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

By Michael Barr, Charles Wells

The basic options of type 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 observe their development. a large insurance of themes in class thought and desktop technology is constructed together with introductory remedies of cartesian closed different types, sketches and trouble-free specific version idea, and triples. The presentation is casual with proofs integrated simply after they are instructive, supplying a extensive assurance of the competing texts on classification conception in machine technological know-how.

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0 FF {3 For f : C ¡ ! D in and u : Y ¡ ! 9 Proposition Let P : ¡ ! be a ¯bration with cleavage ·. For any arrow f : C ¡ ! D in , F f : F (C) ¡ ! F (D) as de¯ned by FF{1 through FF{3 is a functor. Moreover, if ° is a splitting, then F is a functor from op to Cat. 8 and is left as an exercise. 10 Exercises 1. Verify that for any functor P : object C is a subcategory of . 2. 9. ¡ ! 2 The Grothendieck construction 43 3. Let Á : Z4 ¡ ! 9. a. Show that the functor from C(Z4 ) to C(Z2 ) induced by Á is a ¯bration and an op¯bration.

If has pullbacks, this functor is a ¯bration. For a given f : C ¡ ! D in and object k : B ¡ ! D of , a cartesian arrow for f and k is any (u; f ) given by a pullback P u - u0 ? C f - B k ? 2) The veri¯cation is left as an exercise (Exercise ES 4). 6 Fibers For any functor P : ¡ ! , the ¯ber over an object C of is the set of objects X for which P (X) = C and arrows f for which P (f) = idC . It is easy to verify that this ¯ber is a subcategory of (Exercise ES 1). 4, the ¯bers are all the same: each one is isomorphic to the category .

It follows that P : ¡ ! is a ¯bration if and only if P op : op ¡ ! op is an op¯bration. If P : ¡ ! is a ¯bration, one also says that is ¯bered over . In that case, is the base category and is the total category of the ¯bration. 3 De¯nition A cleavage for a ¯bration P : ¡ ! is a function ° that takes an arrow f : C ¡ ! D and object Y such that P (Y ) = D to an arrow °(f; Y ) of that is cartesian for f and Y . D and X such that P (X ) = C to an arrow ·(f; X) that is opcartesian for f and X. The cleavage ° is a splitting of the ¯bration if it satis¯es the following two requirements.

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