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5, we discuss some of the ways in which categories arise in mathematics. 15 Exercises 1. Prove that sets (as objects) and injective functions (as arrows) form a category with functional composition as the composition operation c. 2. Do the same as Exercise 1 for sets and surjective functions. 3. 14) is associative. 4. Prove the following for any arrow u : A − → A of a category C . 3 characterize the identity arrows of a category. a. If g ◦ u = g for every object B of C and arrow g : A − → B, then u = idA .

3, where we discussed certain mathematical structures as categories. Here, we discuss categories whose objects are mathematical structures. 1) as arrows. It is denoted GRF. 4) and homomorphisms between them is denoted Grf . Let us check that the composite of graph homomorphisms is a graph homomorphism (identities are easy). Suppose φ : G − → H and ψ : H − →K are graph homomorphisms, and suppose that u : m − → n in G . Then by definition φ1 (u) : φ0 (m) − → φ0 (n) in H , and so by definition ψ1 (φ1 (u)) : ψ0 (φ0 (m)) − → ψ0 (φ0 (n)) in K Hence ψ ◦ φ is a graph homomorphism.

The word ‘primitive’ means given in the definition of the language rather than constructed by a constructor. Some authors use the word ‘constructor’ for the primitive operations. 3 If we make two assumptions about a functional programming language and one innocuous change, we can see directly that a functional programming language L corresponds in a canonical way to a category C(L). A–1 We must assume that there is a do-nothing operation idA for each type A (primitive and constructed). When applied, it does nothing to the data.