Locus 1.3.0

I published the next big version of Locus to github. It features the completed topos theory of categories based upon presheaf representations and compositional quivers. That is the main feature implemented in this new version. This is apparently original work, but once you think about it its pretty obvious there is no other way to thinking of categories except as presheaves of the compositional quiver type.

• Quivers, unital quivers, permutable quivers, dependency quivers, etc are all significantly improved so that you can run computations with their subobject and congruence lattices. Some of that has already been displayed here.
• Ternary quivers are now being implemented in Locus. These are like the familiar quivers used to describe graphs, except their edges have three arrows coming out of them. They are to a large extent responsible for the topos theory of abstract algebra, as binary operations are ternary quivers. Their three components are the first component, the second component, and the composite for any ordered pair. So magmas for example are ternary quivers.
• Categories are composition quivers, which are defined by a ternary quiver heading into a binary quiver consisting of morphisms and edges. The ternary quiver defines the composition binary operation of the category, and the composition of the underlying index category ensure that the ternary quiver operation is compositional on the binary quiver. This leads to the topos theory of categories, which is a great asset to us because it is best to think of categories as objects of a presheaf topos.
• By the same token we can study partial magmoids, which are the horizontal categorification of partial binary operations, magmoids, semigroupoids, groupoids, and all other related compositional structures via the topos of composition quivers. This justifies by implementation of these categorical algebraic structures in Locus, and my decision to consider them based upon topos theory to be special types of presheaves.
• Locus is going to be absolutely categorical. So for example, magmas will be magmoids, rings will be ringoids, ordered monoids will be two categories, etc. The only question is what type of structures you want to add on to a category. This new implementation of Locus sets the groundwork for that change.
• The next versions of Locus will study other presheaf related constructions on categories. The entire project is based upon presheaf foundations, and that now comes through by representing categories as presheaves.