
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 ncategory." An ncategory is a wonderfully recursive sort of thing in which there are objects, 1morphisms between objects, 2morphisms between morphisms, and so on up to nmorphisms. In a "cobordism ncategory" the objects are 0manifolds, the 1morphisms are 1dimensional manifolds that go between 0manifolds (as the unit interval goes from one endpoint to another), the 2morphisms are 2dimensional manifolds that go between 1manifolds (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 cuttingandpasting constructions in ndimensional topology can be described algebraically in the cobordism ncategory. To wax rhapsodic for a moment, we can think of an ncategory as exemplifying the notion of "ways to go between ways to go between ways to go between..... ways to go between things," and cobordism ncategories are the particular ncategories that algebraically encode the possibilities along these lines that are implicit in the notion of ndimensional spacetime.
Now, the problem is that the correct definition of an ncategory is a highly nontrivial affair! And it gets more complicated as n increases! A 0category 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 1category 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
Now, a 2category is more complicated. There are objects, 1morphisms, and 2morphisms, and one can compose morphisms and also compose 2morphisms. There is, however, a choice: one can make ones 2category "strict" and require that the rules 2) and 4) above hold for the 1morphisms and 2morphisms, or one can require them "literally" only for the 2morphisms, and allow the 1morphisms some slack. Technically, one can choose between "strict" 2categories, usually just called 2categories, or "weak" ones, which are usually called "bicategories."
What do I mean by giving the 1morphisms some "slack"? This is a very important aspect of the ncategorical philosophy... I mean that in a 2category one has the option of replacing equations between 1morphisms by isomorphisms  that is, by 2morphisms 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" 2category one doesn't have associativity of 1morphisms. Instead, one has "associators", which are 2morphisms 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 1morphisms 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" 2category 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 2categories, which as I said people usually call bicategories, try:
2) J. Benabou, Introduction to bicategories, Lect. Notes in Math., vol. 47, Berlin, SpringerVerlag, 1968, pp. 171.
Now, one can continue this game, but it gets increasingly complex if one goes the "weak" route. In a "weak ncategory" the idea is to replace all basic identities that one might expect between jmorphisms, 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 3categories" 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 4categories, which happen to be called "braided monoidal 2categories" (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 ncategory. In particular, ncategories have a lot to do with ndimensional space. A weak 3category with only one object and one 1morphism 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 2categories" of Kapranov and Voevodsky are really just weak 4categories with only one object and one 1morphism. (The reason for the term "2category" here is that since all one has is 2morphisms, 3morphisms, and 4morphisms, one can pretend one is in a 2category in which those are the objects, morphisms, and 2morphisms.)
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 ncategorical 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 ntruncated homotopy types (of semisimplicial sets, or of topological spaces) was essentially equivalent to the study of socalled ngroupoids (where n is any natural integer). This is expected to be achieved by associating to any space (say) X its "fundamental ngroupoid" Π_{n}(X), generalizing the familiar Poincare fundamental groupoid for n = 1. The obvious idea is that 0objects of Π_{n}(X) should be the points of X, 1objects should be "homotopies" or paths between points, 2objects should be homotopies between 1objects, etc. This Π_{n}(X) should embody the ntruncated homotopy type of X, in much the same way as for n = 1 the usual fundamental groupoid embodies the 1truncated homotopy type. For two spaces X, Y, the set of homotopyclasses of maps X → Y (more correctly, for general X, Y, the maps of X into Y in the homotopy category) should correspond to nequivalence classes of nfunctors 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 ngroupoid, which should imply that any ngroupoid is nequivalent to a Π_{n}(X). Moreover when the notion of an ngroupoid (or more generally of an ncategory) is relativized over an arbitrary topos to the notion of an ngerbe (or more generally, an nstack), 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 "ωgroupoids". 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, 3845, 1992.
F.AlAgl, Aspects of multiple categories, University of Wales PhD Thesis, 1989.
F.AlAgl and R.J.Steiner, Nerves of multiple categories, Proc. London Math. Soc., 66, 92128, 1992.
N.Ashley, Simplicial Tcomplexes, University of Wales PhD Thesis, 1976, published as Simplicial Tcomplexes: a nonabelian version of a theorem of DoldKan, Diss. Math. 165, 1158 (1988).
H.J.Baues, Algebraic homotopy, Cambridge University Press, 1989.
H.J.Baues, Combinatorial homotopy and 4dimensional complexes, De Gruyter, 1991.
L.Breen, Bitorseurs et cohomologie nonAbélienne, 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, 401476, 1990.
R.Brown, Higher dimensional group theory, in Lowdimensional topology, ed. R.Brown and T.L.Thickstun, London Math. Soc. Lect. Notes 46, Cambridge University Press, 215238, 1982.
R.Brown, From groups to groupoids: a brief survey, Bull. London Math. Soc., 19, 113134, 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 nonAbelian homological and homotopical algebra, Homotopy theory and related topics: Proceedings Kinosaki, 1988, Edited M.Mimura, Springer Lecture Notes in Math. 1418, 105129, 1990.
R.Brown, P.J.Higgins, The equivalence of ωgroupoids and cubical Tcomplexes, Cah. Top. G\eom. Diff. 22, 349370, 1981.
R.Brown, P.J.Higgins, The equivalence of ∞groupoids and crossed complexes, Cah. Top. G\eom. Diff. 22, 371386, 1981.
R.Brown, P.J.Higgins, The algebra of cubes, J. Pure Appl. Algebra, 21, 233260, 1981.
R.Brown, P.J.Higgins, Tensor products and homotopies for ωgroupoids and crossed complexes, J. Pure Appl. Algebra, 47, 133, 1987.
R.Brown, J.Huebschmann, Identities among relations, in Lowdimensional topology, ed. R.Brown and T.L.Thickstun, London Math. Soc. Lect. Notes 46, Cambridge University Press, 153202, 1982.
R.A.Brown, Generalised group presentations, Trans. Amer. Math. Soc., 334, 519549, 1992.
M.Bullejos, A.M.Cegarra, J.Duskin, On cat^{n}groups and homotopy types, J. Pure Appl. Algebra 86 (1993) 135154.
M.Bullejos, P. Carrasco, A.Cegarra, Cohomology with coefficients in symmetric cat^{n}groups. An extension of EilenbergMac 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 Tcomplexes, University of Wales PhD Thesis, 1984; published as A general theory of polyhedral sets and their corresponding Tcomplexes, Diss. Math. 266, 1988.
M.M.Kapranov, V.Voevodsky, Combinatorialgeometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results), Cah. Top. Geom. Diff. Cat. 32, 1127, 1991.
M.M.Kapranov, V. Voevodsky, ∞groupoids and homotopy types Cah. Top. G\eom. Diff. Cat. 32, 2946, 1991.
M.M.Kapranov, V. Voevodsky, 2categories and Zamolodchikov tetrahedra equations, preprint, 102pp, 1992.
J.L.Loday, Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra, 24, 179202, 1982.
G.Nan Tie, Iterated W and Tgroupoids, J. Pure Appl. Algebra, 56, 195209, 1989.
T.Porter, A combinatorial definition of ∞types, Topology 22 (1993) 524.
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) 687716.
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, 8387, 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) 99107.
© 1994 John Baez
baez@math.removethis.ucr.andthis.edu
