Monday, November 11, 2002

Adjointness refresher! A pair of functors F:A->X, G:X->A is an adjunction with F the left adjoint and G the right adjoint if there is an isomorphism between all Hom(Fa,x) in X and Hom(a,Gx) in A, for all x in X and a in A. An adjunction has a unit which you can visualize as the set of (universal) arrows you get in A by "pinching" Fa and x in X until they become the same object, i.e. it's the arrow in each set Hom(a,Gx) (in A) that is isomorphic to id(Fa) in X. You can also think of these as a set of arrows n:a->GFa for each a in A, i.e. the arrows you get when you take each object, run it through F, then run the resulting object "back" through G. Similarly, if you "pinch" a and Gx together in A, the counit of the adjunction is the set of ("contravariant") arrows you get from FGx to x in X. I.e. take an object x in X, run it "up" through G, then "back down" through F. Now take an arrow from the resulting object back to your original x. Those arrows for all x in X is the counit, which (like the unit) is a natural transformation. finish... 11:47:11 AM