# 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)

## Publications

Sebastian Posur, On free abelian categories for theorem proving, Journal of Pure and Applied Algebra (2022). |

This article explores the following natural idea:
every additive category freely generates an abelian one,
and computing in free mathematical structures amounts to theorem proving.
Thus, computing in free abelian categories should enable us to computationally
prove theorems in homological algebra. For a quick overview see my slides of a talk that I gave at the
representation theory and related topics seminar.

Martin Bies, Sebastian Posur, Tensor products of finitely presented functors, Journal of Algebra and its Applications (2021). |

We give a unified approach for the implementation of tensor products in various contexts
via the study of right exact monoidal structures on the category of finitely presented functors.
In contrast to the well-known Day convolution of arbitrary functors, the monoidal structures in the finitely presented case
do not need to be closed in general, but we find some useful sufficient criteria.

Sebastian Posur, Methods of constructive category theory, Representations of Algebras, Geometry and Physics, Contemporary Mathematics, vol. 769, Amer. Math. Soc., Providence, RI, (2021), pp. 157-208. |

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.

Sebastian Posur, A constructive approach to Freyd categories, Applied Categorical Structures (2021). |

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. The implementation of the so-called Freyd categories
in CAP makes 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, Closing the category of finitely presented functors under images made constructive, Compositionality (2020). |

An abstraction of the module data structure used by the CAS Macaulay2 to the level of functors clarifies when the category of finitely presented functors admits images: this is the case if and only if it is abelian.

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.

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, 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.

## Publications (non-refereed)

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

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

## Preprints

Johannes Flake, Robert Laugwitz and Sebastian Posur, Indecomposable objects in Khovanov-Sazdanovic’s generalizations of Deligne’s interpolation categories. |

We discuss categorical tools for the classification problem of indecomposable objects in Karoubian tensor categories.
These tools include: filtrations, gradings, field extensions and Galois descent for such categories.
We apply these tools to *Khovanov-Sazdanovic’s recent generalizations of Deligne’s interpolation categories*.
Our chosen categorical point of view allows us to deduce results with a high level of generality:
for example, we are able to determine the graded Grothendieck ring of Deligne’s interpolation categories over arbitrary fields,
and the case of positive characterstic provides a natural setup for symmetric functions in the modular case.
For a quick introduction, see the following talk notes (fifth annual meeting of the DFG collaborative research center SFB-TRR 195).

Sebastian Posur, A constructive approach to Fourier-Mukai transforms for projective spaces via A-infinity-functors between pretriangulated dg categories. |

The deficiencies of triangulated categories, like the failure of cones to be functorial,
make it impossible to specify an exact functor only on generators in a naive sense,
which is unsatisfactory from a computational point of view.
However, the world of dg and A-infinity categories nicely rectifies the situation. In this preprint,
we provide an explicit formula for the universal property of pretriangulated hulls and apply it to the creation of Fourier-Mukai transforms.

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.

## 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. |