Date: Mon, 16 Nov 1998 17:08:09 -0500 (EST) From: James Stasheff Subject: categories: query The language of higher category theory in more than analogous to homotopy theory, especially in the cellular version. What is the appropriate reference for a non-categorical reader? Have the A_\infty categories of Smirnov or Fukaya been treated in the categorical literature?? thanks .oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds From: john baez Subject: categories: Re: query Date: Tue, 17 Nov 1998 20:12:45 -0800 (PST) Jim Stasheff writes: > The language of higher category theory in more than analogous > to homotopy theory, especially in the cellular version. What > is the appropriate reference for a non-categorical reader? > Have the A_\infty categories of Smirnov or Fukaya been treated > in the categorical literature?? Unfortunately the truly appropriate reference has not yet been written, because the equivalence between weak infinity-groupoids and homotopy types has not yet worked out in full detail, at least not in the cellular version. The *dream* of translating all of homotopy theory into higher category theory is outlined in: John Baez and James Dolan, Categorification, to appear in Proceedings Workshop on Higher Category Theory and Mathematical Physics at Northwestern University, Evanston, Illinois, March 1997, eds. Ezra Getzler and Mikhail Kapranov, preprint available as math.QA/9802029. This also has lots of references to different places where various bits of the dream have been realized. Basically the dream consists of working out the following correspondence: HIGHER CATEGORY THEORY HOMOTOPY THEORY omega-groupoids homotopy types n-groupoids homotopy n-types k-tuply groupal omega-groupoids homotopy types of k-fold loop spaces k-tuply groupal n-groupoids homotopy n-types of k-fold loop spaces k-tuply monoidal omega-groupoids homotopy types of E_k spaces k-tuply monoidal n-groupoids homotopy n-types of E_k spaces stable omega-groupoids homotopy types of infinite loop spaces stable n-groupoids homotopy n-types of infinite loop spaces Z-groupoids homotopy types of spectra How do A_infinity categories fit in? As far as I can tell they should correspond to omega-categories where all j-morphisms are invertible for j > 1. Date: Thu, 19 Nov 1998 11:15:46 +1000 From: street@mpce.mq.edu.au (Ross Street) Subject: categories: Re: query >The language of higher category theory in more than analogous >to homotopy theory, especially in the cellular version. What >is the appropriate reference for a non-categorical reader? >Have the A_\infty categories of Smirnov or Fukaya been treated >in the categorical literature?? Fortunately, since the A_\infty categories (described sketchily in the preprint I have of Fukaya) use chain complexes rather than topological spaces (so that I believe a 1-object A_\infty category is an algebra for the A_\infty non-permutative operad on chain complexes - an A_\infty DG-algebra), we do not need to realise the whole homotopy types dream (as described by John Baez) to give a categorical description of A_\infty categories. Several years ago I remember discussing this by email with Jim Stasheff. Dominic Verity was here at the time. I cannot remember all the details but the two basic ingredients were some free strict n-categories made from the set (whose elements are to be the objects of the A_\infty category) in much the way the orientals are constructed, and the construction (below) mentioned in my Oberwolfach Descent Theory notes of September 1995. There is a connection between the Gray tensor product and ordinary chain complexes. Each chain complex R gives rise to a (strict) omega-category J(R) whose 0-cells are 0-cycles a in R, whose 1-cells b : a --> a' are elements b in R_1 with d(b) = a'- a, whose 2-cells c : b --> b' are elements c in R_2 with d(c) = b'- b, and so on. All compositions are addition. This gives a functor J : DG --> omega-Cat from the category DG of chain complexes and chain maps. In fact, J : DG --> omega-Cat is a monoidal functor where DG has the usual tensor product of chain complexes and omega-Cat has the Gray tensor product. By applying J on homs, we obtain a (2-) functor J_* : DG-Cat --> V_2-Cat, where V_2 is omega-Cat with the Gray-like tensor product (extending the natural tensor product of oriented cubes as described in Sjoerd Crans thesis). In particular, since DG is closed, it is a DG-category and we can apply J_* to it. The V_2-category