I did not understand number theory or characteristic p-algebraic geometry at all. I just know a little about frobenius homomorphism between two schemes. On the other hand, when I learned something on triangulated category. I found there was also a definition of "frobenius morphism" the definition is as follows:
There are two categories C and D. f_:D--->C, f^ :C--->D is left adjoint to f_, we call f_ is a Frobenious morphism if there exists an auto-equivalence G of C such that composition f^* G is right adjoint to f_*.
First question is:is there any relationship between this two frobenius morphism?
Second question is:does frodenius category play roles in algebraic geometry?
All the comments related to this are welcomed.