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Extra resources for A Categorical Primer

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Y y ?? y sum diagonal diagonal product On the left, denotes the unique arrow A + B ! X guaranteed by the ' UMP of a sum whenever we have arrows A ! X B, whereas on the right, A CATEGORICAL PRIMER 43 denotes the unique arrow X ! A B guaranteed by the UMP of a product whenever we have arrows A ' X ! B. '; 3. Verify that the coequalizer and equalizer objects of A ?! B (if they exist) are obtained as the left and right adjoints (respectively) of the diagonal functor from C to the category of pairs C## , as indicated by the diagrams: C C## C C## K ('; ) A ????!

A Grp-object of C is an object G together with: 1. a group product operator : G G ! G, 2. an group inversion operator : G ! G, 3. an group identity element " : 1 ! G, such that 1. e. (???! G G (1??? 1 ! commutes, x? 1G G ????! y G x? ! 1G ??? G 2. e. (???! G G (1??? G ! commutes, ?? y " 1 ????! y G ?? y! " ? 1 ??? 3. e. 1G (G G) G ????! G G ????! G ? '? y 1G G (G G) ????! G G ????! G commutes. A Grp-arrow of C is an arrow ' : G ! H between two group objects such that 1. e. "G ! G 1 ???? '?

Y! " ? 1 ??? 3. e. 1G (G G) G ????! G G ????! G ? '? y 1G G (G G) ????! G G ????! G commutes. A Grp-arrow of C is an arrow ' : G ! H between two group objects such that 1. e. "G ! G 1 ???? '? y commutes, "H ! H 1 ???? 2. e. G G G G ????! ' '? y commutes. '? y H H H H ????! Exercise: verify that Grp-arrows between Grp-objects form a subcategory of C, denoted GrpC . Show that this category has products and equalizers (and thus kernels in the usual sense of group theory). Verify that GrpTop is the category of A CATEGORICAL PRIMER 59 topological groups, while GrpMan is the category of Lie groups, GrpSh X is the category of sheaves of groups over X, and AbgSh X is the category of sheaves of abelian groups over X.

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