By Chris Hillman
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Additional info for A Categorical Primer
9 dom ????! X ?? y 1 > ????! y Note that if we choose a di erent representative of A = ], the diagram dom ????! X ?? y 1 >! ?? y must still be a pullback square, so = . In short, characteristic arrows describe subobjects (as they should), and we can write A instead of . As this little argument tends to emphasize, the condition that (15) be a pullback square is much stronger than merely requiring that (15) commute. As usual, a classifying object might not exist, but if it does, it is unique up to isomorphism.
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. De ne "(('jEx ; x); e) = '(e) and verify this is an arrow of Bn X.
B 1X 1X X X ???? X ????! X X X ???? X ????! X 1 1 A ????! T ???? B ' T^ ?? y ?? y ?? y X X ?? y ?? y ?? y pushout diagonal diagonal pullback 5. Verify that the direct limit over J and the inverse limit over J (if they exist) de ne respecively right and left adjoints to the diagonal functor from C to CJ , as indicated in the diagrams: CJ C CJ C lim ! Xj ?? y jk ! Xk Xj ???? 'j ? y ?? y'k N N ???? N 1 direct limit diagonal N 1M M M ????! 'j ? y ?? y'k Xj ????! Xk jk diagonal M ?? y lim Xj inverse limit Exercise: let C be a category.
A Categorical Primer by Chris Hillman