Horizontal categorification of binary operations

A fundamental first step in our understanding of abstract algebra is the process by which we can translate from a single-object structure like a monoid to get its many-object variant such as a category. In particular, it is by this process that we can consider generalisations of categories like magmoids.

Operation	Oidification
Partial magma	Partial magmoid
Magma	Magmoid
Semigroup	Semigroupoid
Monoid	Category
Group	Groupoid

For example, it is by this process by which I have managed to consider additional generalisations of categories like partial magmoids. In this context, a partial magmoid is simply the horizontal categorification of a partial magma. Partial magmoids are useful in the study of quotients.

I emphasize this comparison to demonstrate that semigroups and categories are not profoundly different subjects. Categories and monoids are alike in almost every way as they lie together on a common axis of odification. Either one can be used to study the other for all intents and purposes. Categories are just the nicer way of looking at things is all.

A far greater difference actually lies between order theory on the one hand and either category theory / semigroup theory on the other. The later subjects have far more algebraic flavour and can be seen as ways of studying higher forms of preorders, enriched with extra algebraic structure. Categories are like higher preorders. So these are far more genuinely different subjects.

Horizontal categorification is a nice tool that we can use to group mathematical subjects together. Subjects that are on the same line of horizontal categorification are the most similar to one another, and those subjects that are not are the most genuinely different from one another.

References:
Horizontal categorification