Also available at http://math.ucr.edu/home/baez/week35.html
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.math.jussieu.fr/~leila/mathtexts.php
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 (see
"categories", available by ftp as described at the end of this post):
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!
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" Pi_n(X), generalizing the familiar Poincare fundamental
groupoid for n = 1. The obvious idea is that 0-objects of Pi_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 Pi_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 Pi_n(X) to Pi_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 Pi_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 "omega-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-Ab\elienne, 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
$\omega$-groupoids and cubical T-complexes, Cah. Top.
G\eom. Diff. 22, 349-370, 1981.
R.Brown, P.J.Higgins, The equivalence of
$\infty$-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
$\omega$-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 cat$^n$-groups and
homotopy types, J. Pure Appl. Algebra 86 (1993) 135-154.
M.Bullejos, P. Carrasco, A.Cegarra, Cohomology with
coefficients in symmetric cat$^n$-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. G\eom. Diff. Cat. 32, 11-27,
1991.
M.M.Kapranov, V. Voevodsky, $\infty$-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 n-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.
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Quote of the Week:
"Since the month of March last year, so nearly a year ago, the greater
part of my energy has been devoted to a work of reflection on the
*foundations of non-commutative (co)homological algebra*, or what is
the same, after all, of homotopic algebra. These reflections have
taken the concrete form of a voluminous stack of typed notes, destined
to for the first volume (now being finished) of a work in two volumes
to be published by Hermann, under the overall title "Pursuing Stacks".
I now foresee (after successive extensions of the initial project)
that the manuscript of the whole of the two volumes, which I hope to
finish definitively in the course of this year, will be about 1500
typed pages in length. These two volumes are moreover for me the
first in a vaster series, under the overall title "Mathematical
Reflections", in which I intend to develop some of the themes sketched
in the present report." - Alexander Grothendieck,
Sketch of a Program (1983)
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