| Well, to make it as simple as possible. This kinda thing comes up when
you transform some difficult and inaccesable objects to something
managable. One example is the study of topological spaces.
These things are difficult because there are very few tools you can use
in topological spaces. So you assign (with something called a
morphism) each topological space to a group which is easier to deal with.
Now you prove something about the group and then you can conclude
something about the original topological space. Now you are dealing
with morphism that "maps" from category of topological space to the
category of groups. These categories are not sets (because you get
things like Russell's paradox, if you call them sets).
Eugene
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| A book I found was a good, fun introduction to the subject is
by Arbib & Manes. It's called "Intro to Category Theory" or something
imaginative like that.
It starts off with the idea of sets and the definition of
morphisms between sets. It then shows how you can obtain a more
powerful and general theory by dispensing with the idea of elements
in the sets. So you move to the idea of categories: but the key thing
is the idea of morphism.
I found that it was a lot easier to come to grips with the theory
of Object-Oriented Programming having had some encounters with the
principles of Category Theory, but I wouldn't want to push the connections
too far.
The theorems in the subject are quite pretty. It is very much a
subject where a picture is worth a thousand words.
CHeers,
Andrew.
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| C.A.R. Hoare,
Notes on an Approach to Category Theory for Computer Scientists;
in
Constructive Methods in Computing Science
NATO ASI Series, Vol. F55
Edited by M. Broy
Springer-Verlag 1989
ISBN 0-387-51369-8 (U.S.)
ISBN 3-540-51369-8
R. Goldblatt,
Topoi: The Categorial Analysis of Logic;
North Holland, Revised Edition, 1969.
E.G. Manes and M.A. Arbib,
Algebraic Approaches to Program Semantics;
Springer-Verlag, 1986.
Saunders MacLane,
Categories for the Working Mathematician;
Springer-Verlag, 1971.
J. Lambek and P.J. Scott,
Introduction to Higher Order Categorical Logic;
Cambridge University Press, 1986.
H. Herrlich and G.E. Strecker,
Category Theory, Second Edition,
Helderman Verlag, Berlin 1979.
D.E. Rydeheard and R.M. Burstall,
Computational Category Theory.
Prentice-Hall, 1988.
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