Peter J. Veger veger at cistron.nl
Sun Sep 28 00:45:22 EDT 2003

```> I'm not sure if anyone mentioned the examples of a poset and a monoid as
categories.
> There is no "internal" structure in these. In the former, the objects are
the
> elements and there is a morphism between a and b iff a <= b. A functor
then becomes
> an order preserving map. In the latter, there is one object and the
morphisms are
> the elements. The identity is the identity map and if x and y are two
elements /
> morphisms then composition is xy. A functor is then a homomorphism.
> Dominic.

It is probably better to use the example of a preordered set.
A Preordered set is a set with a relation <=, that is
reflexive (x<=x) and transtive (x<=y & y<=z --> x<=z)
(but not necessarily antisymmetric, x<=y & y<=x --> x=y
as for a poset).

If, in a preordered set,
a is the unique morphism: x -> y and
b the unique morphism: y -> x, then
(writing x for the identity morphism id(x))
a.b=y and b.a=x
Categorically, x and y are isomorphic.
Concluding that x=y is not in the categorical language.

Of course, you should distinguish the category of preorders (posets,
monoids) from preorder (poset, monoid) as a category.

As for literature, you have the classic:
Saunders Mac Lane:  Categories for the Working mathematician

but the working computer scientist may prefer:
Andrea Asperti and Guiseppe Longo: Categories, Types, and Structures (MIT
Press, 1991)
Michael Barr and Charles Wells: Category Theory for Computing Science (3rd
ed., Les Publication CRM, 1999)
idem, Introduction to Category Theory
(www.let.uu.nl/esslli/Courses/barr-wells.html)

Peter J. Veger, Best, Netherlands

```