An algebraic structure is a set with some functions associated to it that meet some axioms. Semi-groups, fields, loops, and magmas are all examples of algebraic structures. All algebraic structures can be placed on a gradient that goes from “general structures” to “specific structures.” An interesting way to measure this is to see how many different structures one can make for a single “order” of that kind of algebraic structure. One of the most strict structures is a finite field. Finite fields have so many requirements imposed onto them that the amount of finite fields for a given order is either 0 or 1. There are 0 finite fields when the order is not the power of a prime and 1 when it is. This is opposed to one of the most general structures is a partial magma.

The number of magmas with 9 elements is: 68119360559090181752360568039079551890168488158332691488172212319110842810368

Here’s a magma with 12 elements:

The set is the set of positions of eggs in an egg carton with the function being where the egg is placed after making breakfast.

Categories, I believe, sit in a very good spot on this gradient as they let you create many different kinds of categories:

• the category of sets and functions
• the category of types and functions
• a topos
• the category of an F-algebra with F-homomorphisms
• the Hegelian taco

Categories let you extend them a lot, so much so that there have been several mathematicians who have tried rewriting the foundations of Mathematics in terms of Category Theory (CT). All this, while also not getting in your way.

A very nice generalisation of categories is something called a groupoid. Groupoids are categories where every morphism is an isomorphism. The theory of groupoids has been developed so much that ideas like Kant’s dialectic from philosophy can be modeled using some rather interesting groupoids.

Other Structures

You might be asking yourself:

Why does this not happen with other structures?

Good question, I don’t know the answer myself. But I know that the perspective that many mathematicians have about CT is that it is used as a tool for generalizations and to link different branches of Mathematics together. I have yet to have seen someone else have this perspective about some other field of Mathematics. I don’t mean to say that there isn’t anyone out there with a perspective like that. If there are, there definitely aren’t as many as those who view CT this way.

I would also like to point out that Set Theory (ST) has also been used to develop math foundations and is therefore used frequently in other branches of Mathematics. In this case though, many things in ST can be reframed as basic concepts of CT. You can read more about this on my post about ETCS.

Motivation

So, why is this important?

Well first we have to look at my perspective on how to organize solutions for problems. I believe that one should let problems develop organically, and only after seeing the organic pattern develop, impose a structure that reflects the organic development. For example, many researchers will let their minds race on a new problem landscape, to see if there are any new or missing perspectives. If this seems promising, formalisations come into play. After formalising, the most natural question to make is: has this been done before? That single question is the imposition of a structure onto a problem that developed organically. The consequence of finding out that someone else has worked on something similar to the theory you have developed is to compare and contrast your theory to their theory. One can see this same sequence of steps play out in many places, not just Mathematics research.

In this case, CT seems to be like the most natural way to continue extending ideas in some places in Mathematics.

Abrupt ending so that the food for thought gets force-fed.