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