J*(DG) has chain complexes as 0-cells and chain maps as 1-cells; the 2-cells are chain homotopies and the higher cells are higher analogues of chain homotopies. Best regards, Ross Date: Thu, 19 Nov 1998 10:31:09 +0100 From: grandis@dima.unige.it (Marco Grandis) Subject: categories: Re: query I have not seen Fukaya's paper. If the problem is to formalise higher homotopies for, say, topological spaces or chain complexes, together with their operations, there is a simple solution based on the path endofunctor (or, dually, the cylinder endofunctor; or their adjunction) and its powers. The path endofunctor P is constructed so that a homotopy a: f -> g: X -> Y amounts to a map a: X -> PY; the maps f, g are recovered by means of the two faces d-, d+: PY -> Y. (For Top, PY is obviously the space Y^[0, 1] of paths in Y, with the compact-open topology; for chain complexes (PY)_n = Y_n + Y_(n+1) + Y_n, with suitable differential.) P comes equipped with various natural transformations (faces, degeneracy, connections, symmetries, concatenation...), satisfying "algebraic" coherence axioms (a sort of "cubical comonad" with additional structure). These transformations represent the basic structure of lower order homotopies. But now you have, practically for free: - n-tuple homotopies, represented by the power endofunctor P^n, - n-homotopies, represented by a subfunctor P_n of P^n, - their operations, via the usual algebra of natural transformations, - the deduced coherence relations of the latter. Eg, a double homotopy, represented by a map X -> P^2(Y), has four faces given by the four natural transformations d-P, d+P, Pd-, Pd+: P^2 -> P; if k is the concatenation of homotopies, kP and Pk are the vertical and horizontal concatenation of double homotopies, and so on; if k is associative, as is the case for chain complexes but not for spaces, so are all higher concatenations. A double homotopy is said to be a 2-homotopy when its "vertical" faces (say) are trivial, i.e. factor through the degeneracy 1 -> P. (For Top, P^2(Y) is the space of maps from the standard square [0, 1]^2 to Y; P_2(Y) is the subspace of those maps which are constant on the vertical faces of the square.) This way of deducing higher homotopies and their operations from the lower ones can be found in: M. Grandis, Categorically algebraic foundations for homotopical algebra, Appl. Categ. Structures 5 (1997), 363-413. M. Grandis, On the homotopy structure of strongly homotopy associative algebras, J. Pure Appl. Algebra, 134 / 1 (1999 ?), 15-81. Regards, Marco Grandis Dipartimento di Matematica Universita' di Genova via Dodecaneso 35 16146 GENOVA, Italy e-mail: grandis@dima.unige.it http://pitagora.dima.unige.it/webdima/STAFF/GRANDIS/ ftp://pitagora.dima.unige.it/WWW/FTP/GRANDIS/ From: Andre Joyal Subject: categories: category-homotopy Date: Thu, 19 Nov 1998 21:34:42 +1100 (EST) This is in reply to a message of James Stasheff and of other participants. I agree that the language of category is more than analogous to homotopy theory, and think that the connection is deep. The work on A-infinity spaces is a striking early illustration of the connection between homotopy coherence and categorical coherence. Homotopy theory is the main guideline for the emergence of higher category theory. For example, the first correct concept of 3-category was discovered as a direct outcome of modelling homotopy 3-types with Gray groupoids. Before this example, all existing 3-categories were equivalent to strict ones. In my opinion, after a first stage of developpement, higher category theory should induce new progress in homotopy theory. This is the real test. In particular, the new theory should help understanding homotopy groups of spheres. I am including an abstract of my talk on a related subject. Regards, Andre Joyal > University of Sydney Algebra Seminar > Friday 20th November, 12-1pm, Carslaw 273 > > Title: The homology of symmetric and braided monoidal categories > Abstract: > We wish to comment on some aspects of the connection between > category theory, topology and homological algebra. The > connection is at the root of higher K-theory and it is guiding > much of the actual research on higher dimensional categories. > We shall concentrate on the relation between monoidal categories > and iterated loop spaces. To each category C we can associate a > space BC called the (Milgram) classfying space of C. The > homology of C is defined to be the homology