The Locus project

Locus is a project for creating a mathematical expert system based upon topos theory. The key to creating a mathematical model to computation is in the congruence lattices of objects of copresheaf topoi. We develop the first topos based computer algebra system in order to realise this new model and vision. A copresheaf based ontology and knowledge based system is provided to bring everything together.

User manual overview:

• Preliminaries
  • Introduction
  • Copresheaf viewer
• Topoi as foundations
  • Mathematical overview
  • The topos of sets
  • The topos of functions
  • The topos of pairs of sets
  • The topos of bijections
• Topos theory of computation
  • The geometry of computation
  • Memory locations and data dependencies
  • Abstract formalisation of lenses
  • Product decompositions and flow models
  • Representing knowledge about computation
• Elementary topos theory
  • The topos of quivers
  • Lattices and partial orders
  • Semigroups, monoids, and groups
  • MSet topoi
  • Category theory
  • Copresheaves over arbitrary categories
  • Incidence functors and set systems
  • Higher order quivers and relations
  • Functional dependencies as copresheaves
  • Simplicial methods
• Grothendeick topos theory
  • Sites
  • Sheaves
• Topos theory of structure
  • Multi-sorted structures
  • Topos theoretic foundations of algebra
• Ontology
  • Classification of copresheaves
• Interfaces
  • Apache commons math