A supplement for Category theory for computing science by Michael Barr, Charles Wells

By Michael Barr, Charles Wells

The basic recommendations of class concept are defined during this textual content which permits the reader to boost their figuring out progressively. With over three hundred routines, scholars are inspired to watch their development. a large assurance of themes in class thought and laptop technological know-how is built together with introductory remedies of cartesian closed different types, sketches and effortless specific version thought, and triples. The presentation is casual with proofs incorporated purely after they are instructive, offering a huge assurance of the competing texts on type concept in computing device technology.

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Cat, G®(x; C) = (®Cx; C) for (x; C) an object of G( ; F ) (so that C is an object of object of F C), and G®(u; f) = (®C 0u; f ) for (u; f) an arrow of G( ; F ) (so that f : C ¡ ! C 0 in in F C 0 ). and x is an and u : F f x ¡ ! x0 Note that in GR{2, ®C 0 u has domain ®C 0 (F f x), which is Gf (®Cx) because ® is a natural transformation. The veri¯cation that G® is a functor is omitted. 7 Theorem The functor F : SO( ) ¡ ! 4 is an equivalence of categories with pseudo-inverse G. There is a similar equivalence of categories between split ¯brations and contravariant functors.

This theorem allows one to think of the wreath product as generalizing triangular matrices bigger than 2 £ 2. 8 Proposition Let G : ¡ ! Cat, H : ¡ ! Cat and K : ¡ ! Cat be functors. Then there is an isomorphism of categories I making this diagram commute. wrH @ G wr(H wr K)@ ) I- ( wrG ( @ @R ª¡ Cat ¡ wrG ) wrG wr H ¡ ¡ (G wr H ) wr K Note that the standard wreath product is not associative. 9 Applications of the wreath product It is natural to wish to simulate complicated state transition systems using systems built up in some way from a small stock of simpler ones.

Thus if you wanted to require a = b ) g(f(a; b)) = h(f (a; b)) where a and b are of type A, f is an operation of type C and g and h are operations of type D, you build the graph of your sketch with arrows f : A £ A ¡ ! C, g; h : C ¡ ! D, e : E ¡ ! C, u : A ¡ ! E and d : A ¡ ! A £ A, with cones requiring A £ A to be the required product, e to be the equalizer of g and h, d to be the diagonal map, and this diagram A u ? E d- A £ A f ? - C e which requires f ± d to factor through the equalizer of g and h.

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