of BC. The space BC > is a monoid when C has a tensor product, and it has the > structure of an E-infinity space (resp. of an E-2 space) if the > tensor product is symmetric (resp. braided). We shall briefly > discuss the work of P. May and F. Cohen on the homology of E-n > spaces. It shows that the homology of a symmetric (resp. of a > braided) monoidal category is a graded-commutative algebra > admitting Dyer-Lashof operations (resp. is a poisson algebra). > These structures play a crucial role in determining the homology > of the symmetric groups and of the braid groups. The poisson > algebra structure also appears in the recent work of Lehrer and > Segal on the rational homology of classical regular semisimple > varieties. > > ------------------------------ > > Best wishes, > Andrew Mathas > Date: Fri, 20 Nov 1998 14:44:43 +0000 From: "T.Porter" Subject: categories: Jim's query This is not really an adequate reply to Jim's query. The reason is that as I understand it, he is asking for a source that will explore the homotopy theory of the `globular' models of weak infty categories. My approach to the area is in some sense the dual. Starting with the various models for homotopy types or bits of them, try to see what is mirrored in the weak infty category theory by the homotopy structure. This is in some sense the converse to his query but may none-the-less be relevant. My intuition was, and still is, that the Kan condition on simplicial sets gives a composition/pasting up to coherent homotopy. Moreover the `filler' gives the justification for the composite. (Compare the filler structure of the nerve of a category with that in an arbitrary Kan complex.) Accepting that as a starting point, and the idea that the `category' of weak infinity categories should be a weak infinity category, Jean-Marc Cordier and I looked at `locally Kan' simplicially enriched categories (e.g. Trans AMS 349(1997)1-54). With that viewpoint, it becomes clear that the structure of an A_\infty category is needed to make things really `coherent', but that many of the constructions of `ordinary' category theory have A_\infty or homotopy coherent analogues in this setting, which thus serves as a test-bed' for the development of the more general theory. In part this relates to Michael Batanin's paper in the Cahiers where explicit consideration of A_\infty structure is given. That theory looks at the `global' structure to some extent, but simplicially enriched groupoids model all homotopy types, so a corollary of the simplicial to globular type of transition should be that one should be able to construct weak \infty categroies DIRECTLY from the algebra of a simplicially enriched groupoid. The obvious place to look for this is in the Moore complex which carries a hypercrossed complex structure in the sense of Pilar Carrasco and Antonio Cegarra. (This is related to the n-hypergroupoid structures of Jack Duskin.) Exploring the \infty category structure, potentially in their definition, is the aim of another line of research and in low dimensions, this has been attacked by Ali Mutlu and myself, (see very recent articles in TAC or Bangor's preprint list on the web). Getting nearer to Jim's query, any bridge between homotopy theory and higher dimensional category theory should I feel aim to be approachable by algebraic topologists and therefore should start with a recognisable model for homotopy types. Another approach that must be mentioned is that of Tamsamani and Simpson using multisimplicial objects. Presumably this also links in with the cat^n-groupoid approach pioneered some 14 years ago by Loday. This only handles n-types but can be extended to a model that has higher information but in those dimensions above n, the Whitehead products are trivial. (Has anyone looked at the Whitehead and Samelson products from a globular or weak \infty category viewpoint?) The question of simplicial rather than cubical theory is a difficult one. Marco Grandis made a good case for the cubical formulation the other day, and the use of Kan filler conditions in a cubical setting has been for a long time a speciality of Heiner Kamps (see the recent book by him and me!). That theory does not directly address the question of A_\infty category structures, but it does raise a question that deserves some attention. Suppose you want an analogue of a given theorem from homotopy theory but in another non-topological context and you can get a weak