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From Wiki:

"In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself."

Question: Are applications of category theory in computer science based primarily on the use of concrete categories?

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I don't think so. Anyway, all small categories can be made concrete. – Berci Oct 10 '12 at 14:29
Chu spaces are more applicable to computer science, and according to Vaughan Pratt's Chu realizes all small concrete categories paper every small category (thus every concrete category) can be "turned into" a Chu space. – Alex Nelson Oct 10 '12 at 16:25

In the programming language Haskell, the category Hask is the category with Haskell types as objects and functions between types as values. A functor from Hask to Set is one that takes each type to a set of each possible value to that type and functions to functions. This is concrete.

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