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Stanisław Szawiel

Publications and source records attributed to Stanisław Szawiel.

6 recordsLinked to original sources
arXiv2017

Categories of Physical Processes

We study the mathematical foundations of physics. We reconstruct textbook quantum theory from a single symmetric monoidal functor $GNS : \mathbf{Phys} \longrightarrow \ast\mathbf{Mod}$, based on the Gelfand-Naimark-Segal construction and the notion of representability. We derive the probabilistic interpretation of quantum mechanics, including the Born rule, the Schrödinger and Heisenberg pictures, the relation between symmetries and group representations, and a theory of quantum Markov processes, including wave function collapse. Inclusion of the classical limit and deformation quantization is briefly sketched. Gauge symmetry and extended locality cannot currently be accommodated, due to conceptual difficulties discussed in an appendix.

math-ph↗
arXiv2012

Monads of regular theories

We characterize the category of monads on $Set$ and the category of Lawvere theories that are equivalent to the category of regular equational theories.

math.CT↗
arXiv2012

Theories of analytic monads

We characterize the equational theories and Lawvere theories that correspond to the categories of analytic and polynomial monads on Set, and hence also the categories of the symmetric and rigid operads in Set. We show that the category of analytic monads is equivalent to the category of regular-linear theories. The category of polynomial monads is equivalent to the category of rigid theories, i.e. regular-linear theories satisfying an additional global condition. This solves a problem A. Carboni and P. T. Johnstone. The Lawvere theories corresponding to these monads are identified via some factorization systems.

math.CT↗
arXiv2012

Rigidity is undecidable

We show that the problem `whether a finite set of regular-linear axioms defines a rigid theory' is undecidable.

math.LO↗
arXiv2010

The Web Monoid and Opetopic Sets

We develop a new definition of opetopic sets. There are two main technical ingredients. The first is the systematic use of fibrations, which are implicit in most of the approaches in the literature. Their explicit use leads to certain clarifications in the construction of opetopic sets and other constructions. The second is the "web monoid", which plays a role analogous to the "operad for operads" of Baez and Dolan, the "multicategory of function replacement" of Hermida, Makkai and Power. We demonstrate that the web monoid is closely related to the "Baez-Dolan slice construction" as defined by Kock, Joyal, Batanin and Mascari.

math.CT↗