11 research outputs found
Cofibrant complexes are free
We define a notion of cofibration among n-categories and show that the
cofibrant objects are exactly the free ones, that is those generated by
polygraphs.Comment: 16 page
A folk model structure on omega-cat
We establish a model structure on the category of strict omega-categories.
The constructions leading to the model structure in question are expressed
entirely within the scope of omega-categories, building on a set of generating
cofibrations and a class of weak equivalences as basic items. All object are
fibrant while cofibrant objects are exactly the free ones. Our model structure
transfers to n-categories along right-adjoints, for each n, thus recovering the
known cases n = 1 and n = 2.Comment: 33 pages, expanded version of the original 17 pages synopsis, new
sections adde
Operads, clones, and distributive laws
International audienceWe show how non-symmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a (pseudo-)monad on the bicategory of categories and profunctors. We also explain how other previous categorical analyses of operads (via Day's tensor products, or via analytical functors) fit with the profunctor approach
Operads, clones, and distributive laws
International audienceWe show how non-symmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a (pseudo-)monad on the bicategory of categories and profunctors. We also explain how other previous categorical analyses of operads (via Day's tensor products, or via analytical functors) fit with the profunctor approach
A diagrammatic approach to networks of spans and relations
In this thesis we exhibit nondeterministic semantics for various classes of circuits. Motivated initially by quantum circuits, we also give nondeterministic semantics for circuits for classical mechanical systems and Boolean algebra. More formally, we interpret these classes of circuits in terms of categories of spans or relations: in less categorical terms these are equivalent to matrices over the natural numbers or the Boolean semiring. In the relational picture, we characterize circuits in terms of which inputs and outputs are jointly possible; and in the spans picture, how often inputs and outputs are jointly possible. Specifically, we first show that the class of circuits generated by the Toffoli gate as well the states |0⟩, |1⟩, √2|+⟩ and their adjoints is characterized in terms of spans of finite sets. We also give a complete axiomatization for these circuits. With this semantics in mind, we discuss the connection to partial and reversible computation. Shifting to the phase-space picture we also characterize circuits in terms of how they relate abstract positions and momenta. We show how this gives a unifying relational semantics for certain classes circuits for classical mechanical systems, as well as for stabilizer quantum circuit
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Formalizing graphical notations
The thesis describes research into graphical notations for software engineering, with a principal interest in ways of formalizing them. The research seeks to provide a theoretical basis that will help in designing both notations and the software tools that process them.
The work starts from a survey of literature on notation, followed by a review of techniques for formal description and for computational handling of notations. The survey concentrates on collecting views of the benefits and the problems attending notation use in software development; the review covers picture description languages, grammars and tools such as generic editors and visual programming environments. The main problem of notation is found to be a lack of any coherent, rigorous description methods. The current approaches to this problem are analysed as lacking in consensus on syntax specification and also lacking a clear focus on a defined concept of notated expression.
To address these deficiencies, the thesis embarks upon an exploration of serniotic, linguistic and logical theory; this culminates in a proposed formalization of serniosis in notations, using categorial model theory as a mathematical foundation. An argument about the structure of sign systems leads to an analysis of notation into a layered system of tractable theories, spanning the gap between expressive pictorial medium and subject domain. This notion of 'tectonic' theory aims to treat both diagrams and formulae together.
The research gives details of how syntactic structure can be sketched in a mathematical sense, with examples applying to software development diagrams, offering a new solution to the problem of notation specification. Based on these methods, the thesis discusses directions for resolving the harder problems of supporting notation design, processing and computer-aided generic editing. A number of future research areas are thereby opened up. For practical trial of the ideas, the work proceeds to the development and partial implementation of a system to aid the design of notations and editors. Finally the thesis is evaluated as a contribution to theory in an area which has not attracted a standard approach
Higher operads, higher categories
Higher-dimensional category theory is the study of n-categories, operads,
braided monoidal categories, and other such exotic structures. It draws its
inspiration from areas as diverse as topology, quantum algebra, mathematical
physics, logic, and theoretical computer science. This is the first book on the
subject and lays its foundations.
Many examples are given throughout. There is also an introductory chapter
motivating the subject for topologists.Comment: Book, 410 page
Polygraphs: From Rewriting to Higher Categories
Polygraphs are a higher-dimensional generalization of the notion of directed
graph. Based on those as unifying concept, this monograph on polygraphs
revisits the theory of rewriting in the context of strict higher categories,
adopting the abstract point of view offered by homotopical algebra. The first
half explores the theory of polygraphs in low dimensions and its applications
to the computation of the coherence of algebraic structures. It is meant to be
progressive, with little requirements on the background of the reader, apart
from basic category theory, and is illustrated with algorithmic computations on
algebraic structures. The second half introduces and studies the general notion
of n-polygraph, dealing with the homotopy theory of those. It constructs the
folk model structure on the category of strict higher categories and exhibits
polygraphs as cofibrant objects. This allows extending to higher dimensional
structures the coherence results developed in the first half
Homologie polygraphique des syst\`emes locaux
In this article, we introduce a notion of polygraphic homology of a strict
-category with coefficients in a local system, generalizing the
polygraphic homology with coefficients in , introduced by
Fran\c{c}ois M\'etayer. We show that the homology of a simplicial set with
coefficients in a local system coincides with the polygraphic homology of its
image by the left adjoint of the Street nerve with coefficients in the
corresponding local system. We define in this framework a comparison morphism
between the polygraphic homology of a strict -category and the homology
of its Street nerve, and we show that this morphism is an isomorphism for
(1-)categories. This is not true for an arbitrary -category.
Nevertheless, we conjecture that for an analogous construction in the framework
of weak -categories ``\`a la Grothendieck'' we would always obtain an
isomorphism.Comment: 55 pages, in French languag