Functorial characterizations of internal magmoids

An internal quiver in a category $C$ is just a diagram $F: T_2^* \to C$. These diagrams consist of a pair of parallel arrows in the category. Then if we are given such an internal quiver, we can always form an equalizer of its to arrows to get a subobject of the arrow object $A$. The problem is that this subobject $S$ is not going to contain all the elements we want for a composition domain. For example, for a category $C$ this consists of all endomorphisms instead of all composable pairs of arrows. Instead we have to work with pullbacks, and get the pullback of the first and second arrow. This is why internal categories are always defined over categories that have all pullbacks. Then this pullback has the property that it is the equalizer of the first and second arrows as they filter through the product object $A^2$. This makes $R$ into an internal relation in the category $C$, embedded in the object $A$. If we take a look at the above diagram, its not hard to see that the first and second projection arrows can be made to filter through $A \times_O A$ to make them into a ternary quiver consisting of three parallel arrows going to $A$. This leads to a diagram that looks like this. Then diagrams of this type form a category $C$. The axioms of this category $C$ can be used to ensure the laws specifying the source and target of composite morphisms. In particular, we can ensure that the source of the composite is the source of the second morphism and the target of the composite is the target of the first morphism. All that can be encoded in the index category. We then have the idea of a compositional quiver, which is a further over this index category $CQ$.

The importance of this functorial characterization is that we can form a functor category $C^{CQ}$. The problem, however, is that internal compositional quivers don't necessarily have to have nice input domains $A \times_O A$. This is why to have an internal magmoid, we must have a compositional quiver diagram with the property that the composition domain is the pullback $A \times_O A$.

In other words, this means that in the compositional quiver diagram $A \times_O A$ and the morphisms $1$ and $2$ form a universal limiting cone for the pullback. This leads me to an interesting idea: functors with the property that certain subcones within the functor, described as subdiagrams, are universal. This can be used for example to characterize whenever an object in a diagram is a product or a coproduct. Something like this would make for an interesting data structure for those interested in implementing category theory on the machine.

To form an internal category is a little bit more involved. We can always add identities to an internal magmoid diagram, by adding another identity morphism going from objects to arrows. To make that unital magmoid into a category also requires that we satisfy associativity and the unit laws. Rest assured, however, that internal categories can be formed in $Sets^{\to}$ because it has all pullbacks, and if there is anything interesting to be said there I will be saying it eventually. For now these are just some of my thoughts on internal magmoids and related notions.

A key takeaway is that everything should be represented by a functor. Natural transformations for example are simply functors from a category in to an arrow category, so they also have nicely defined functorial semantics. If we take categories to be something like functors or diagrams internal to some category, that is a step forwards too. While internal categories could be just some collection of objects and arrows satisfying some axioms, if we can make them in to special types of functors that is most convenient. Functors are the ideal structures we should work towards dealing with.

References:
internal categories