## June 26, 2009

### This Book Needs a Title

#### Posted by John Baez

You may be familiar with Raymond Smullyan’s delightful books packed with puzzles and paradoxes. One of them — not my favorite — is called This Book Needs No Title.

Peter May and I are almost done editing a book that’s quite the opposite. It does need a title.

Background Essays Towards Higher Category Theory would be an accurate description, but it’s not very snappy. Towards Higher Category Theory is overly ambitious. Can you think of something better?

To help you dream up an appropriate title, here’s a draft of the preface. And if you spot mistakes in this preface, I’d like to hear about them. (The bibliography makes no pretensions to completeness, so surely many people will be offended by how we have neglected their work. If you’re one of those people, I apologize.)

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.

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### Re: This Book Needs a Title

Prolegomena to any Future Higher Category Theory.

Posted by: Graham on June 26, 2009 8:54 PM | Permalink | Reply to this

### Re: This Book Needs a Title

“Approaching Higher Category Theory”

That works in a couple of ways and it is short and simple.

Posted by: Gregor N. Purdy Sr. on June 26, 2009 9:22 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Approaching…That gets my vote

Barwick gets no citation??

Posted by: jim stasheff on June 27, 2009 3:56 AM | Permalink | Reply to this

### Re: This Book Needs a Title

I like Approaching Higher Category Theory.

Barwick has not yet made his thesis available, but maybe I can at least find out the title.

Posted by: John Baez on June 27, 2009 7:38 AM | Permalink | Reply to this

### Re: This Book Needs a Title

“Discourses on Higher Category Theory” Wiki:

“In the work of Michel Foucault, and social theorists inspired by him, discourse has a special meaning. It is “an entity of sequences of signs in that they are enouncements (enoncés)” (Foucault 1969: 141). An enouncement (often translated as “statement”) is not a unity of signs, but an abstract matter that enables signs to assign specific repeatable relations to objects, subjects and other enouncements (Ibid: 140). *Thus, a discourse constitutes sequences of such relations to objects, subjects and other enouncements (statements)*. A discursive formation is defined as the regularities that produces such discourses. Foucault used the concept discursive formation in relation to his analysis of large bodies of knowledge, such as political economy and natural history.(Foucault: 1970)”

I think for literate readers, there is an element of self-reference which I think was the point of the Smullyan book title and for the eclectic set there is a pointer to Descartes: “Discourse on the Method of Rightly Conducting the Reason, and Seeking [Searching For] Truth in the Sciences” title. “Formative Discourses of Higer Category Theory” ?! Webster: Discursive
Medieval Latin discursivus, from Latin discursus, past participle of discurrere to run about — more at discourse Date: 1598
1 b: proceeding coherently from topic to topic 2: marked by analytical reasoning
3: of or relating to discourse

Posted by: Stephen Harris on June 27, 2009 5:38 PM | Permalink | Reply to this

### Re: This Book Needs a Title

“Formative Discourses of Higer Category Theory”

Maybe it would be better to use the preposition “on” rather than “of” above.

Posted by: Stephen Harris on June 27, 2009 5:42 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Barwick gets no citation??

John Baez replies:

Barwick has not yet made his thesis available,

What one could (maybe should) cite instead is

JL, $(\infty,2)$-categories and the Goodwillie calculus (pdf)

In definition 1.3.6 this defines $(\infty,1)$-categories $Cat_{(\infty,n)}$ of $(\infty,n)$-categories iteratively as complete Segal space objects inside $Cat_{(\infty,(n-1))}$.

On p. 6 (second line of section 1) this is attributed as “due to Barwick”.

Pre-Luriean Higher Category Theory?

;-)

Posted by: Urs Schreiber on June 29, 2009 10:12 AM | Permalink | Reply to this

### Re: This Book Needs a Title

Prolegomenon to an Introduction to Higher Category Theory?

Early Developments in Higher Category Theory?

Introduction to Bicategories and Higher Categories?

OK, the first was a bit of a joke. The third sounds closer to what you’re aiming at, i.e. more detail on 2-categories and general surveys of n-categories. My only concern is that, in my experience, “Introduction To …” is most usually the title of a textbook, rather than a collection of papers. But for a collection of expository and review papers, it might do.

