In this thesis we design a framework for computing in (abelian) categories in a structured manner, inspired by constructions in category theory.
We start by giving necessary definitions for a category to be computable in the sense of this thesis. This includes the requirements on the data structure for objects and morphisms, and the specifications of categorical operations which need to be implemented.
As a first example, we provide data structures and algorithms to show how the category of finitely presented graded modules over a graded computable ring can be implemented in this context.
Then we describe the category of Serre morphisms of an abelian category. It provides an example of the flexibility a categorical framework offers for the implementation of abelian categories. The category of Serre morphisms will then be used, together with the previously described implementation of f.p. graded modules, to implement the category of coherent sheaves over a normal toric variety. To achieve this, we present an algorithm to compute the graded parts of a f.p. graded module over a Laurent polynomial ring, the latter graded by a finitely presented abelian group.
As application of this axiomatic computational setup for both f.p. graded modules and coherent sheaves over toric varieties, we describe a categorical algorithm to compute a grade-compatible presentation of a f.p. graded module and a coherent sheaf.
A realization of the categorical framework to implement computable categories was created alongside this thesis: CAP (Categories, Algorithms, Programming). All concepts and algorithms presented in this thesis are implemented in CAP. In the last chapter of the thesis, some technical concepts of CAP are explained and motivated.