## Sebastian Gutsche

### Constructive Category Theory and Applications to Algebraic Geometry

Many structures in abstract algebra and computer algebra can be
organized as categories and many algorithms boil down to basic
categorical constructions. Using these categorical constructions as a
programming language leads to highly abstract algorithms which can be
applied to various algebraic structures. We call the concept of
writing algorithms in basic categorical constructions categorical
programming. An implementation of a categorical programming language
is realized in the Gap package CAP.

In this talk I will first describe what is necessary for an algebraic
structure to form a computable category. For many examples, e.g.,
vector spaces over certain fields, modules over certain rings, etc.,
the implementations are already available in various packages of the
CAP-project. I will then describe how such categorical organization of
data can be used to write abstract algorithms and to produce more
complicated data structures.

As an application of the latter I will use the category of finitely
presented graded modules over a graded polynomial ring to model the
category of coherent sheaves over a toric variety, and then apply the
same generic algorithms to both f.p. graded modules and coherent
sheaves.