Posted by: Tim Silverman on June 26, 2009 10:39 PM | Permalink | Reply to this

### Re: This Book Needs a Title

My only concern is that, in my experience, “Introduction To …” is most usually the title of a textbook, rather than a collection of papers.

Posted by: Toby Bartels on June 27, 2009 1:22 AM | Permalink | Reply to this

### Re: This Book Needs a Title

A slight modification of your second suggestion:

Early Essays in Higher Category Theory

It’s also a slightly more descriptive version of Towards Higher Category Theory.

Posted by: Eric on June 27, 2009 4:47 PM | Permalink | Reply to this

### Re: This Book Needs a Title

“the low road to higher category theory”?

Posted by: anona on June 26, 2009 11:46 PM | Permalink | Reply to this

### Re: This Book Needs a Title

A launchpad for higher category theory.

Posted by: Anon on June 26, 2009 11:58 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Bring me a higher category theory.

Posted by: Allen K. on June 27, 2009 1:15 AM | Permalink | Reply to this

### Errors

$(\infty,n)$-categories, which are roughly weak $\infty$-groupoids where all morphisms above dimension $n$ are invertible

You mean $\infty$-categories where all morphisms above dimension $n$ are invertible; an $\infty$-groupoid is an $(\infty,0)$-category.

Posted by: Toby Bartels on June 27, 2009 1:26 AM | Permalink | Reply to this

### Re: Errors

Simona Paoli’s paper turns to the role of “internal” structures higher category theory

This is missing a preposition, probably ‘in’.

Posted by: Toby Bartels on June 27, 2009 1:54 AM | Permalink | Reply to this

### Re: Errors

Wow, real corrections instead of joke titles! Thanks, Toby.

Posted by: John Baez on June 27, 2009 7:39 AM | Permalink | Reply to this

### “A Higher Category Theory Reader”

A standard approach in other fields is to call a collection of foundational papers a ‘reader’.

A Reader for Higher Category Theory

Posted by: RodMcGuire on June 27, 2009 2:29 AM | Permalink | Reply to this

### Re: “A Higher Category Theory Reader”

That’s got possibility. Thanks!

Posted by: John Baez on June 27, 2009 7:40 AM | Permalink | Reply to this

### Re: “A Higher Category Theory Reader”

I should have also mentioned that a title that contains “a reader” or “readings in” generally implies a collection of papers that many people in the field are familiar with, and for the convince of students they have all been collected into one book so that students don’t have to hunt them up and (in the old days) make Xeroxes of them.

Posted by: RodMcGuire on June 28, 2009 1:51 AM | Permalink | Reply to this

### Re: “A Higher Category Theory Reader”

That sounds exactly right!

Posted by: Tim Silverman on June 27, 2009 2:40 PM | Permalink | Reply to this

### Re: “A Higher Category Theory Reader”

The main problem with A Higher Category Reader is that people might read this book hoping to learn what a higher category is — that is, an $n$-category. But a main point of the preface above is that the essays in this book don’t explain $n$-categories: we’re not ready for that yet. “Instead of an introduction to higher category theory, we have decided to publish a series of papers that provide useful background for this subject.”

That’s why Peter May suggested the title Background Essays Towards Higher Category Theory — with “towards” meaning “heading in that direction, but not yet getting there”.

Approaching Higher Categories also conveys this idea.

Anyway, I sent Peter a list of possibilities, including A Higher Categories Reader, and we’ll see what he says.

Posted by: John Baez on June 27, 2009 5:26 PM | Permalink | Reply to this

### Re: “A Higher Category Theory Reader”

John said:

… the essays in this book don’t explain n-categories

Not that I’m wedded to the word Reader, but my feeling on this is

a) That’s why should you should look at the preface before buying the book!

b) It’s not like they’re going to get a better definition of n-categories elsewhere—if they’re looking for a definition, this is about as close as they’re going to come.

