In this thesis we create a purely categorical framework for cohomology computations of G-equivariant coherent sheaves on projective space for a finite group G. For this, we develop three different sub-frameworks: First, we construct a skeletal tensor category SRep(G) equivalent to the representation category Rep(G) of G. Second, we design, in the context of an arbitrary abelian category, an algorithm for computing spectral sequences which is suitable for a direct computer implementation, i.e., it only uses categorical constructions
provided by the axioms of an abelian category. Last, we describe how to internalize the exterior algebra E and its modules in a tensor category.
Combining our three sub-frameworks yields an algorithm for computing spectral sequences within the category of E-modules internal to SRep(G). Thanks to an equivariant version of the famous BGG-correspondence, we can use such an algorithm for computing cohomology groups of G-equivariant sheaves on projective space. Furthermore, this algorithm allows us to compute a new invariant called
spectral cohomology table which in this thesis is proven to be stronger than the classical cohomology table.
Since our framework can be described in purely categorical language, a software project in GAP facilitating the implementation of abstract categories and categorical algorithms was born during the writing of this thesis: Cap (Categories, Algorithms, Programming). The categorical framework along with all algorithms presented in this thesis is implemented in Cap.