By Michael Barr, Charles Wells

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**Example text**

Composition must be associative in the sense that, if either of (f ; g); h or f ; (g; h) is defined, then so is the other and they are the same operation. We 22 Categories must also require, for f : A − → B, that f ; idB and idA ; f are defined and are the same operation as f . That is, we impose the equations f ; idB = f and idA ; f = f on the language. Both these requirements are reasonable in that in any implementation, the two operations required to be the same would surely do the same thing.

A. Show that sets with relations and homomorphisms between them form a category. b. Show that if (S, α) and (T, β) are both posets, then f : S − → T is a homomorphism of relations if and only if it is a monotone map. 2. Show that (strict) ω-complete partial orders and (strict) continuous functions form a category. 3. Let R+ be the set of nonnegative real numbers. Show that the poset (R+ , ≤) is not an ω-CPO. 4. Show that for every set S, the poset (P(S), ⊆) is a strict ω-CPO. † Give an example of ω-CPOs with a monotone map between them that is not continuous.

4 Categories of sets with structure The typical use of categories has been to consider categories whose objects are sets with mathematical structure and whose arrows are functions that preserve that structure. The definition of category is an abstraction of basic properties of such systems. Typical examples have included categories whose objects are spaces of some type and whose arrows are continuous (or differentiable) functions between the spaces, and categories whose objects are algebraic structures of some specific type and whose arrows are homomorphisms between them.