Friday, April 1, 2022

Locus 0.81 changes

In order to further develop the new elementary topos theoretic foundations of computation, I have produced a new version of the Locus project. Here are some of the changes.
  • The topos theoretic foundations of computation are defined in terms of $Sets$ and $Sets^{\to}$. An abstraction layer over that is provided by set relations and flow models. This new version contains a new and complete implementation of this fundamental research into the foundations of computation. At the same time you can get the lattice of subobjects of a set relation by calling the $sub$ multimethod.
  • We provide an implementation of relational functors which are functors of the form $F : C \to Rel$ and several techniques for dealing with them. In particular, an algorithm is provided for converting a family of disets into a relational functor over the two arrow category.
  • Support for partial transformation semigroups, such as the action by atomic charts of a preorder is provided. The action of a partial transformation semigroup is defined by a relational functor, with partial transformations described as set relations
  • The conversion of morphisms of functions into copresheaves is defined basically based upon the presence of identities in either the input or output function. As a result, a morphism of functions could be presented as a triangle copresheaf if one of the two functions is an identity. This makes for a more pleasant experience with the copresheaf viewer.
  • I wrote a function for creating module categories, defined as enriched categories with each hom class being an additive group. At the same time, the category of set relations is described as a 2-category with each hom class being a poset.
  • I wrote a new function for converting topological spaces into sites. This will be useful when we further develop our Grothendeick topos theory subsystem.
  • Half the functionality of the Locus project was moved into the elementary folder, dealing strictly with elementary topoi of copresheaves. This will make the project more organized when I add support for some things related to grothendeick toposes and schemes.

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