Hi Rafael,
I have developed many perspectives on categories
As the study of algebraic structures with several objects As the study of primitive mathematical universe or space (not as fancy as a topos) As an unifying tool in mathematics As a foundation of mathematics (that is structural) As an abstarction of an abstarction of an abstarction of ... (if you go to higher categories) As a generalized theory of representations
If i have missed someone please let me know.
I came from Algebraic Specifications to Category Theory. When my students (computer science, software engineering) ask me about the benefit of categories I'm referring often to the "Categorical Manifesto" of Jo Goguen. If they insist more and ask "Be honest! What is the REAL reason that some theoreticians like categories so much?" Then I'm trying to be honest and say: Because categories are the winner of the competition "What mathematical structure is closed under the maximal number of "reasonable" constructions". They have just the right amount of structure - not too few, as graphs for example, and not to much, as cpo's for example. And if there is time I'm telling them about the "Erlanger Programm" of Felix Klein. Of course we can turn this statement and say: If a construction doesn't provide a category when the "inputs" are categories, then this construction can not be considered to be a "reasonable construction". Best regards Uwe Wolter [For admin and other information see: http://www.mta.ca/~cat-dist/ ]