Re: terminology

[John Baez <baez@math.ucr.edu>]

Colin wrote:

It is an interesting impulse in higher category theory to avoid identity in
> favor of isomorphism on the level of objects, and to avoid isomorphism in
> favor of equivalence on the level of categories.   But
> so far as I know no one has yet articulated a way to avoid ever using
> identity of objects and identity of categories.
>

I think Michael Makkai has done it.  He has formulated a foundational
approach to mathematics based on infinity-categories, in which equality
plays no fundamental role:

http://www.math.mcgill.ca/makkai/mltomcat04/mltomcat04.pdf

I think some approach along these general lines might ultimately become
quite popular.  However, to think in an easy intuitive way about a
mathematical world without equality, we need new definitions of words such
as "the" and "is". Those who find it unpleasant to change the definition of
words such as "autonomous" may think it absurd to consider a such a radical
shift in basic terminology. However, we can already see these words changing
their meanings as we pass from reasoning within sets - where we say "the"
product 2 x 3 "is" 6 - to reasoning within categories - where we say "the"
product of "the" 2-element set and "the" 3-element set "is" "the" 6-element
set.

For a readable introduction to some of Makkai's ideas, try:

http://www.math.mcgill.ca/makkai/equivalence/equivinpdf/equivalence.pdf

Best,.
jb

[For admin and other information see: http://www.mta.ca/~cat-dist/ ]