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