- #1
Don Aman
- 73
- 0
I'm looking for help constructing the natural isomorphism between [itex]V\otimes V^*[/itex] and [itex]\operatorname{End}(V)[/itex], with V a vector space.
So far, I think I should have functors F and G which take [itex]V \mapsto V\otimes V^*[/itex] and [itex]V \mapsto \operatorname{End}(V)[/itex]. I'm having a little trouble figuring out how the functors should act on morphisms though. For example, the only sensible thing that I can get F(f) to be is the morphism [itex]v\otimes \sigma \mapsto f(v)\otimes (f^{-1})^*\sigma[/itex]. Only, here I have to assume that f is invertible, which I don't want. The functor should be defined for all morphisms, right?
thanks
-Don
So far, I think I should have functors F and G which take [itex]V \mapsto V\otimes V^*[/itex] and [itex]V \mapsto \operatorname{End}(V)[/itex]. I'm having a little trouble figuring out how the functors should act on morphisms though. For example, the only sensible thing that I can get F(f) to be is the morphism [itex]v\otimes \sigma \mapsto f(v)\otimes (f^{-1})^*\sigma[/itex]. Only, here I have to assume that f is invertible, which I don't want. The functor should be defined for all morphisms, right?
thanks
-Don