By Chris Hillman
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3. Prove the identity 1M 1N = 1M N . 4. Show that Hom(X; M N) is in bijection with Hom(X; M) Hom(X; N). 5. Now suppose there is a nal object F in C, and show that M F is always isomorphic to M. 6. Conclude that the objects 1M ; 1M 1F are isomorphic in X=C. 7. Show that the isomorphism classes of objects in X=C form a commutative monoid. Exercise: Fix an arrow E ! F of C. A CATEGORICAL PRIMER 31 1. Suppose that pushouts always exist in C. Show that we obtain a functor from E=C to F=C, called the coslice change functor, as follows.
Y ?? y Verify that here ; are the components of a natural transformation. (The rst functor takes the nonidentity arrows of the category Pair to the arrows X ';! Y of; C, while the second takes the nonidentity arrows of Pair to the ' arrows A ! ) Verify that the map taking an arrow X ! 1???! ?? y' '? 1???! Y Y ;1Y de nes a functor, called the diagonal functor, from C to C## . 2. De ne a category with three objects U; V; W and ve arrows 1U ; 1V ; 1W , and U ! W V . Verify that this is in fact a category.
G F A A ???? y commutes. GB ?? yG GB This provides the categorical way of understanding the hierarchy of structure in mathematics. In general, whenever we have a forgetful functor G from B to A, we have a left adjoint F which augments (if neccessary) and places just the right structure on an object X of A to make it into an object of B. Such left adjoints are called free constructions. Exercise: let G be the forgetful functor from RMod to Set. Show that it has a left adjoint F , where F X is the free R-module over the set X.