I agree it’s tricky. But there is kind of a contrary pull between, the one hand, the kind of Preliminary Steps Towards … /Prolegomenon To … aspect (which kind of implies being at the start), and the aspect that this book will get you up to date with the best attempts so far (which kind of implies being right up at the front). Difficult to get both into half a dozen words…

Posted by: Tim Silverman on June 27, 2009 6:46 PM | Permalink | Reply to this

### Re: “A Higher Category Theory Reader”

Tim wrote:

It’s not like they’re going to get a better definition of $n$-categories elsewhere—if they’re looking for a definition, this is about as close as they’re going to come.

Not really! It’s not as if definitions of $n$-category don’t exist, or haven’t been explained well. The most popular definitions are explained quite nicely here:

But these definitions are a lot of work to learn, and they haven’t yet been connected by a web of theorems. So, Peter and I are deliberately holding back and explaining the preliminaries, rather than plunging into the thick of battle as these younger, braver authors do.

Posted by: John Baez on June 27, 2009 7:34 PM | Permalink | Reply to this

### Space Elevator; Re: This Book Needs a Title

From Ground Floor to Infinitely Higher Categories

Posted by: Jonathan Vos Post on June 27, 2009 6:03 AM | Permalink | Reply to this

### Re: This Book Needs a Title

The first rung: how to get started in higher category theory

(obvious, but here it is) Some things you wanted to know about n-categories, but were afraid to ask

So you want to define an n-category? (thanks, Glenn Gould)

Posted by: David Roberts on June 27, 2009 10:16 AM | Permalink | Reply to this

### Re: This Book Needs a Title

The first rung: how to get started in higher category theory

I like this one! (Possibly change ‘how to get’ to ‘getting’.)

Posted by: Toby Bartels on June 28, 2009 1:11 AM | Permalink | Reply to this

### Re: This Book Needs a Title

One obvious possibility is:

‘Higher Categories for Dummies’.

Posted by: Phil Harmsworth on June 27, 2009 12:02 PM | Permalink | Reply to this

### Re: This Book Needs a Title

My wife suggested that one too, right around when I decided that conversation was going nowhere.

Posted by: John Baez on June 27, 2009 5:32 PM | Permalink | Reply to this

### Re: This Book Needs a Title

OK, I can see why you would be dismissive, but presumably you do want the book to stand out from the crowd. Lightly humorous titles (such as your ‘joke’ title of ‘n-Categories for the Mathematician of Leisure’) are not necessarily a bad idea. This title is a nice play on ‘Categories for the Working Mathematician’, and conveys a feeling of accessibility. Judging by the reaction, you’d have one customer who’s not even in your target audience.

Posted by: Phil Harmsworth on June 28, 2009 6:03 AM | Permalink | Reply to this

### Re: This Book Needs a Title

Personally I like joking around, but I know Peter May will want a title with a bit more gravitas — and the authors of the papers would probably agree with him. Remember, academic publishing is not about sales — it’s about prestige. I’m not sure a paper in a book For Dummies will help my junior colleagues get tenure. So, I’ll save the joke titles for books where I’m the only one to bear the consequences.

Now Peter has suggested On The Way Towards Higher Category Theory. I would like to remove a few syllables, for example by deleting ‘on the way’, or changing ‘higher category theory’ to ‘higher categories’, or ‘towards’ to ‘to’. People don’t actually say long book titles.

Posted by: John Baez on June 28, 2009 1:05 PM | Permalink | Reply to this

### Re: This Book Needs a Title

I still prefer Approaching..
it’s more readily distinguished from more classical’ titles

Posted by: jim stasheff on June 28, 2009 1:49 PM | Permalink | Reply to this

### Re: This Book Needs a Title

People don’t actually say long book titles.

No, they’ll just call the book ‘Baez and May’ or ‘the Baez-May volume’, probably regardless of which title you pick.

Posted by: Mark Meckes on June 28, 2009 3:19 PM | Permalink | Reply to this

### Re: This Book Needs a Title

If we called it Higher Categories for Lower Life Forms, I know that’s what people would actually say.

Posted by: John Baez on June 28, 2009 11:06 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Now Peter has suggested On The Way Towards Higher Category Theory. […] People don’t actually say long book titles.

