Terminology: Why is it called sheaf COhomology? I just learned the definition of sheaf cohomology as the derived functors of the global sections functor.  I have a question about terminology - why is it called sheaf COhomology and not just homology - since it is after all a sequence of covariant functors (like homology!).  I was under the impression that the CO in cohomology stood for NOT covariant....
Thanks!

Because sheaf cohomology is a direct generalization of singular cohomology with coefficients.  The idea is that instead of taking cohomology with (global) coefficients in some fixed abelian group $A$, we take cohomology with (local) coefficients in some sheaf.
To be precise, if $X$ is a (sufficiently nice) topological space, $A$ is an abelian group, and we write $\underline{A}$ for the constant $A$-valued sheaf on $X$, then the sheaf cohomology $H^i(X, \underline{A})$ agrees with the singular cohomology $H_{\rm sing}^i(X, A)$.
Or, if you prefer: by comparing the functoriality of $H_{\rm sing}^i(-, A)$ with the functoriality of $H^i(X, -)$, you're comparing apples to oranges.
By the way, cohomology is called cohomology because it's (Poincare) dual to homology.  The term predates category theory by at least a decade.