composition structure in your setting (typically given by fillers for boxes in a cubical `enrichment'). What fillers/composites do you need for your particular theorem (e.g.Dold's theorem on homotopy equivalence between cofibrations, or long Dold-Puppe type sequences)? This last question also raises that of the motivation for the generalisations. I am convinced these are useful, even important, but perhaps some debate on directions to explore and goals to seek out might help in providing a fuller answer to Jim's question. Tim. ************************************************************ Timothy Porter |tel direct: +44 1248 382492 School of Mathematics |mathematics office: 382475 University of Wales, Bangor | fax: 383663 Dean St. |World Wide Web Bangor |homepage http://www.bangor.ac.uk/~mas013/ Gwynedd LL57 1UT |Mathematics and Knots : exhibition United Kingdom |http://www.bangor.ac.uk/ma/CPM/ Date: Thu, 19 Nov 1998 19:06:47 +0000 (GMT) From: Ronnie Brown Subject: categories: higher order category theory and homotopy theory There is a more down to earth and historically rooted approach. A concern of the early topologists (Dehn, Hopf, Alexandroff,...) was to find a higher dimensional version of the fundamental group, since the were irritated by the facts that (i) \pi_1 gave good geometric information (e.g. for complex variable theory) which often involved the non commutativity, (ii) H_1 was \pi_1 made abelian, for connected spaces, (iii) H_n existed for all n \ge 0 (and was abelian). There was also the intuition of big things made out of composing little bits (simplices in the Poincare formulation). The `solution' was to use formal sums (homology), and this gave abelian results. Cech's 1932 submission on Higher homotopy groups to the Zurich ICM was withdrawn at the request of Alexandroff and Hopf once they had proved these groups abelian for n \ge 2. The whole aim was to get immediate geometric results and computations. My suggestion in 1967 was to use `higher homotopy groupoids' since these did not have to be abelian. It was only in 1974 that Philip Higgins and I found a definition of a homotopy double groupoid of a pair; with this and the work with Chris Spencer we could prove a 2-D Van Kampen theorem. This is still giving actual computations of homotopy 2-types and invariants not previously known. For an exposition see my notes Groupoids and crossed objects in algebraic topology from the 1997 School in Algebraic Topology at Grenoble http://www.bangor.ac.uk/~mas010/brownpr.html which also go on briefly to the higher dimensional case (for strict higher groupoids) parts of which were announced in 1977 (with Philip Higgins, Compte Rendue Acad. Sci. Paris S\'er. A. 285 (1977) 997-999, 286 (1978) 91-93. Also, with the use of n-fold groupoids one can recover all n-types (Loday) a result Grothendieck described to me in 1986 (in Montpellier) as `absolutely beautiful'. Again, the intricate structure of these can give new computations of homotopy invariants and homotopy types, while studying pushouts of these (strict again!) one can find new algebraic constructions, such as a non abelian tensor product of groups (which act on each other). We have to develop new methods to compute even with these strict objects, so there is still a lot to be done in getting explicit information on specific examples using the non strict ones. One overall setting is to have functors \Pi ----------> (topological data) <--------- (Algebraic data) IB | | U | | B | | V V ======== (spaces)====== where IB (\mathhbb{B}) and B are classifying data or space functors, U is forget, and \Pi is a functor which should satisfy Van Kampen Theorem (preserve certain colimits), and also \Pi IB is equivalent to 1. At the level of 3-types we have such a set up with topological data = squares of (pointed) spaces, algebraic data = crossed squares (a kind of triple groupoid). We do not (so far) have such a setup when (algebraic data) is say the candidate proposed by Andre (which are equivalent to 2-crossed modules). Graham Ellis has shown how to compute 3-types and homotopy classes of maps using the above set up (Math. Z., 461 (1993) 93-110. ). The existence of the functor \Pi (satisfying VKT, so one can apply free algebraic objects) helps a lot. So I find it interesting that even strict 3-fold groupoids have an intricate structure (they model 3-types!). The relations and uses of all these models needs lots of work. The tensor product of