Re: Some books on category and topos theory

From: Bruno Marchal <marchal.domain.name.hidden>
Date: Tue, 9 Jul 2002 16:41:22 +0200

Wei Dai asks some question to Tim May which I would like to comment
taking into account some other posts.

Wei Dai:

>Suppose I had the time for only one book, which would you recommend?


I think you (Wei) decide to look for the book by Lawvere. Good choice
but you should know it is just an introduction. Now, that book is useful
even for learning algebra. Some who knows algebra (i.e. groups, rings,
fields, topological spaces and more importantly their morphism) could
look for more advanced material perhaps.


>Also,
>can you elaborate a bit more on the motivation behind category theory? Why
>was it invented, and what problems does it solve? What's the relationship
>between category theory and the idea that all possible universes exists?


Tim makes a very genuine remark (but he writes so much I fear that has
been unnoticed!). He said: read Tegmark (Everything paper), then learn
category, then read again Tegmark. Indeed I would say category theory has
emerged from the realisation that mathematical structures are themselves
mathematically structured. Categorist applies the every-structure principle
for each structure. Take all groups, and all morphism between groups: you
get the category of groups. It is one mathematical structure, a category
(with objects = groups and arrows = homomorphism) which, in some sense
capture the essence of group.
Note that the category of groups is too large to be defined in Zermelo
Fraenkel set theory (as almost any so called
large category). The usual trick of categorist is to mention the Von Neumann
Bernay Godel theory of set which has classes (collection of sets which are not
sets themselves). A much modern view is to make category theory in a well
chosen topos!
Note that some common mathematical structure *are* categories. A group is
a category (with one object: the set of group elements, and the arrows are the
action of the group on itself). Partially ordered set (set with a transitive,
reflexive, and antisymmetric relation on it) give other simple exemples:
object of the category are the element of the partially ordered set, and
the unique (here) arrow between object is the order relation). Automatically
Boolean Algebra, but also Heytingian lattices, etc... are categories.
Of course just a set can be made into a trivial category.
Other categories lives in between groups and lattices. They have lot of objects
and lot of arrows between objects.
Categories arises naturally when mathematician realised that many proofs
looks alike so that it is easier to abstract a new structure-of-structures,
then makes proofs in it, then apply the abstract proof in each structure
you want. So they define universal constructions in category (like the
"product"), which will correspond automatically to
    - "and" in boolean algebra
    - "and" in Heyting algebra
    - group product in cat of groups
    - topological product in cat of topological spaces
    - Lie product in cat of Lie groups, etc.
So Category theory helps you to make a big economy of work ... once you
invest in it, if you are using algebra. It saves your time.
But, to come back to Tim remark, it hints that a giant part whole of
mathematics is naturally mathematically structured, and this should be taken
into account. Also, as I have explained before, the whole of math cannot
be entirely mathematically structured in any consistent way (that's too big).
This can be shown with logic, but categories can give you a concrete
feeling of the bigness and endlessness of such an enterprise.
Another motivation for category is the realisation that elements of structures
are not necessary for defining those structures. Objects behavior are defined
(up to isomorphism) by their relationship (arrows) with other objects.
That's a sort of functional or relational philosophy not so different from
comp. As exercice you could try to define injection and surjection
between sets without mentioning the elements!



>
>Does it help understand or formalize the notion of "all possible
>universes"?


I don't think categories can help in any direct way, although I doubt
indeed we can live without categories (nor without logic-modalities) in the
long run. You could try to define the category of (multi)universes.
What would be a morphism between universes?
Note that lawvere has try to provide math foundation through the category
of all categories (with functors as arrows) but this has not succeed.
He discovers toposes instead.

Note that categories are difficult to marry with ... recursion theory.
(Despite so-called Dominical Categories, which does the job, but that is
too heavy math for me ...).


>I know in logic there is the concept of a categorical theory
>meaning all models of the theory are isomorphic. Does that have anything
>to do with category theory?


Not really. It is one of the reason it is better to use the adjective
categorial in algebra and categorical in logic. But not all scientist
follow this custom.


Bruno
Received on Tue Jul 09 2002 - 07:38:22 PDT