String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of monoids, allowing computations on those and their manipulation by a computer. Even better, when the presentation is confluent and term
inating, they provide one with a notion of canonical representative of the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. One of the main purposes of this article is to give a progressive introduction to the notion of higher-dimensional rewriting system provided by polygraphs, and describe its links with classical rewriting theory, string and term rewriting systems in particular. After introducing the general setting, we will be interested in proving local confluence for polygraphs presenting 2-categories and introduce a framework in which a finite 3-dimensional rewriting system admits a finite number of critical pairs.
When working with distant collaborators on the same documents, one often uses a version control system, which is a program tracking the history of files and helping importing modifications brought by others as patches. The implementation of such a sy
stem requires to handle lots of situations depending on the operations performed by users on files, and it is thus difficult to ensure that all the corner cases have been correctly addressed. Here, instead of verifying the implementation of such a system, we adopt a complementary approach: we introduce a theoretical model, which is defined abstractly by the universal property that it should satisfy, and work out a concrete description of it. We begin by defining a category of files and patches, where the operation of merging the effect of two coinitial patches is defined by pushout. Since two patches can be incompatible, such a pushout does not necessarily exist in the category, which raises the question of which is the correct category to represent and manipulate files in conflicting state. We provide an answer by investigating the free completion of the category of files under finite colimits, and give an explicit description of this category: its objects are finite sets labeled by lines equipped with a transitive relation and morphisms are partial functions respecting labeling and relations.
Generating multimedia streams, such as in a netradio, is a task which is complex and difficult to adapt to every users needs. We introduce a novel approach in order to achieve it, based on a dedicated high-level functional programming language, calle
d Liquidsoap, for generating, manipulating and broadcasting multimedia streams. Unlike traditional approaches, which are based on configuration files or static graphical interfaces, it also allows the user to build complex and highly customized systems. This language is based on a model for streams and contains operators and constructions, which make it adapted to the generation of streams. The interpreter of the language also ensures many properties concerning the good execution of the stream generation.
Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide
one with a notion of canonical representative for the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. Here, we are interested in proving confluence for polygraphs presenting 2-categories, which can be seen as a generalization of term rewriting systems. For this purpose, we propose an adaptation of the usual algorithm for computing critical pairs. Interestingly, this framework is much richer than term rewriting systems and requires the elaboration of a new theoretical framework for representing critical pairs, based on contexts in compact 2-categories.
Game semantics provides an interactive point of view on proofs, which enables one to describe precisely their dynamical behavior during cut elimination, by considering formulas as games on which proofs induce strategies. We are specifically intereste
d here in relating two such semantics of linear logic, of very different flavor, which both take in account concurrent features of the proofs: asynchronous games and concurrent games. Interestingly, we show that associating a concurrent strategy to an asynchronous strategy can be seen as a semantical counterpart of the focusing property of linear logic.