crossed complexes (corresponding to a Gray tensor product of strict \infinity groupoids) yields interesting models for loop spaces, by considering monoids for this tensor: (see for example Baues and Tonks On the twisted cobar construction. Mathematical Proceedings of the Cambridge Philosophical Society 121 (1997) pp.229-245 ). But the nice thing about crossed complexes is one can write down explicitly the Eilenberg-Zilber theorem (Tonks' thesis), which I have not seen for infinity groupoids! Ronnie Brown Date: Fri, 20 Nov 1998 18:58:35 GMT From: carlos@picard.ups-tlse.fr (Carlos Simpson) Subject: categories: Joyal's message + query A. Joyal writes: >In particular, the new theory should help understanding >homotopy groups of spheres. I think it is safe to say that the ``n-category crowd'' (myself included) would love to make some progress on this. The main thing I am wondering about is: what exactly does one want to know about the homotopy groups of spheres? for example, is there a concrete question which needs answering? Up until not too long ago, I was under the impression that the question was how to calculate them; but it turns out that E. Brown in 1956 gave a perfectly good algorithm for calculating homotopy groups; another was subsequently given by Kan and refined by Curtis: one just has to calculate the ``nonabelian homology groups'' of a simplicial complex of free groups, and by Curtis one can divide out by a higher commutator subgroup so this actually only concerns a simplicial complex of nilpotent groups. (I give another algorithm in q-alg/9710011 using n-categorical type ideas, but this doesn't necessarily seem particularly useful and in any case I have more recently learned that it is basically the same thing as the ``Milgram model'' see the Handbook, see also an article by Baues for the double loop space etc.etc....) One might complain that these are ``algorithms'' rather than ``formulae'' but I have never quite understood the distinction. One generally needs a computer algorithm to evaluate one of Zagier's formulas! Maybe there is a subtle type of distinction about the type of machine the algorithm is supposed to run on? Or a quantitative question about the asymptotics of the computing time? Another possible version of the question (related to the computing time question in the previous sentence) might be: does one know how to bound the size of the $\pi _i (S^k)$? I can imagine applying n-categorical ideas to give a bound here, but it seems to be that there must be known bounds using spectral sequences or other. If there are known bounds, does someone know what the best one is? Other than that, does Prof. Joyal or anyone else have an idea of the type of question about the $\pi _i(S^k)$ which could/should be attacked by n-categories? -------------------- On an introduction to $n$-categories: it would be great if J. Baez could bundle together his ``This week's finds in math. phys.'' which concern $n$-categories. This would make a really nice introduction specially for friends and family. For the more mathematically minded, a recommended ``bapteme de feu'' would be Grothendieck's ``Pursuing stacks''. Another good place to look is Benabou's LNM 47. ---- On A_{\infty}-categories and \infty-categories: the response of R. Street looks quite adequate to the question (and in fact is a nice illustration of what the ``Gray tensor product'' really means, a point I had never clearly understood up to now...). Here are a few more remarks, specially related to J. Baez's comment. The notion of Segal category (which, again I recently found out, first appeared in the paper of Dwyer-Kan-Smith...) corresponds to the notion (that J. Baez mentionned) of \infty-category in which the $i$-morphisms are invertible for $i>1$ (I call this condition ``$1$-groupic''). A first remark is that Segal categories are equivalent to strict simplicially enriched categories, as was already shown by Dwyer-Kan-Smith. (in particular, you can read ``simplicially enriched category'' wherever I write ``1-groupic $\infty$-category'' below...). This suggests that A_{\infty}-categories are also probably equivalent to strict i.e. DG categories, but it isn't a proof (see below for why not) and it would be interesting to know if someone has an answer to that question. Segal categories or $1$-groupic $\infty$-categories, are not actually totally equivalent to A_{\infty} categories, because they don't encode the ``spectrum'' structure