A Categorical Primer by Chris Hillman

By Chris Hillman

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Maximal element 1 ! , called false, is the characteristic arrow of the unique Here, the arrow 1 ! arrow 0 ! 1, where 0 is of course an initial object of T. ) Sketch of proof: the basic idea is to apply the Yoneda Embedding Theorem. Begin by observing that we have a mapping t Sub(X) Sub(X) Sub(X) ?! But Sub(X) is in bijection with Hom(X; ), so this may be considered a mapping t Hom(X; ) Hom(X; ) Hom(X; ) ?! or, since Hom(X; ) Hom(X; ) is in (natural) bijection with Hom(X; ), a mapping t \X ( )\ X ?!

Next, dualize the diagram de ning an exponential. Observe that in P A E is a sum. Show that the sum functor ( ) E has a left adjoint ( ) n E which is \dual" to exponentiation in P. The natural bijections are indicated by the diagrams P P P P A\E ?? y A ?? y B Ec ( ) \ E ( )E B AnE ?? y A ?? y B E B ( )nE ( ) E 42 CHRIS HILLMAN 2. Now consider the product category P P. Verify that the map taking the arrow A B of P to the arrow (A; A) (B; B) of P P de nes a functor, called the diagonal functor. Show that union and intersection (both de ned as functors from P P to P) give left and right adjoints, respectively, of the diagonal functor, as indicated by diagrams P P P P P P A B (A; B) (A; A) A C (C; C) (B; C) B\C # # # union diagonal # diagonal intersection Exercise: we will show that all the operations within a given category C which were discussed in Sections 4 and 5 are all examples of adjoint functors.

Exercise: suppose E ! X and F ! X are objects of Bn X. We de ne a new bundle F E ! X as follows. First, given x 2 X, the stalk (F E )x is (F E )x = ('jEx; x) : Ex ! Fx ; such that ' 2 Hom (E; F) Bn X In other words, (F E )x consists of the restrictions to the stalk Ex of the various bundle morphisms ', where each such restriction has been labled by the point x. ) Next de ne F E = ]x2X (F E )x and set ('jEx ; x) = x. Give F E the pullback topology from X via . Verify F E ! X is an object of Bn X.

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