Re terminology:

["Joyal, André" <joyal.andre@uqam.ca>]

Dear Urs and Ronnie,

As you know, there are important differences between category theory and classical algebra. 
One lies in the fact that equivalent categories are considered to be the "same", 
even if when they are not isomorphic.
In category theory most constructions are yielding an object
which is not unique, but only unique up to some kind of equivalence, 
at best unique up to a unique isomorphism.
The general idea seems to be that an object is well defined
if its different incarnations are connected by a contractible
network of equivalences.


It seems to me that the challenge of higher dimensional algebra 
is to learn how to handle constructions whose output
are not unique but only unique only up to a contractible network.
Of course, we may decide to replace these constructions by ones
producing a truly unique object, but the replacement seems often artificial.
For example, we may decide to choose a representative for
the cartesian product of every pair of objects in a category. 
We are then lead to distinguish between two kinds of product
preserving functors. The functors preserving the 
product strictly are given a role, but this seems artificial to me.

I do not want to be negative about the idea of turning higher
dimensional algebra into ordinary algebra, because we may learn
something in the process. Also, Quillen homotopical algebra
can be regarded as a method for reducing higher categorical
and homotopy algebra to ordinary categorical algebra.
However, there was a real gain in moving from a purely algebraic description
of higher categories to one based on simplicial sets and homotopical algebra.
The category of quasi-categories is cartesian closed, a property which appears
to be false for the category of fibrant objects in the "algebraic" models.
This is also true for the category of n-quasi-category (Rezk).
 

Best,
André



-------- Message d'origine--------
De: categories@mta.ca de la part de Ronnie Brown
Date: ven. 21/05/2010 13:00
À: Urs Schreiber; categories@mta.ca
Objet : categories: Re terminology:
 
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.

........


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