73 research outputs found
Natural Associativity and Commutativity
Paper presented in three lectures in Anderson Hall on September 23, 24, 26, 196
Symmetry groupoids and patterns of synchrony in coupled cell networks
A coupled cell system is a network of dynamical systems, or âcells,â coupled together. Such systems
can be represented schematically by a directed graph whose nodes correspond to cells and whose
edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that
preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized
cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only
mechanism that can create such states in a coupled cell system and show that it is not.
The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information
about the input sets of cells. (The input set of a cell consists of that cell and all cells
connected to that cell.) The admissible vector fields for a given graphâthe dynamical systems with
the corresponding internal dynamics and couplingsâare precisely those that are equivariant under
the symmetry groupoid. A pattern of synchrony is ârobustâ if it arises for all admissible vector
fields. The first main result shows that robust patterns of synchrony (invariance of âpolydiagonalâ
subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an
equivalence relation on cells is âbalanced.â The second main result shows that admissible vector
fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled
cell network, the âquotient network.â The existence of quotient networks has surprising implications
for synchronous dynamics in coupled cell systems
Innocent strategies as presheaves and interactive equivalences for CCS
Seeking a general framework for reasoning about and comparing programming
languages, we derive a new view of Milner's CCS. We construct a category E of
plays, and a subcategory V of views. We argue that presheaves on V adequately
represent innocent strategies, in the sense of game semantics. We then equip
innocent strategies with a simple notion of interaction. This results in an
interpretation of CCS.
Based on this, we propose a notion of interactive equivalence for innocent
strategies, which is close in spirit to Beffara's interpretation of testing
equivalences in concurrency theory. In this framework we prove that the
analogues of fair and must testing equivalences coincide, while they differ in
the standard setting.Comment: In Proceedings ICE 2011, arXiv:1108.014
Categorical Models for a Semantically Linear Lambda-calculus
This paper is about a categorical approach to model a very simple
Semantically Linear lambda calculus, named Sll-calculus. This is a core
calculus underlying the programming language SlPCF. In particular, in this
work, we introduce the notion of Sll-Category, which is able to describe a very
large class of sound models of Sll-calculus. Sll-Category extends in the
natural way Benton, Bierman, Hyland and de Paiva's Linear Category, in order to
soundly interpret all the constructs of Sll-calculus. This category is general
enough to catch interesting models in Scott Domains and Coherence Spaces
Formalizing of Category Theory in Agda
The generality and pervasiness of category theory in modern mathematics makes
it a frequent and useful target of formalization. It is however quite
challenging to formalize, for a variety of reasons. Agda currently (i.e. in
2020) does not have a standard, working formalization of category theory. We
document our work on solving this dilemma. The formalization revealed a number
of potential design choices, and we present, motivate and explain the ones we
picked. In particular, we find that alternative definitions or alternative
proofs from those found in standard textbooks can be advantageous, as well as
"fit" Agda's type theory more smoothly. Some definitions regarded as equivalent
in standard textbooks turn out to make different "universe level" assumptions,
with some being more polymorphic than others. We also pay close attention to
engineering issues so that the library integrates well with Agda's own standard
library, as well as being compatible with as many of supported type theories in
Agda as possible
Categories for the working mathematician
Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general ized monoid. Chapters VI and VII explore this notion and its generaliza tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure (given by a tensor product) lead inter alia to the study of more convenient categories of topological spaces
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