Object and morphism structures of categories

In partial algebra, we get to study a lot of structures that wouldn't be available to us otherwise in total algebra. In particular this is true of categories, which are often best studied using partial algebraic structures. Partial algebra can be used to establish structures on both the morphism and object sets of a category. To demonstrate this we can first create a category:

; the index category of the topos of quivers
(def example-category
  (n-arrow-category 2))

Then we can get the object and morphism structures of the category using Locus and its new support for partial algebraic structures. The morphic structure is a partial magma and the object structure is a partial action.

Morphism structure:
The morphic structure of the category is a partial magma which can be acquired by using the composition-operation routine. The composition operation of a category is a partial magma with a domain $R$ in which $(f,g) \in R$ provided that $Out(g) = In(f)$.

; the morphic structure of the example category
(def op 
  (composition-operation example-category))
  
; every partial algebraic operation is a partial magma
(partial-magma? op)
;=> true

Every single partial magma is now a special type of a partial magmoid. This is part of the same categorification process by which every magma is a magmoid, every semigroup is a semigroupoid, every group is a groupoid, and so on. But this is not an arbitrary notion, because partial magmoids are actually an important part of the theory of the topos of compositional quivers $Sets^{CQ}$.

(partial-magmoid? op)
;=> true

So everything in Locus is based upon the most categorical of foundations. Even when you are considering a partial magma of a category, you are just considering the single object version of a partial magmoid. With this partial composition magma, we can consider categories as types of partial algebraic operations.

Object structure:
In total algebra we focus on MSets and their topoi $Sets^M$. All the actions of such an MSet are total transformations, defined by the topos $Sets$ rather then the category of sets and partial functions. To generalize MSets we have to engage in a process that is two fold: we can replace the monoid with a partial algebra and we can replace the total transformations by partial transformations.

The result of this process is a PSet which is the action of a partial magma by partial transformations on a set. Now consider a category $C$ then its object set $Ob(C)$ is a PSet. The partial magma of this PSet is the composition operation on morphisms of a category we just defined, and its action on objects is by atomic partial transformations. Every arrow $f: A \to B$ is actually just a partial transformation $A \to B$ on $Ob(C)$ that is defined on only the one element $A$ and hence it is called atomic. We can get this PSet in Locus using the to-pset method.

; the object structure of the example category
(def example-action 
  (to-pset example-category))
  
; the resulting pset is an action by atomic charts
(atomic-chart-pset? example-action)
;=> true

The implementation of PSets in Locus now largely coincides with that of MSets, so a number of multimethods can be freely applied to either PSets or MSets and you will get the expected results. Examples include action preorders, action homomorphisms, action equality, and so on. We can use this to get the object preorder of a category, which is part of the object structure of a category induced by its PSet.

; the object preorder of the example category
(def object-preorder
  (action-preorder example-action))

So the object preorder of a category which has $A \subseteq B$ whenever there exists a morphism $f \in Hom(A,B)$ is actually the action preorder of a PSet. So its just another case of an action preorder. On the other hand, the action preorders of the PSets induced by the action of morphisms on themselves to the left, right, or in both directions are what produce the generalized Green's relations of a category.

Its not hard to see that a partial magmoid is actually equivalent to a PSet by atomic partial transformations, and a category is just a special case that has a number of conditions: totality, the existence of identities, and associativity. So a category can be recovered from its PSet in the same way that a transformation semigroup can be recovered from its MSet.

See also:
Action preorders