Cubical Sets and Trace Monoid Actions

Ahmet A Husainov

doi:10.1155/2013/285071

Abstract

This paper is devoted to connections between trace monoids and cubical sets. We prove that the category of trace monoids is isomorphic to the category of generalized tori and it is a reflective subcategory of the category of cubical sets. Adjoint functors between the categories of cubical sets and trace monoid actions are constructed. These functors carry independence preserving morphisms in the independence preserving morphisms. This allows us to build adjoint functors between the category of weak asynchronous systems and the category of higher dimensional automata.

Highlights

In this paper, it is established that the category of generalized tori is isomorpic to the category of trace monoids and basic homomorphisms
The map M(E, I) 󳨃→ TM(E, I) extends to a functor T : FPCM → Set◻op which assigns to each basic homomorphism f : M(E, I) → M(E󸀠, I󸀠) a morphism of cubical sets given by a family of maps Tnf : TnM(E, I) → TnM(E󸀠, I󸀠) defined as
We have considered the category of trace monoids and basic homomorphisms and proved that this category has all colimits

Summary

Introduction

It is established that the category of generalized tori is isomorpic to the category of trace monoids and basic homomorphisms. Adjoint functors between the category of cubical sets and the category of trace monoids acting on sets are constructed. B s2 consisting of three states and two transitions with the independence relation I = {(a, b), (b, a)} does not satisfy the confluence condition and is not an automation with concurrency relation It remains an open problem of existence of a left adjoint functor from the category of higher dimensional automata in the category containing asynchronous systems which are not confluent. We construct the adjoint functors between the categories of trace monoids and cubical sets. Section we start with the construction of adjoint functors between the category of trace monoids acting on sets and the category of cubical sets. We build adjoint functors between the categories of asynchronous systems and higher dimensional automata

Trace Monoids and Cubical Sets

Trace Monoid Actions

Conclusion