Re: calculus, homotopy theory and more (corrected)

[Andre Joyal <joyal.andre@uqam.ca>]

Dear Michael,

A basic ingredient in my approach to higher categories is the notion
of complete Segal space introduced by Rezk.
I have learned Rezk theory in proving the Quillen equivalence 
between quasi-categories and complete Segal spaces.
In my "Notes on Quasi-categories" I am introducing
an abstract notion of complete Segal space called
*Rezk category*, or *reduced category*.
A category object (internal to a quasi-category) is said 
to be *reduced* if its object of objects is 
*isomorphic* to its object of isomorphisms via the unit map. 
(an isomorphism in a quasi-category is an arrow which
is invertible in the homotopy category).
An ordinary category (in set) is reduced iff every
isomorphism is a unit, a very stringent condition.
Ordinary categories are seldom reduced (posets are).
Every reduced category is skeletal.
An equivalence between reduced categories is necessarly 
an isomorphism. In contrast, there are plenty of 
reduced categories in homotopy theory.
In fact every category internal to the quasi-category of spaces
is *equivalent*  to a reduced category (via a fully faith ess surj functor).
This key result was proved by Rezk for complete Segal spaces:
every Segal category is *equivalent* to a complete Segal space.
The theory of reduced categories is essentially (homotopy) algebraic 
(unlike ordinary category theory in which we need to expand the notion 
of isomorphism (of categories) with that of equivalence).

I do not have the time to explain more of the idea of my proof now. 
A sketch can be found in my "Notes on Quasi-categories".

You wrote:

>I can not reproduce it in this post because it requires some 
>pictures. But I remember, Andre, we discussed it with you in Montreal in
>2004. I'll be happy to explain it again in Genoa.
 
I hope I will understand this time!
I always find our conversation very stimulating!

See you in Genoa,

André


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