Can anyone please give me examples of:
1.- An exact functor other than taking the Galois group from the category of fields.
2.- A half exact functor.
3.- A contravariant right exact functor.
I know you can form some of these examples by playing with the tensor product, the Hom functor and their derived functors. I'm looking for other examples.
Let $X$ be a topological space and let $x\in X$. The operation of assigning to each sheaf $\mathscr F$ on $X$ its stalk $\mathscr F_x$ at $x$ defines an exact functor $$ (-)_x:\operatorname{Sh}(X)\to\Bbb Z\text{-}\operatorname{Mod} $$ where $\operatorname{Sh}(X)$ is the category of sheaves on $X$ and $\Bbb Z\text{-}\operatorname{Mod}$ is the category of abelian groups.
Let $\operatorname{C}(\mathcal A)$ be the category of cochain complexes in an abelian category $\mathcal A$. Then the process of taking the cohomology $H^n(C^\bullet)$ of a cocomplex $C^\bullet$ defines a half-exact functor $$ H^n:\operatorname{C}(\mathcal A)\to\mathcal A $$
For examples of contravariant right-exact functors you might find this post on mathoverflow useful.
