Subject: categories: Joyal's message + query

From: [email-masked] (Carlos Simpson)
Date: Fri, 20 Nov 1998 18:58:35 GMT

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