By Richard Bird, Oege de Moor

Describes an algebraic method of programming that enables the calculation of courses. Introduces the basics of algebra for programming. provides paradigms and methods of application building that shape the center of set of rules layout. Discusses services and different types; functions; family members and allegories; datatypes; recursive courses, optimization concerns, thinning algorithms, dynamic programming and grasping algorithms. applicable for all programmers.

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Denote by pt I ∧ the constant functor from I op to Set, deﬁned by pt I ∧ (i) = {pt} for all i ∈ I . Note that pt I ∧ is a terminal object of I ∧ . We deﬁne a set, called the projective limit of β, by lim β = Hom I ∧ (pt I ∧ , β) . 2) lim β ←− i {xi }i ∈ β(i), i∈I , β(i), and it is immediately checked that: β(i) ; β(s)(x j ) = xi for all s ∈ Hom I (i, j) . i Since I and β(i) are small, lim β is a small set. The next result is obvious. 1. Let β : I op − → Set be a functor and let X ∈ Set. There is a natural isomorphism ∼ → lim Hom Set (X, β) , Hom Set (X, lim β) − ←− ←− → Set, i → Hom Set (X, β(i)).

7. 12) Y G X0 X1 G Z. 12) is co-Cartesian if X 0 Y X 1 − → Z. 12) is Cartesian if Y −→ X 0 × Z X 1 . Assume that C admits ﬁnite coproducts. 2): Y ⇒ X0 X1 − →Z. Assume that C admits ﬁnite products. 12) is Cartesian if and only if the sequence below is exact: Y − → X0 × X1 ⇒ Z . 8. Let f : X − → Y be a morphism in a category C. (i) Assume that C admits ﬁber coproducts and denote by i 1 , i 2 : Y ⇒ Y X Y → Y (or simply σ ) the natural the coprojections. We denote by σY : Y X Y − → Y , that is, σY ◦ i 1 = σY ◦ i 2 = idY .

Let I be a category and assume that C admits inductive limits indexed by I . , (i) If A : C op − lim A(X i ) for any inductive system {X i }i∈I in C), then C A A(lim X i ) ←− −→ i∈I i∈I admits inductive limits indexed by I and j A : C A − → C commutes with such limits. (ii) If a functor F : C − → C commutes with inductive limits indexed by I , →C then for any Y ∈ C , CY admits inductive limits indexed by I and CY − commutes with such limits. Proof. (i) Let {(X i , u i )}i∈I be an inductive system in C A indexed by I .