No, but if you can get people to call it ‘On the Way’ instead of ‘Baez and May’, then that's a bonus. (But I agree that you should change ‘Towards’ to ‘to’).

And it rhymes! On the Way, with Baez and May. Now if only the Preface had something about the Tao of Mathematics ….

Posted by: Toby Bartels on June 28, 2009 11:22 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Your situation seems similar to that which inspired David Eisenbud to write “Commutative Algebra with a view toward Algebraic Geometry”. All the actual theorems in that book are pure commutative algebra, but the exposition is meant to prepare and motivate the reader to study algebraic geometry.

So I’d like a title like “X with a view toward Higher Category Theory”. But I’m not sure what X is!

Posted by: David Speyer on June 27, 2009 4:35 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Higher Categories for the Higher Mathematician

Sorry. Couldn’t resist :)

Posted by: Eric Forgy on June 27, 2009 4:53 PM | Permalink | Reply to this

### Re: This Book Needs a Title

That reminds me of a book I’ve imagined writing just as a joke: something like n-Categories for the Mathematician of Leisure.

Posted by: John Baez on June 27, 2009 5:31 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Oh well. I’ll put in my 2bits:

Preliminary essays on higher categories

Posted by: Minhyong Kim on June 27, 2009 7:29 PM | Permalink | Reply to this

### Mathematicians of Leisure, Unite!

As a researcher in the biosciences, I would buy a book with this title for real. Just saying.

Posted by: Hunter Washburne on June 27, 2009 7:30 PM | Permalink | Reply to this

### Re: This Book Needs a Title

A Prelude to n-Categories

Posted by: Aaron Bergman on June 27, 2009 6:55 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Elegant — and since Peter May’s mother was just as operatic as he is operadic, it strikes a musical note he may appreciate.

Posted by: John Baez on June 27, 2009 7:40 PM | Permalink | Reply to this

### A Prelude to n-Categories

Of all the suggestions I’ve seen so far, this one has my vote.

Or maybe a slight variation

A Prelude to Higher Category Theory

Next would be

Approaching Higher Category Theory

I also like

Essays in Higher Category Theory

Posted by: Eric Forgy on June 28, 2009 5:25 PM | Permalink | Reply to this

### Re: This Book Needs a Title

By the way, I rather liked that book of Smullyan’s when I first read it, especially the story at the end with some title like ‘planet without laughter.’ Alas, I don’t like it anymore either.

Posted by: Minhyong Kim on June 27, 2009 7:34 PM | Permalink | Reply to this

### Re: This Book Needs a Title

I don’t remember it very well, but it seemed a bit heavy-handed compared to his books of puzzles.

By the way: in case someone out there hasn’t read Smullyan’s puzzle books, here’s one to try. I’ve adapted the wording a bit, to suit the current economic climate.

Unable to find an academic job, a hungry logician enters the cafeteria of a prestigious university and starts talking to one of the undergraduate students working there. “I’d like to make a deal. I see you’re serving bean soup today. I’ll say a sentence. If it’s true, you have to promise to give me a bean. If it’s not true, you have to promise to not give me a bean. Okay?”

The student sees no harm in this. “Okay, sure.”

Then the logician says a sentence… and the student, upon reflection, realizes the only way he can fulfill his promise is to give the logician a free lunch every day for the whole year!

What could the logician have said?

Posted by: John Baez on June 27, 2009 7:52 PM | Permalink | Reply to this

### Re: This Book Needs a Title

A hyperlinked version of the above preface and reference list is now at [[Approaching Higher Category Theory]]

Posted by: Urs Schreiber on June 27, 2009 9:17 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Hi! I saw a version where only a few of the references were included. I put in a complete set of references — not in the officially approved nLab format.

I’d like a copy of this on my personal web… but the hyperlinks to nLab webpages don’t work there. Does someone know how to make a link from one web to another within the nLab?

Posted by: John Baez on June 28, 2009 11:10 AM | Permalink | Reply to this

### Re: This Book Needs a Title

To link to a page on the main nLab grid from your personal web, you can either add “nlab:” to the link, e.g.