on Hom sets. In fact the notion of $A_{\infty}$-category in the DG world probably has a topological analogue where the Hom's are indeed spectra (cf Vogt et al on E_{\infty} ring spaces and the like???). Here is a proposition for an $\infty$-categorical notion which should correspond to this placing of a spectra structure on the Hom spaces. Suppose $C$ is a 1-groupic $\infty$-category. We say that $C$ is {\em spectric} if it satisfies the following conditions: ---C has an initial and final object and these coincide (call them *); ---if $X\in C$ then the fiber product $\Omega X := * \times _{X} *$ exists (as a homotopy-limit) in $C$; (also that $\Omega$ becomes a functor...?) ---the functor $\Omega$ is an equivalence of categories. These conditions are inspired by M. Hovey's definition of ``stable model category'' in ``Model categories''. (In particular, if M is a stable model category then the simplicial category, Dwyer-Kan localization of M inverting wk equivs L(M), will be a spectric $\infty$-category.) It follows from this definition that the $Hom$ spaces in C (i.e. the $\infty$-groupoids which are the Hom's of the $\infty$-category C) have structures of spectra: Hom(x,y)=\Omega Hom(x, \Omega ^{-1}y) = \Omega Hom(\Omega x, y) etc. Thus, C will correspond to the version of A_{\infty} category mentionned above where the Homs are spectra. On the other hand, given an A_{\infty}-category B, formally adjoin the shifts of objects, by setting for example Hom(X, Y[n]):= \Omega^nHom(X,Y) (for integers n) etc. It looks likely that this will give a spectric $\infty$-category C, and that these two constructions will be (as much as possible) a correspondence between A_{\infty}-categories and spectric $\infty$-categories. .... Sorry to clog up the e-waves so... Carlos Simpson Date: Fri, 20 Nov 1998 17:40:38 -0500 (EST) From: James Stasheff Subject: categories: Re: Jim's query Amazing what deep interpretations have been given to my shallow question though i appreciate the answers. Al I was after was some place for a novice reader to see how category theory and homtopy theory interacted in the concept of A_\infty-category cf. Smirnov or Fukaya .oooO Jim Stasheff jds@math.unc.edu (UNC) Math-UNC (919)-962-9607 \ ( Chapel Hill NC FAX:(919)-962-2568 \*) 27599-3250 http://www.math.unc.edu/Faculty/jds Date: Sat, 21 Nov 1998 10:06:29 +1100 From: Michael Batanin Subject: categories: Re: query This is a summary of my correspondence with J.Stasheff. James Stasheff wrote: > As monoids can be described as categories with one object, > one can consider \Aoo structures on categories with a notion of > homotopy, e.g. topological categories or differential graded categories. > To be more precise, the set of objects and the set of morphisms > carry a notion of homotopy. As usual, one deals with composable > morphisms and then weakens the axiom of associativity up to homotopy > in the strong sense in order to > define \Aoo - categories. This was first done by Smirnov by 1987 > \cite{smirnov:baku} to handle functorial homology operations and > their dependence on choices (cf. indeterminacy). More recently, Fukaya > \cite{fukaya:1} reinvented \Aoo -categories with remarkable applications to > Morse theory and Floer homology. > Inspired by this work, Nest and Tsygan have proposed an \Aoo -category > with automorphisms of an associative algebra as objects and for > the space of morphisms, a twisted version of the Hochschild complex > of the corresponding endomorphism algebras. Michael Batanin: One can generalize "ordinary" category theory in the different ways. One can consider internal category theory, enriched category theory. We can also consider a category as a special sort of simplicial set. All this points of view have their own A_{\infty}-analogues. I realize, that the approach of Smirnov, Fukaya and others is a generalization of "internal" category theory. In my paper "Monoidal globular categories as a natural environment ..."