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 (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.
CAP Days (conference)
|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
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.
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.
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
an implementation of a skeletal version of the semisimple tensor category of finite dimensional group representations
an implementation of the category of internal modules over an internalized version of the exterior algebra
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.
One can eliminate using an arbitrary monomial order.
In particular, no elimination order is needed.
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.
We give a unified approach for the implementation of tensor products in various contexts.
|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.