Re terminology:

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

Dear Urs,

Thanks for your friendly and detailed reply.

I should say that I also feel responsible for defending and advertising
the work of my long time collaborator, Philip Higgins, without whom of
course much of the work would not have got done, certainly not so
quickly. His last contribution to maths was in 2005; I helped with the
presentation of his TAC paper, but insisted that it showed `you know the
lion from his claw', as all the ideas were his. He is happily playing
the violin and making string instruments from bare blocks of wood! (That
shows his craftmanship.) He remembers the project  as very hard work but
a lot of fun!

You mention the process from category to infinity-category. Actually
that was why we introduced the term infinity-category in
34.  (with P.J. HIGGINS), ``The equivalence of $\infty$-groupoids and
crossed  complexes'', {\em Cah. Top. G\'eom. Diff.} 22 (1981)
371-386.
See also:
33.  (with P.J. HIGGINS), ``The equivalence of $\omega$-groupoids and
cubical  $T$-complexes'', {\em Cah. Top. G\'eom. Diff.} 22
(1981) 349-370.

The paper [34] also gives a definition of what is now called (rightly) a
globular set. I am curious to know whether there are earlier definitions
of these terms. Joachim Lambek once asked me  at a category theory
meeting: `Why don't people from xxxx refer to these papers?' What could
I say? People do write about 2-groupoids without referring to analogous
work on crossed complexes.

The points Andre makes are very interesting. In the 1970s we were very
puzzled by the Kan condition, and still are, for the following reason.
The fact that the simplicial singular complex of a space is a Kan
complex is due to a fact about the models, namely a geometric simplex
retracts onto all faces but one. So we can have in effect fillers for
S(X) natural in X, by making choices on the models. Also these fillers
are clearly related to multiple compositions of the remaining faces. The
problem is that there is no unique choice of such retractions, nor is it
clear what might be the relations between iterates of such fillers.
These considerations led Keith Dakin to the notion of T-complex  for his
1976 thesis; somehow `T-complex' has more recently become `complicial
set', but nobody asked me. (Groan! Groan!) So it seems that the notion
of quasi category as a weak Kan complex still has not captured something
about the basic example; but how to repair that is quite unclear.

As I have said before, we found it necessary for certain aspects to work
cubically, as expressing in a manageable way `algebraic inverses to
subdivision', and also to get monoidal closed structures. Again, it is
not clear how to capture axiomatically the properties of the cubical
singular complex, as some kind of weak cubical infinity groupoid. I have
been unable to cope with the complications of multiple compositions in
globular or simplicial terms. Is there an operad view of the cubical case?

I have no wish to hold things up or disparage work developing these
ideas in different ways, just the contrary, and indeed I wrote that I
was thinking of higher dimensional group theory, rather than category
theory. The contrast and relations between such views could, perhaps
should, be illuminating.

We are putting a photo from Macquarie of John Robinson's sculpture
`Journeys' as a frontispiece to the new book.

Best wishes to all for the future of this great adventure.

Ronnie


Urs Schreiber wrote:
> Dear Ronnie Brown,
>
> you write::
>
>
>> My own problem is with term `infinity groupoid' which
>>  is used to describe something which is not even a groupoid,
>>
>
> It seems to follow the well established terminology in higher category
> theory, which proceeds: category, 2-category, 3-category, ....
> infinity-category and  groupoid, 2-groupoid, 3-groupoid, ...
> infinity-groupoid.
>
>
>> It seems to be an example of these confusions is the way the simplicial
>> singular complex of a space is called an infinity-groupoid, even the
>> `fundamental infinity groupoid', when what seems to be referred to is that
>> it is a Kan complex, i.e. satisfies the Kan extension condition, studied
>> since 1955.
>>
>
> The notion of Kan complex is one model for the notion of
> infinity-groupoid. There are other, equivalent models. And there are
> models that model stricter subclasses of infinity-groupoids, such as
> those you are famous for having studied. Part of the point of saying
> "infinity-groupoid" instead of "Kan complex" or else is to amplify the
> general notion over its concrete implementation.
>

....

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