By Chris Hillman

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HomC (X; A B) / Moreover, this bijection is \natural" in the sense that it respects \perturbations" of X and (A; B). ) (A0 ; B 0) in C, we have the new arrows (12) ' 0 ' 0 A0 ???? X ????! B in C; applying the UMP for A0 B 0 guarantees that there is a unique arrow A0 B 0 X 0 through which the arrows (12) factor. Naturality means that this arrow is 0 ' precisely the composition A0 B 0 ???? X we would \naturally" expect, where 0 A0 B 0 A B is the unique arrow induced by the arrows A A B ????! B B0 A0 ????

Models in a Topos A startling aspect of topos theory is that it uni es two seemingly wholly distinct mathematical subjects: on the one hand, topology and algebraic geometry, and on the other hand, logic and set theory. Saunders Mac Lane and Ieke Moerdijk 20] In this nal (rather sketchy) section, we explore brie y the idea that a (not quite arbitrary) topos can be used to model any mathematical concept whatsoever. 58 CHRIS HILLMAN We begin by examining in some detail how the notion of a group can be recast in categorical form.

X and T ! X, de ne an arrow S + T ! X, where S + T is the sum of the objects S; T of C. ) 2. Prove the identity 1S +T = 1S + 1T . 3. Show that + is isomorphic to + in C=X. 4. Show that Hom(S + T; X) is in bijection with Hom(S; X) ] Hom(T; X). 5. Now suppose that there is an initial object I in C. Show that S must be isomorphic to S + I. (Hint: show that 1S + , where I ! X S + I. ) 6. Conclude that the objects + 1X ; are isomorphic in C=X. 7. Show that isomorphism classes of objects in C=X form a commutative monoid.