[[nlab:category]]

or

[category](http://ncatlab.org/nlab/show/category)

Posted by: Eric Forgy on June 28, 2009 7:14 PM | Permalink | Reply to this

### Re: This Book Needs a Title

… and just to state the obvious, a search and replace

[[ -> [[nlab:

is the easiest in this case.

Posted by: Eric Forgy on June 28, 2009 7:16 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Thanks, Eric!

Posted by: John Baez on June 28, 2009 9:27 PM | Permalink | Reply to this

### Re: This Book Needs a Title

I put in a complete set of references

Thanks. I was hoping somebody would. After “B” I was running out of time…

not in the officially approved nLab format.

No such thing as yet (and not high in priority, I prefer content over form).

Personally, though, I am sticking to

bullet author, title hard copy references (arXiv, web, pdf , blog)

Posted by: Urs Schreiber on June 28, 2009 8:13 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Since it never hurts to add pages that do not “take off”, I will also add the references to the experimental

Bibliography

Posted by: Eric Forgy on June 28, 2009 8:55 PM | Permalink | Reply to this

### Re: This Book Needs a Title

I have taken the liberty of moving most of the stuff about Approaching Higher Categories to my personal web on the $n$Lab. I feel a bit better having ‘my’ stuff there.

Posted by: John Baez on June 28, 2009 9:19 PM | Permalink | Reply to this

### Re: This Book Needs a Title

I have taken the liberty of moving most of the stuff about Approaching Higher Categories to my personal web on the nLab. I feel a bit better having ‘my’ stuff there.

I understand the rational of removing hyperlinks inside paper titles, which maybe was a bad idea of mine.

But since I also still feel it is a pity to send the reader away from the $n$Lab to try read a paper on, say, simplicial weak $\omega$-catgeories while we keep an entry on that (just as an example) I thought it would be worthwhile to create more “reference” $n$Lab entries for such cases.

I started with [[Lectures on $n$-Categories and Cohomology]] for which a lot of $n$Lab material already exists (as linked to there).

I took the liberty of inserting a link to that on your page.

Posted by: Urs Schreiber on June 29, 2009 9:54 AM | Permalink | Reply to this

### Re: This Book Needs a Title

The Evolution of Higher Category Theory

Posted by: Charlie C on June 28, 2009 2:35 PM | Permalink | Reply to this

### Evolution of “Evolution”; Re: This Book Needs a Title

I like this, but it raises a question. I’m not joking. I’m trying to connect the entitling criteria to Philosophy of Mathematics.

Is Higher Category Theory the result of an adaptive process, winning over other theories because it has Higher Darwinian Fitness? Or is it being (collectively) Created? Specifically, Created by Intelligent Design?

More generally, is Mathematics Invented, or is Mathematics Discovered?

Experts differ.

Posted by: Jonathan Vos Post on June 29, 2009 1:29 AM | Permalink | Reply to this

### Re: This Book Needs a Title

Pursuing Higher Categories?

Posted by: Tom Hirschowitz on June 29, 2009 8:52 AM | Permalink | Reply to this

### Re: This Book Needs a Title

From the candidates put forth, Peter May has chosen Towards Higher Categories.

Posted by: John Baez on June 29, 2009 12:04 PM | Permalink | Reply to this

### Re: This Book Needs a Title

Of course ‘towards’ is a fine word with which to begin a title.

Something I’ve never quite grasped is when you Americans say ‘toward’ as opposed to ‘towards’.

Posted by: David Corfield on June 29, 2009 2:19 PM | Permalink | Reply to this

### Re: This Book Needs a Title

There seem to be certain language police (at least in this country), or perhaps those in editorial positions with entrenched and often demonstrably wrong-headed opinions, who generally decry ‘towards’. I have absolutely no idea why; it seems kinda weird. I used to write this word all the time, and now I’ve stopped because I find it a nuisance dealing with language cops (preferring to pick the battles I’m more interested in).

The blog “grammarcops” (yeesh) quotes Fowler, who seems far from helpful in this matter, here.

