June 5, 1994

This Week's Finds in Mathematical Physics (Week 35)

John Baez

1) Pursuing stacks (A la poursuite des champs), 1983 letter from Alexandre Grothendieck to Daniel Quillen, 593 pages. Scanned version available from the Grothendieck Circle at http://www.grothendieckcircle.org/

I owe somebody enormous thanks for sending this to me, but I won't mention his name, since I don't want people pestering him for copies. (This no longer matters, now that it's available online.) Grothendieck is mainly famous for his work on algebraic geometry, in which he introduced the concept of "schemes" to provide a modern framework for the subject. He was also interested in reformulating the foundations of topology, which is reflected in "Pursuing Stacks". This is a long letter to Quillen, inspired by Quillen's 1967 book "Homotopical Algebra". It's a fascinating mixture of visionary mathematics, general philosophy and a bit of personal chat. Let me quote a bit:

I write you under the assumption that you have not entirely lost interest for those foundational questions you were looking at more than fifteen years ago. One thing which strikes me, is that (as far as I know) there has not been any substantial progress since - it looks to me that understanding of the basic structures underlying homotopy theory, or even homological algebra only, is still lacking - probably because the few people who have a wide enough background and perspective enabling them to feel the main questions, are devoting their energies to things which seem more directly rewarding. Maybe even a wind of disrepute for any foundational matters whatever is blowing nowadays! In this respect, what seems to me even more striking than the lack of proper foundations for homological and homotopical algebra, is the absence I daresay of proper foundations for topology itself! I am thinking here mainly of the development of a context of "tame" topology, which (I am convinced) would have on the everyday technique of geometric topology (I use this expression in contrast to the topology of use for analysts) a comparable impact or even a greater one, than the introduction of the point of view of schemes had on algebraic geometry. The psychological drawback here I believe is not anything like messyness, as for homological and homotopical algebra (or for schemes), but merely the inrooted inertia which prevents us so stubbornly from looking innocently, with fresh eyes, upon things, without being dulled and emprisoned by standing habits of thought, going with a familiar context - too familiar a context!

One reason why I'm interested in this letter is that Grothendieck seems to have understood the importance of "higher algebraic structures" before most people. Recently, interest in these has been heating up, largely because of the recent work on "extended topological quantum field theories." The basic idea is that, just as a traditional quantum field theory is (among other things) a representation of the symmetry group of spacetime, a topological quantum field theory is a representation of a more sophisticated algebraic structure, a "cobordism n-category." An n-category is a wonderfully recursive sort of thing in which there are objects, 1-morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms. In a "cobordism n-category" the objects are 0-manifolds, the 1-morphisms are 1-dimensional manifolds that go between 0-manifolds (as the unit interval goes from one endpoint to another), the 2-morphisms are 2-dimensional manifolds that go between 1-manifolds (as a cylinder goes from on circle to another), etc. In practice one must work with manifolds admitting certain types of "corners", and equipped with extra structures that topologists and physicist like, such as orientations, framings, or spin structures. The idea is that all the cutting-and-pasting constructions in n-dimensional topology can be described algebraically in the cobordism n-category. To wax rhapsodic for a moment, we can think of an n-category as exemplifying the notion of "ways to go between ways to go between ways to go between..... ways to go between things," and cobordism n-categories are the particular n-categories that algebraically encode the possibilities along these lines that are implicit in the notion of n-dimensional spacetime.

Now, the problem is that the correct definition of an n-category is a highly nontrivial affair! And it gets more complicated as n increases! A 0-category is nothing but a bunch of objects. In other words, it's basically just a set, if we allow ourselves to ignore certain problems about classes that are too big to qualify as sets. A 1-category is nothing but a category. Recall the definition of a category:

A category consists of a set of objects and a set of morphisms. Every morphism has a source object and a target object. (A good example to think of is the category in which the objects are sets and the morphisms are functions. If f:X → Y, we call X the source and Y the target.) Given objects X and Y, we write Hom(X,Y) for the set of morphisms from X to Y (i.e., having X as source and Y as target).

The axioms for a category are that it consist of a set of objects and for any 2 objects X and Y a set Hom(X,Y) of morphisms from X to Y, and

1. Given a morphism g in Hom(X,Y) and a morphism f in Hom(Y,Z), there is morphism which we call f o g in Hom(X,Z). (This binary operation o is called composition.)

2. Composition is associative: (f o g) o h = f o (g o h).

3. For each object X there is a morphism id|X from X to X, called the identity on X.

4. Given any f in Hom(X,Y), f o id|X = f and id|Y o f = f.

Now, a 2-category is more complicated. There are objects, 1-morphisms, and 2-morphisms, and one can compose morphisms and also compose 2-morphisms. There is, however, a choice: one can make ones 2-category "strict" and require that the rules 2) and 4) above hold for the 1-morphisms and 2-morphisms, or one can require them "literally" only for the 2-morphisms, and allow the 1-morphisms some slack. Technically, one can choose between "strict" 2-categories, usually just called 2-categories, or "weak" ones, which are usually called "bicategories."

