Subject: categories: Ancient history

From: Peter May <[email-masked]>
Date: Wed, 16 Nov 2005 22:26:16 -0600 (CST)

In his posting today, John Baez advertised the slogan:

             FIBRATIONS OVER THE BASE SPACE B WITH FIBER F
                          ARE "THE SAME" AS
         HOMOMORPHISMS SENDING LOOPS IN B TO AUTOMORPHISMS OF F.

He hedged it with a ``dose of vagueness'', but in fact I proved a
completely precise and general version of exactly this result in
``Classifying spaces and fibrations'', Memoirs AMS 155, Jan. 1975. Using
Moore loops on B, LB, one has a topological monoid, and one also has the
topological monoid Aut(F) of homotopy equivalences of $F$. A ``transport''
is a homomorphism of topological monoids from LB to Aut(F).  Allowing F to
vary by a homotopy equivalence, one can define an equivalence relation on
transports such that the equivalence classes are in natural bijective
correspondence with the equivalence classes of `fibrations over the base
space B with fiber F'.  One can generalize the context by allowing fibers
in some nice category and prove the same result.  See opus cit, Theorem
14.2, page 83. That was over 30 years ago, so naturally I wasn't thinking
about categorification, but I would imagine that the methods categorify.

Some questions from more recent work (in progress in fact):

1.  In work on (equivariant, stable) parametrized homotopy theory,
Johann Sigurdsson and I need and develop duality in ``symmetric
bicategories B'', which are not to be confused with the reasonably
standard symmetric monoidal bicategories.  Rather there must be a
prescribed involution on the bicategory B, a pseudo-equivalence t
between  B  and  its opposite bicategory (not completely general:
we find it helpful to require tt = id on 0-cells).  For example, the
standard bicategory whose 0-cells are rings, whose 1-cells R >--> S are
(S,R)-bimodules, and whose 2-cells are maps of bimodules is symmetric;
t takes R to its opposite ring and takes an (S,R)-bimodule to the same
Abelian group regarded as a (tR,tS)-bimodule.  Is there a pre-existing
theory of such bicategories and their duality theory, analogous to
duality theory in symmetric monoidal categories?

2.  The example in 1 is additionally a symmetric monoidal bicategory
under the tensor product over Z, and there is an analogous bicategory
starting with a commutative ground ring replacing Z.  These assemble
nicely into a tricategory of commutative rings, algebras, bimodules,
and maps of bimodules.  Moreover, the bicategory in 1 is actually part
of a pseudo double category with maps of algebras as vertical 1-cells.
Promoting this to the tricategory just mentioned, one has maps of
commutative rings as vertical 1-cells and maps of algebras as
vertical 2-cells.  I don't know a name for the resulting notion,
something like a pseudo triple category.  Here again, what is most
important is duality theory.  Has anybody studied such structures?
There are derived versions of the cited example, and such structures
also appear naturally in our work on parametrized homotopy theory.
Thankfully, we do not (yet) seem to need tetracategories!

---

From: jim stasheff <[email-masked]>
Date: Thu, 17 Nov 2005 10:08:02 -0500

Even more ancient:

Parallel transport in fibre spaces," Bol. Soc. Mat. Mexicana (1968), 68-86.

Unfortunately that's a hard paper to get a hold of

somewhat related is

Associated fibre spaces," Michigan Math. Journal 15 (1968), 457-470.

and at the survey level

 H-spaces and classifying spaces, I-IV", AMS Proc. Symp. Pure Math.
22 (1971), 247-272.

Of course, as you might expect, I describe things in terms of
A_\infty-morphisms
from the space of loops into

Aut(F) of homotopy equivalences of $F$

Now that some of us are comfortable with A_\infty-cats,
categborification should proceed
perhaps with some technical details.

jim

---

From: Marco Grandis <[email-masked]>
Date: Fri, 18 Nov 2005 19:59:20 +0100

In reply to P. May's message.

I do not know of any work on "pseudo triple category".

On the other hand, the pseudo double category  Rng  of rings,  
homomorphisms and bimodules
is briefly considered in our first work on pseudo double categories,  
subsection 5.3

    M. Grandis - R. Paré, Limits in double categories, Cah. Topol.  
Géom. Diff. Catég. 40 (1999), 162-220

as a substructure of the  pseudo double category of Ab-categories, Ab- 
functors and Ab-profunctors.

As an interesting construction, in this pseudo double category:
- the "tabulator" of a bimodule  u: R -+-> S  (u  is a left-R,  right- 
S  bimodule) (i.e., its double limit) can be constructed as a ring of  
triangular 2x2 matrices, with "matrix product"

   r  x     r'  x'        rr'   rx'+xs'
         .           =
   0  s     0   s'        0       ss'

(with  r. r' in R,  s, s' in S  and  x, x'  in  u).

Tabulators are crucial for double limits, since all of them can be  
constructed from double products, double equalisers and tabulators.
[  In a bicategory, the tabulator (of the vertical identity of A) is  
the cotensor  2*A,  and the previous result amounts to the  
construction of weighted limits, in
   R.H. Street, Limits indexed by category valued 2-functors, J. Pure  
Appl. Algebra 8 (1976), 149-181.  ]

Best regards

Marco Grandis