we do meet isomorphisms of categories

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

Dear Marco,

Perhaps I can advocate the importance of isomorphism between categories.
An equivalence between skeletal categories is necessarly an isomorphism.
There are many examples of skeletal categories in mathematics.
The category of matrices over a ring is skeletal.
The category Delta in homotopy theory is also skeletal.

The category Delta(+) of all finite ordinals and order preserving maps
is an interesting example because, as everyones know, it is
freely generated as a monoidal category by a monoid object.
It is not free in the category of strict monoidal functors
but free in the category of (strong) monoidal functors.
There is of course another category, FatDelta(+), which is 
freely generated by a monoid object in the category of strict 
monoidal functors, but it is seldom used.
The two categories FatDelta(+) and Delta(+) are equivalent,
but the category Delta(+) is simpler because it is skeletal.

The category of finite cardinals and all maps is also skeletal.
It is freely generated by one object as a category with finite coproducts.
It is also freely generated by a commutative monoid as a
symmetric monoidal category.

A category C is skeletal iff every equivalence A-->C has a section.
This property characterises minimal models in algebraic topology.
For example, a Kan complex Y is minimal iff every homotopy equivalence
X-->Y, with X a Kan complex, has a section. Minimal models are important
in topology. Sullivan's rational homotopy theory is essentially a technique 
for constructing minimal models of graded commutative algebras. The rational 
homotopy groups of a space can be read directly form its minimal Sullivan model. 

Minimal models exists in higher category theory too.
Every quasi-category has a minimal  model (should I say skeletal?).
This not a property shared by all types of (infty,1)-categories.
Some are better than others. For example, simplicial categories 
do not admit minimal models (in general).

Strict monoidal categories do not admit minimal models either.
This is because strict monoidal structures cannot be transported 
(in general) along equivalence of categories.
Of course, non-strict monoidal structures can.
There is an obstruction for transfering a strict monoidal structure 
to its skeletal model. It is represented by a cohomology class 
of degree 3 when the category is groupoidal.
It is a small miracle of nature that the category Delta(+) is both
strict monoidal and skeletal. 
Similarly for the category of finite cardinals and maps. 

Best wishes,
André


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