Posted by: Todd Trimble on June 29, 2009 4:11 PM | Permalink | Reply to this

### Re: This Book Needs a Title

I also found this, here:

Either toward or towards is correct, although the toward is preferred in formal, academic prose. I would use toward unless your ear tells you otherwise; consistency is important, however. The British, I believe, lean untowardly toward towards.

Authority: The Little, Brown Handbook by H. Ramsay Fowler, Jane E. Aaron, and Kay Limburg. 6th ed. HarperCollins: New York. 1995. By permission of Addison-Wesley Educational Publishers Inc.

Posted by: Todd Trimble on June 29, 2009 4:19 PM | Permalink | Reply to this

### Re: This Book Needs a Title

I was feeling a tinge of confusion regarding ‘toward’ versus ‘towards’ in this title. Somehow ‘towards’ sounds better… maybe the sibilant ‘s’ conveys more of a sense of motion, and anyway it creates a nice consonance with the ‘s’ in ‘categories’.

I hadn’t realize it was a British-vs-American thing.

Todd wrote:

There seem to be certain language police (at least in this country), or perhaps those in editorial positions with entrenched and often demonstrably wrong-headed opinions, who generally decry ‘towards’.

It’ll be a pleasure to piss off some ‘grammar cops’ with a phrase that’s perfectly common in British titles.

Posted by: John Baez on June 29, 2009 10:04 PM | Permalink | Reply to this

### Re: This Book Needs a Title

You might find this observation a bit odd but I think there is another analogical bias operating here, which is whether one scored higher on their verbal aptitude or math aptitude. SATs tend to correlate to measuring this.

The finger points to a category.
The fingers point to a category.
[The fingers point to categories.]

That is about subject and verb “s” agreement. Although toward(s) is a preposition it has a verb/motion nuance that you mentioned. Which do you like better?
Toward a theory of higher consciousness or
Towards a theory of higher consciousness

Toward theories of higher consciousness or
Towards theories of higher consciousness

Posted by: Stephen Harris on June 30, 2009 9:49 AM | Permalink | Reply to this

### Re: This Book Needs a Title

If it is ‘perfectly common’ we (= the upper crust) should avoid using it!!!! (Sorry I couldn’t resist that. Unfortunately I cannot find a typeface that indicates a ‘posh’ accent.)

Posted by: Tim Porter on June 30, 2009 3:55 PM | Permalink | Reply to this

### Re: This Book Needs a Title

JB: “It’ll be a pleasure to piss off some ‘grammar cops’ with a phrase that’s perfectly common in British titles.”

TP: “If it is ‘perfectly common’ we (= the upper crust) should avoid using it!!!!”

SH: Well, since there were 79 titles listed with “towards a” and 96 titles listed with
“toward a” then that makes “toward a” more common.

Posted by: Stephen Harris on June 30, 2009 7:08 PM | Permalink | Reply to this

i haven’t been able to keep up with the deluge of replies on this topic,
so others may have caught some typos’.

p.2 …cannot easily be proven in another [May] - namely?

and some other relevant citations?

Posted by: jim stasheff on June 29, 2009 3:23 PM | Permalink | Reply to this

Thanks, Jim! Peter May added the relevant reference here:

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].

[May] J. P. May, What precisely are $E_\infty$ ring spaces and $E_\infty$ ring spectra?, Geometry and Topology Monographs 16 (2009), 215–284.

I also added a reference to Barwick’s forthcoming work, taken from his CV:

C. Barwick, Weakly Enriched M-Categories, work in progress.

Posted by: John Baez on June 29, 2009 10:12 PM | Permalink | Reply to this

Hi, John.

Can you replace two last references to my work by up to date references:

1. Batanin M., The symmetrisation of $n$-operads and compactification of real configuration spaces, Adv. Math. (211),(2007), pp. 684-725.

2. Batanin M., The Eckmann-Hilton argument and higher operads, Adv. Math. (217)(2008), pp. 334-385.

Michael.

Posted by: Michael Batanin on July 2, 2009 4:53 AM | Permalink | Reply to this

Sure, I’ll update those references. Thanks!