What do I mean by giving the 1-morphisms some "slack"? This is a very important aspect of the n-categorical philosophy... I mean that in a 2-category one has the option of replacing equations between 1-morphisms by isomorphisms --- that is, by 2-morphisms that have inverses! The basic idea here is that in many situations when we like to pretend things are equal, they are really just isomorphic, and we should openly admit this when it occurs. So, for example, in a "weak" 2-category one doesn't have associativity of 1-morphisms. Instead, one has "associators", which are 2-morphisms like this:

a_f,g,h: (f o g) o h → f o (g o h)

In other words, the associator is the process of rebracketing made concrete. Now, when one replaces equations between 1-morphisms by isomorphisms, one needs these isomorphisms to satisfy "coherence relations" if we're going to expect to be able to manipulate them more or less as if they were equations. For example, in the case of the associators above, one can use associators to go from

f o (g o (h o k))

to

((f o g) o h) o k

in two different ways: either

f o (g o (h o k)) → (f o g) o (h o k) → ((f o g) o h) o k

or

f o (g o (h o k)) → f o ((g o h) o k) → (f o (g o h)) o k → ((f o g) o h) o k

Actually there are other ways, but in an important sense these are the basic two. In a "weak" 2-category one requires that these two ways are equal... i.e., this is an identity that the associator must satisfy, known as the pentagon identity. This is one of the first examples of a coherence relation. It turns out that if this holds, all ways of rebracketing that get from one expression to another are equal. (Here I'm being rather sloppy, but the precise result is known as Mac Lane's theorem.)

To learn about weak 2-categories, which as I said people usually call bicategories, try:

2) J. Benabou, Introduction to bicategories, Lect. Notes in Math., vol. 47, Berlin, Springer-Verlag, 1968, pp. 1-71.

Now, one can continue this game, but it gets increasingly complex if one goes the "weak" route. In a "weak n-category" the idea is to replace all basic identities that one might expect between j-morphisms, such as the associative law, by (j+1)-isomorphisms. These, in turn, satisfy certain "coherence relations" that are really not equations, but (j+2)-morphisms, and so on... up to level n. This becomes so complicated that only recently have "weak 3-categories" been properly defined, by Gordon, Power and Street, who call them tricategories (see "week29").

A bit earlier, Kapranov and Voevodsky succeeded in defining a certain class of weak 4-categories, which happen to be called "braided monoidal 2-categories" (see "week4"). The interesting thing, you see, which justifies getting involved in this business, is that a lot of topology automatically pops out of the definition of an n-category. In particular, n-categories have a lot to do with n-dimensional space. A weak 3-category with only one object and one 1-morphism is usually known as a "braided monoidal category," and the theory of these turns out to be roughly the same as the study of knots, links and tangles! (See "tangles".) The "braided monoidal 2-categories" of Kapranov and Voevodsky are really just weak 4-categories with only one object and one 1-morphism. (The reason for the term "2-category" here is that since all one has is 2-morphisms, 3-morphisms, and 4-morphisms, one can pretend one is in a 2-category in which those are the objects, morphisms, and 2-morphisms.)

In any event, these marvelous algebraic structures have been cropping up more and more in physics (see especially Crane's stuff listed in "week2" and Freed's paper listed in "week12"), so I got ahold of a copy of Grothendieck's letter and have begun trying to understand it.

Actually, it's worth noting that these n-categorical ideas have been lurking around homotopy theory for quite some time now. As Grothendieck wrote:

At first sight it had seemed to me that the Bangor group had indeed come to work out (quite independently) one basic intuition of the program I had envisioned in those letters to Larry Breen - namely, that the study of n-truncated homotopy types (of semisimplicial sets, or of topological spaces) was essentially equivalent to the study of so-called n-groupoids (where n is any natural integer). This is expected to be achieved by associating to any space (say) X its "fundamental n-groupoid" Π_n(X), generalizing the familiar Poincare fundamental groupoid for n = 1. The obvious idea is that 0-objects of Π_n(X) should be the points of X, 1-objects should be "homotopies" or paths between points, 2-objects should be homotopies between 1-objects, etc. This Π_n(X) should embody the n-truncated homotopy type of X, in much the same way as for n = 1 the usual fundamental groupoid embodies the 1-truncated homotopy type. For two spaces X, Y, the set of homotopy-classes of maps X → Y (more correctly, for general X, Y, the maps of X into Y in the homotopy category) should correspond to n-equivalence classes of n-functors from Π_n(X) to Π_n(Y) - etc. There are some very strong suggestions for a nice formalism including a notion of geometric realization of an n-groupoid, which should imply that any n-groupoid is n-equivalent to a Π_n(X). Moreover when the notion of an n-groupoid (or more generally of an n-category) is relativized over an arbitrary topos to the notion of an n-gerbe (or more generally, an n-stack), these become the natural "coefficients" for a formalism of non commutative cohomological algebra, in the spirit of Giraud's thesis.

The "Bangor group" referred to includes Ronnie Brown, who has done a lot of work on "ω-groupoids". A while back he sent me a nice long list of references on this subject; here are some that seemed particularly relevant to me (though I haven't looked at all of them).

