# Research

My research aims at enhancing the toolbox of computer algebra with methods and concepts of category theory. The resulting powerful combination of these two areas of research is what I like to call **constructive category theory**.

## CAP project

CAP (Categories, Algorithms, Programming) is a software project for constructive category theory written in GAP. It facilitates both the realization of specific instances of categories and the implementation of generic categorical algorithms. With my colleague Sebastian Gutsche I gave an introduction at the Université Paris Diderot to constructive category theory and its applications to homological algebra. CAP is the cornerstone of my current research for experimenting and computing.

### Trying CAP in a Jupyter notebook

### CAP tutorials

- Worksheet: Category theory in computer algebra: first steps in CAP
- Worksheet: Computing with generalized morphisms

### CAP Days (conference)

### CAP related publications

Sebastian Gutsche, Sebastian Posur, Øystein Skartsæterhagen, On the Syntax and Semantics of CAP, In: O. Hasan, M. Pfeiffer, G. D. Reis (eds.): Proceedings of the Workshop Computer Algebra in the Age of Types,Hagenberg, Austria, 17-Aug-2018, published at http://ceur-ws.org. |

Sebastian Gutsche, Sebastian Posur, CAP: categories, algorithms, programming, Computeralgebra-Rundbrief (2019), Ausgabe 64, 14-17. |

## Publications and preprints

Sebastian Posur, A constructive approach to Freyd categories. |

Peter Freyd’s universal way of equipping an additive category with cokernels
can be seen as an abstraction of the usual model of finitely presented modules
in computer algebra. An implementation of the so-called Freyd categories
in CAP will make it possible to compute even with finitely presented functors.
I gave a talk about this idea at the
first annual meeting of the DFG collaborative research center SFB-TRR 195.
The slides are available here.

Sebastian Posur, Linear systems over localizations of rings, Arch. Math. (2018) 111: 23. |

Computable rings are exactly those rings for which
the category of finitely presented modules is computable abelian
(via the Freyd category construction).
In this paper I discuss the computability of localizations of rings.

Sebastian Posur, Constructing equivariant vector bundles via the BGG correspondence, Journal of Symbolic Computation 91 (2019), 57 – 73, MEGA 2017, Effective Methods in Algebraic Geometry, Nice (France), June 12-16, 2017. |

In this paper I apply the computational framework for equivariant graded modules over the exterior algebra that I established in my PhD thesis to the problem of constructing vector bundles on projective space. The corresponding CAP packages are

- GroupRepresentationsForCAP: an implementation of a skeletal version of the semisimple tensor category of finite dimensional group representations
- InternalExteriorAlgebraForCAP:
an implementation of the category of internal modules over an internalized version of the exterior algebra

Sebastian Posur, Atom spectra of graded rings and sheafification in toric geometry, Journal of Algebra (2019), Volume 534, 207 – 227. |

The theory of atom spectra of abelian categories was introduced by Ryo Kanda.
Applied to the category of finitely presented graded modules,
this theory yields a geometric criterion for deciding which objects
sheafify to zero in the context of toric geometry,
and will hopefully lead to improvements of algorithmic approaches
in the future.

Mohamed Barakat, Markus Lange-Hegermann and Sebastian Posur, Elimination via saturation. |

One can eliminate using an arbitrary monomial order.
In particular, no elimination order is needed.

Sebastian Posur, Methods of constructive category theory. |

This paper serves as an introduction to constructive category theory.
It is shown how the philosophy of category constructors makes it
possible to compute sets of natural transformations
between finitely presented functors. Moreover, it is demonstrated how to
perform diagram chases in homological algebra in a constructive way.

Martin Bies, Sebastian Posur, Tensor products of finitely presented functors. |

We give a unified approach for the implementation of tensor products in various contexts.

## Theses

Constructive Category Theory and Applications to Equivariant Sheaves, PhD thesis, Siegen University. |

G-equivariant Coherent Sheaf Cohomology, Master thesis, RWTH-Aachen University. |

Axiomatik für lineare Differentialgleichungssysteme mit konstanten Koeffizienten und Anwendungen auf Invariantentheorie, Bachelor thesis, RWTH-Aachen University. |

## Non-refereed publications

Sebastian Posur, The search for low rank vector bundles, MFO report no. 25 (2013). |