By Andrea Asperti

Type conception is a mathematical topic whose value in numerous components of machine technology, so much particularly the semantics of programming languages and the layout of courses utilizing summary facts forms, is greatly stated. This publication introduces classification concept at a degree acceptable for laptop scientists and offers functional examples within the context of programming language layout.

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Two categories C and D are equivalent if and only if there are two functors F: C®D and G : D®C such that G ° F @ idC and F ° G @ idD (note that C is isomorphic to D iff G ° F = idC and F ° G = idD). 3 Proposition Let F,G : C® D be functors and t : F® G be a natural transformation from F to G. Assume that, for each aÎ ObC , ta Î D[F(a),G(a)] is an isomorphism. Then t is a natural isomorphism. 47 3. Functors and Natural Transformations Proof Define t-1 : G®F by t-1a = (ta)-1. t-1 is natural, since "fÎC[a,b] t-1b ° G(f) = tb-1 ° G(f) ° ta ° ta-1 = tb-1 ° tb ° F(f) ° ta-1 = G(f) ° t-1a.

1 Definition A coherent structure is a pair (|X|,- ), where |X| is a set and - is a binary, reflexive, symmetric relation on |X|. The elements of |X| are called points, and the relation - is called coherence. The coherent domain associated with (|X|,- ) is the collection X of subsets of P(|X|) whose points are pairwise coherent. The elements of X are ordered by set-inclusion. Coherence is extended to X in the obvious way, that is: A - B iff AÈBÎX. Exercise Prove, when X is a coherent domain, that 1.

Note that C´ ´ C[_,(a,b)] ° then < , > is a natural isomorphism from C´ D and C[_,a´b]: C®Set are contravariant functors. Conversely suppose to have for all objects a ´ C[_,(a,b)] ° D @ C[_,a´b]. The naturality and b an object a´b and a natural isomorphism t : C´ of t-1 is expressed by the following commutative diagram: 51 3. Functors and Natural Transformations Let (q1,q2) = t-1a´b(ida´b) . Note that q1: a´b®a, q2: a´b®b . We want to prove that a´b is a product with projections q1 and q2. Indeed, let f = tc(h,g): c®a´b in the above diagram, then we have (q1,q2) ° ( tc(h,g), tc(h,g) ) = t-1c(ida´b ° tc(h,g) ) = (h,g) and, in particular, q1 ° tc(h,g) = h ; q2 ° tc(h,g) = g .