It would be great if you could update your papers on the arXiv and your homepage so they had the same titles and content as your published papers. It’s also easy to add information to the arXiv saying where papers have been published.

The information you just sent me is not available at either the arXiv or your homepage. Instead, both places list papers with different titles:

The Eckmann-Hilton argument, higher operads and $E_n$-spaces

and

The combinatorics of iterated loop spaces.

Posted by: John Baez on July 2, 2009 9:09 AM | Permalink | Reply to this

Thank you, John.

I’ll do the corrections in the arxive asap. There are corrections I have to make.

Michael.

Posted by: Michael Batanin on July 3, 2009 5:45 AM | Permalink | Reply to this

### Re: This Book Needs a Title

“Steal this Category Theory Book”

“Dan Brown’s Da Vinci Code II: An Introduction to Category Theory”

Posted by: Roy on June 29, 2009 8:33 PM | Permalink | Reply to this

### Re: This Book Needs a Title

A primer to higher category theory.

Higher category theory primer.

Posted by: Anon on June 29, 2009 9:20 PM | Permalink | Reply to this

### Primer (2004); Re: This Book Needs a Title

Only if the cover has a still photo from the film:

Primer (2004)
Roger Ebert / October 29, 2004

Shane Carruth’s “Primer” opens with four techheads addressing envelopes to possible investors; they seek venture capital for a machine they’re building in the garage. They’re not entirely sure what the machine does, although it certainly does something. Their dialogue is halfway between shop talk and one of those articles in Wired magazine that you never finish. We don’t understand most of what they’re saying, and neither, perhaps, do they, but we get the drift.
Challenging us to listen closely, to half-understand what they half-understand, is one of the ways the film sucks us in….

Posted by: Jonathan Vos Post on June 30, 2009 2:27 AM | Permalink | Reply to this

“Pursuing Higher Categories”

Posted by: some guy on the street on June 30, 2009 4:25 AM | Permalink | Reply to this

dang… someone beat me to it… need more patience.

Posted by: some guy on the street on June 30, 2009 4:34 AM | Permalink | Reply to this

### Re: This Book Needs a Title

Gavin Wraith sent me an email containing this remark, and allowed me to post it here:

John

If the book were called “On Higher Things” the authors could always use the nom de plume “Dionysius Longinus”. If this classical reference is too obscure, Google is always to hand. The original was in Greek (“peri hypsous”) and was on the higher things in literature. It came to be called in Latin “De Sublimitate”. I think the copyright on the title has run out ;) - nobody knows who the author was, anyway. He could conceivably have been the private secretary of Queen Zenobia of Palmyra, executed by the emperor Aurelian after she made her bid for independence.

To clarify: it was the secretary, not Zenobia herself, who was executed. Wraith explains:

No, she was much too useful. But Longinus was executed, or so it says in Scriptores Historiae Augustae Vol III, section XXX:

“grave inter eos qui caesi sunt de Longino philosopho fuisse perhibetur, quo illa magistro usa esse ad Graecas litteras dicitur, quem quidem Aurelianus idcirco dicitur occidisse, quod superbior illa epistula ipsius diceretur dictata consilio, quamvis Syro esset sermone contexta.”

= “It was considered bad that the philosopher Longinus was among those executed. He was said to have been employed by Zenobia as her teacher in Greek. Aurelian is said to have killed him as responsible for her proud letter [of independence], though in fact it was written in Aramaic.”

Posted by: John Baez on June 30, 2009 10:52 AM | Permalink | Reply to this

### Re: This Book Needs a Title

With no claim to originality, I suggest “An Invitation to n-categories”

Posted by: Carl Futia on June 30, 2009 5:29 PM | Permalink | Reply to this

### Tomorrow is another day; Re: This Book Needs a Title

Dawn of Higher Categories?

Posted by: Jonathan Vos Post on July 7, 2009 1:19 AM | Permalink | Reply to this

### Re: This Book Needs a Title

“Getting High on Category Theory”

Posted by: glenn branca on July 16, 2009 2:44 AM | Permalink | Reply to this

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