By D. E. Rydeheard

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The function which finds the maximum integer in a list of integers. 2. The function which sums a list of integers. 3. The function which takes a list of coefficients a 0 , a1 , . . , an and a value x and evaluates the polynomial a 0 +a1 ×x+. . an ×xn . 10 EXERCISES 4. Use the append function, concatenating lists end to end, to define the function which reverses a list. 5. The function maplist which applies a function to all items in a list returning the list of results. What is its most general type?

Rarely do we consider categories without some additional internal structure. For example, categories may be equipped with products or coproducts of pairs of objects. We shall define these constructs in the next chapter and show that they can be represented as functions. Thus a category with additional structure consists of the four functions mentioned 48 CATEGORIES AND FUNCTORS above together with extra functions recording the additional structure. g. that all pairs of objects have a product) into functions whose existence is the property in question.

We may go further and consider objects again to be sets but arrows to be relations between sets (labelled with their source and target sets). A relation r : a → b is a subset of the cartesian product a × b. The composition sr of r : a → b with s : b → c is defined by sr = {(x, z) : ∃y ∈ b . (x, y) ∈ r ∧ (y, z) ∈ s} Let us call this category SetRel . 2 Graphs We consider directed multi-graphs, that is, pairs of sets N (of nodes) and E (of edges) together with pairs of functions s, t : E → N (source and target respectively).