(Adv.Math. 136, 39-103 (1998)) I also consider a Cat-internal version of A_{\infty}-\omega-category (so it involves a weak form of interchange law)that I call monoidal globular category. A surprising coherence theorem sais that a general monoidal globular category is equivalent to a strict one (the internal category structure on objects aloows to strictify interchange low). In another my paper "Homotopy coherent category theory and A_{\infty}-structures in monoidal categories" (JPAA 123(1998),67-103) I defined an enriched version of A_{\infty}-category. So we have a honnest set of objects but morphisms are objects of a monoidal simplicial categories with a Quillen model structure. I also can define what A_{\infty} functor is and prove an appropriate coherence and homotopy invariance theorems. Another nice theorem sais that A_{\infty}-categories and their A_{-infty}-functors form an A_{infty}-category in a natural way. The simpliocial point of view on A_{\infty}-categories goes back to Boardman and Vogt book. The corresponding notion is a simplicial set satisfying some weak Kan conditions. This approach was extensively used by T.Porter and J.-M.Cordier (see T.Porter's answer on Jim's query). James Stasheff: > Since in a category we are concerned with $n$-tuples of morphisms > only when they are composable, it is appropriate to similarly relax > the composition operations for in defining an operad. the result is > known as a partial operad and appears in two different contexts: > in the mathematical physics of vertex operator algebras (VOAs) > \cite{yizhi} and Mivhael Batanin: In my work "Globular monoidal categories ..." I introduced the n-dimensional operads over trees. A 1-dimensional operad in this sense is not exatly the same as usual non-symmetric operad as every operation may have source and target and we can multiply just composable chains of operations. A one object version of this may be identify with a usual nonsymmetric operad. (In my paper I use \omega-operads). I wonder if a partial operad is the same as my 1-operad? Michael. From: john baez Subject: categories: n-categorical miscellany Date: Tue, 24 Nov 1998 15:28:09 -0800 (PST) 1) Carlos Simpson writes: >A. Joyal writes: >>In particular, the new theory should help understanding >>homotopy groups of spheres. >I think it is safe to say that the ``n-category crowd'' (myself included) >would love to make some progress on this. The main thing I am wondering >about is: what exactly does one want to know about the homotopy groups of >spheres? for example, is there a concrete question which needs answering? There are a lot of complicated and interesting patterns in the stable homotopy groups of spheres. A lot of these patterns are currently studied using a generalized cohomology theory called complex cobordism theory, together with an intricate mess of spectral sequences: Complex cobordism and stable homotopy groups of spheres / Douglas C. Ravenel. Orlando: Academic Press, 1986. Nilpotence and periodicity in stable homotopy theory / by Douglas C. Ravenel. Princeton, N.J. : Princeton University Press, 1992. It's all far too technical for me to understand, so I don't have much hope of suddenly answering some question that homotopy theorists are stuck on! However, I think that the n-category crowd can give more conceptual explanations of some facts that appear as miracles in the current approach. For example, there's an important relation between complex cobordism theory and formal group laws: the complex cobordism of a point is "the universal formal group law". There should be an elegant conceptual proof of this, but as far as I know, there are just elegant *heuristics* for why it should be true, followed by a grungy computation. 2) Carlos Simpson writes: >On an introduction to n-categories: it would be great if J. Baez could >bundle together his ``This week's finds in math. phys.'' which concern >n-categories. This would make a really nice introduction specially for >friends and family. Especially during the holiday season! I should try to bundle them together more nicely at some point, but right now you can start at: http://math.ucr.edu/home/baez/week73.html , skip the bit about left-right asymmetry in amino acids, read the stuff on n-categories, and then keep clicking on the thing that says: "To continue reading the `Tale of n-Categories', click here." It fizzles out around week100, though I plan to resume it with an issue discussing Carlos Simpson's paper on the stabilization hypothesis and Mark Hovey's book on model categories. 3) Some people have noted that http://xxx.lanl.gov/abs/math.QA/9811139 doesn't give them the paper "HDA4 - 2-Tangles". Sorry! - the delay before the preprint server makes papers publicly available is longer than it used to be. It should be working by tomorrow. If you still have trouble after that (it's a big file and may bust your printer), feel free to email me and ask me to send you a copy.