$-groupoide non-strictes via des ensembles multi-simpliciaux, K-Theory 16 (1999), 51–99. Also available as arXiv:alg-geom/9512006.

[Tam2] Z. Tamsamani, Equivalence de la théorie homotopique des $n$-groupoides et celle des espaces topologiques $n$-tronqués. Also available as arXiv:/alg-geom/9607010.

[Trimble] Trimble $n$-category, $n$Lab entry available at http://ncatlab.org/nlab/show/Trimble+n-category.

[Verity1} D. Verity, Complicial Sets: Characterising the Simplicial Nerves of Strict $\omega$-Categories, Memoirs AMS 905, 2005. Also available as arXiv:math/0410412.

[Verity2] D. Verity, Weak complicial sets, a simplicial weak $\omega$-category theory. Part I: basic homotopy theory. Available as arXiv:math/0604414.

[Verity3] D. Verity, Weak complicial sets, a simplicial weak $\omega$-category theory. Part II: nerves of complicial Gray-categories. Available as arXiv:math/0604416.

Table of Contents

• John Baez and Michael Shulman, Lectures on $n$-categories and cohomology.
• Julia Bergner, A survey of $(\infty,1)$-categories.
• Simona Paoli, Internal categorical structures in homotopical algebra.
• Stephen Lack, A 2-categories companion.
• Lawrence Breen, Notes on 1- and 2-gerbes.
• Ross Street, An Australian conspectus of higher categories.