arXiv subjects

Alexandre Fernandez

Publications and source records attributed to Alexandre Fernandez.

3 recordsLinked to original sources

arXiv2021

Accretive Computation of Global Transformations

Global transformations form a categorical framework adapting graph transformations to describe fully synchronous rule systems on a given data structure.In this work we focus on data structures that can be captured as presheaves and study the computational aspects of such synchronous rule systems.To obtain an online algorithm, a complete study of the sub-steps within each synchronous step is done at the semantic level.This leads to the definition of accretive rule systems and a local criterion to characterize these systems.Finally an online computation algorithm for theses systems is given.

cs.DC↗

arXiv2021

Cellular Automata and Kan Extensions

In this paper, we formalize precisely the sense in which the application of cellular automaton to partial configuration is a natural extension of its local transition function through the categorical notion of Kan extension. In fact, the two possible ways to do such an extension and the ingredients involved in their definition are related through Kan extensions in many ways. These relations provide additional links between computer science and category theory, and also give a new point of view on the famous Curtis-Hedlung theorem of cellular automata from the extended topological point of view provided by category theory. These relations provide additional links between computer science and category theory. No prior knowledge of category theory is assumed.

cs.DC↗

arXiv2021

The Bicategory of Open Functors

We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to mean that some external information is taken into account. For the particular use of the authors, such an open functor is described by two components: a presheaf representing the possible external influences for each input, and a classical functor from the category of elements of this presheaf to the category of results. Considering the appropriate notion of composition then leads to a bicategory. This report describes this bicategory with as little auxiliary constructions as possible and gives all the details of all the proofs needed to establish the bicategory, as explicitly as possible. Subsequent reports will give other presentations of this bicategory and compare it to other existing constructions, e.g. spans, fibrations, pseudoadjunctions, Kleisli bicategories of pseudo-monads, and profunctors (or distributors).

math.CT↗