3) G. Abramson, J.-P.Meyer, J.Smith, A higher dimensional analogue of the fundamental groupoid, in Recent developments of algebraic topology, RIMS Kokyuroku 781, Kyoto, 38-45, 1992.

F.Al-Agl, Aspects of multiple categories, University of Wales PhD Thesis, 1989.

F.Al-Agl and R.J.Steiner, Nerves of multiple categories, Proc. London Math. Soc., 66, 92-128, 1992.

N.Ashley, Simplicial T-complexes, University of Wales PhD Thesis, 1976, published as Simplicial T-complexes: a non-abelian version of a theorem of Dold-Kan, Diss. Math. 165, 11-58 (1988).

H.J.Baues, Algebraic homotopy, Cambridge University Press, 1989.

H.J.Baues, Combinatorial homotopy and 4-dimensional complexes, De Gruyter, 1991.

L.Breen, Bitorseurs et cohomologie non-Abélienne, The Grothendieck Festschrift: a collection of articles written in honour of the 60th birthday of Alexander Grothendieck, Vol. I, edited P.Cartier, et al., Birkhauser, Boston, Basel, Berlin, 401-476, 1990.

R.Brown, Higher dimensional group theory, in Low-dimensional topology, ed. R.Brown and T.L.Thickstun, London Math. Soc. Lect. Notes 46, Cambridge University Press, 215-238, 1982.

R.Brown, From groups to groupoids: a brief survey, Bull. London Math. Soc., 19, 113-134, 1987.

R.Brown, Elements of Modern Topology, McGraw Hill, Maidenhead, 1968; Topology: a geometric account of general topology, homotopy types and the fundamental groupoid, Ellis Horwood, Chichester, 1988.

R.Brown, Some problems in non-Abelian homological and homotopical algebra, Homotopy theory and related topics: Proceedings Kinosaki, 1988, Edited M.Mimura, Springer Lecture Notes in Math. 1418, 105-129, 1990.

R.Brown, P.J.Higgins, The equivalence of ω-groupoids and cubical T-complexes, Cah. Top. G\eom. Diff. 22, 349-370, 1981.

R.Brown, P.J.Higgins, The equivalence of ∞-groupoids and crossed complexes, Cah. Top. G\eom. Diff. 22, 371-386, 1981.

R.Brown, P.J.Higgins, The algebra of cubes, J. Pure Appl. Algebra, 21, 233-260, 1981.

R.Brown, P.J.Higgins, Tensor products and homotopies for ω-groupoids and crossed complexes, J. Pure Appl. Algebra, 47, 1-33, 1987.

R.Brown, J.Huebschmann, Identities among relations, in Low-dimensional topology, ed. R.Brown and T.L.Thickstun, London Math. Soc. Lect. Notes 46, Cambridge University Press, 153-202, 1982.

R.A.Brown, Generalised group presentations, Trans. Amer. Math. Soc., 334, 519-549, 1992.

M.Bullejos, A.M.Cegarra, J.Duskin, On catⁿ-groups and homotopy types, J. Pure Appl. Algebra 86 (1993) 135-154.

M.Bullejos, P. Carrasco, A.Cegarra, Cohomology with coefficients in symmetric catⁿ-groups. An extension of Eilenberg-Mac Lanes classification theorem. Granada Preprint, 1992.

P.J.Ehlers and T. Porter, From simplicial groupoids to crossed complexes, UCNW Maths Preprint 92.19, 35pp, 1992.

D.W.Jones, Polyhedral T-complexes, University of Wales PhD Thesis, 1984; published as A general theory of polyhedral sets and their corresponding T-complexes, Diss. Math. 266, 1988.

M.M.Kapranov, V.Voevodsky, Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results), Cah. Top. Geom. Diff. Cat. 32, 11-27, 1991.

M.M.Kapranov, V. Voevodsky, ∞-groupoids and homotopy types Cah. Top. G\eom. Diff. Cat. 32, 29-46, 1991.

M.M.Kapranov, V. Voevodsky, 2-categories and Zamolodchikov tetrahedra equations, preprint, 102pp, 1992.

J.-L.Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra, 24, 179-202, 1982.

G.Nan Tie, Iterated W and T-groupoids, J. Pure Appl. Algebra, 56, 195-209, 1989.

T.Porter, A combinatorial definition of ∞-types, Topology 22 (1993) 5-24.

S.J.Pride, Identities among relations of group presentations, in E.Ghys, A.Haefliger, A. Verjodsky, eds. Proc. Workshop on Group Theory from a Geometrical Viewpoint, International Centre of Theoretical Physics, Trieste, 1990, World Scientific, (1991) 687-716.

R.Steiner, The algebra of directed complexes, University of Glasgow Math Preprint, 29pp, 1992.

A.Tonks, Cubical groups which are Kan, J. Pure Appl. Algebra 81, 83-87, 1992.

A.Tonks and R.Brown, Calculation with simplicial and cubical groups in Axiom, UCNW Math Preprint 93.04.

A.R.Wolf, Inherited asphericity, links and identities among relations, J. Pure Appl. Algebra 71 (1991) 99-107.

© 1994 John Baez
baez@math.removethis.ucr.andthis.edu