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By Barr M.

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2 Deﬁnition A functor F : C − → D is an equivalence of categories if there are: E–1 A functor G : D − → C. E–2 A family uC : C − → G(F (C)) of isomorphisms of C indexed by the objects of C with the property that for every arrow f : C − → C of C , G(F (f )) = uC ◦ f ◦ u−1 C . E–3 A family vD : D − → F (G(D)) of isomorphisms of D indexed by the objects of D, with the property −1 . that for every arrow g : D − → D of D, F (G(g)) = vD ◦ g ◦ vD If F is an equivalence of categories, the functor G of E–1 is called a pseudo-inverse of F .

13 Functors by commutative diagrams We express the deﬁnition of functor using commutative diagrams. Let C and D be categories with sets of objects C0 and D0 , sets of arrows C1 and D1 , and sets of composable pairs of arrows C2 and D2 , respectively. A functor F : C − → D consists of functions → D1 along with the uniquely determined function F2 : C2 − → D2 such that → D 0 , F1 : C 1 − F0 : C0 − C2 ❅ proj1 ✠ C1 ❅ proj2 ❅ ❘ ❅ F2 C1 ❄ D2 F1 proj1 ✠ ❄ D1 ❅ proj2 F1 ❅ ❅ ❘ ❄ ❅ D1 commutes. 14 Diagrams as functors In much of the categorical literature, a diagram in a category C is a functor D : E − → C where E is a category.

The method of categorical deﬁnition is close in spirit to the modern attitude of computing science that programs and data types should be speciﬁed abstractly before being implemented and that the speciﬁcation should be kept conceptually distinct from the implementation. We believe that the method of categorical deﬁnition is a type of abstract speciﬁcation which is suitable for use in many areas of theoretical computing science. This is one of the major themes of this notes. When a category C is a category of sets with structure, with the arrows being functions which preserve the structure, a categorical deﬁnition of a particular property does not involve the elements (in the standard sense of set theory) of the structure.