2,624 research outputs found
Recommended from our members
Category theory : definitions and examples
Category theory was invented as an abstract language for describing certain structures and constructions which repeatedly occur in many branches of mathematics, such as topology, algebra, and logic. In recent years, it has found several applications in computer science, e.g., algebraic specification, type theory, and programming language semantics. In this paper, we collect definitions and examples of the basic concepts in category theory: categories, functors, natural transformations, universal properties, limits, and adjoints
An Algebraic Characterisation of Concurrent Composition
We give an algebraic characterization of a form of synchronized parallel
composition allowing for true concurrency, using ideas based on Peter Landin's
"Program-Machine Symmetric Automata Theory".Comment: This is an old technical report from 1981. I submitted it to a
special issue of HOSC in honour of Peter Landin, as explained in the Prelude,
added in 2008. However, at an advanced stage, the handling editor became
unresponsive, and the paper was never published. I am making it available via
the arXiv for the same reasons given in the Prelud
Signatures and Induction Principles for Higher Inductive-Inductive Types
Higher inductive-inductive types (HIITs) generalize inductive types of
dependent type theories in two ways. On the one hand they allow the
simultaneous definition of multiple sorts that can be indexed over each other.
On the other hand they support equality constructors, thus generalizing higher
inductive types of homotopy type theory. Examples that make use of both
features are the Cauchy real numbers and the well-typed syntax of type theory
where conversion rules are given as equality constructors. In this paper we
propose a general definition of HIITs using a small type theory, named the
theory of signatures. A context in this theory encodes a HIIT by listing the
constructors. We also compute notions of induction and recursion for HIITs, by
using variants of syntactic logical relation translations. Building full
categorical semantics and constructing initial algebras is left for future
work. The theory of HIIT signatures was formalised in Agda together with the
syntactic translations. We also provide a Haskell implementation, which takes
signatures as input and outputs translation results as valid Agda code
- …