{
  "$type": "site.standard.document",
  "description": "It's categories.",
  "path": "/posts/perfect-structure/",
  "publishedAt": "2026-06-16T15:48:48.000Z",
  "site": "at://did:plc:6n2ngs7zpcpwxz3jaoxj56tu/site.standard.publication/3mo6y7ludvn2h",
  "tags": [
    "fun",
    "category theory",
    "mathematics"
  ],
  "textContent": "An algebraic structure is a set with some functions associated to it that meet some axioms.\nSemi-groups, fields), loops, and magmas) are all examples of algebraic structures.\nI won't assume that the reader has dealt with any of these structures but I will assume that you know about set cardinality and the fact that these structures were originally created, mostly, in an abstract way in order to model different things. The best example for this, from a programming perspective, is that of the semigroup. If you want to parallelize a procedure in your program, a way to do this would be to demonstrate it forms a semigroup. If so, parallelizing it is super easy, since a semigroup's operation is associative. I won't go into too much detail on this since this is not the point of this post.\n\nI would like to start building up some intuition by considering one of the simplest algebraic structures: magmas.\nA magma is simply a set with a binary operation.\nThis binary operation does not have to be total or closed (when not total the magma is called a partial magma).\nRequiring that the operation is closed is useful, in fact coming up with examples for non-closed operations is somewhat difficult for me, but closed and total operations are quite simple to create.\nAfter requiring closure, one can add the first real properties: associativity and totality.\nThis is called a semigroup.\nAdding identities to this makes this structure a monoid.\nAdding inverses to this makes it a group.\nAdding commutativity makes it an Abelian group.\n\nAlgebraic structures continue this pattern: keep adding operations and axioms.\nI consider fields to be the epitome of these.\nThe notion of \"uniqueness\" starts creeping in the more you do this though.\nAnd the question whether two different structures of the same kind of algebraic structure are the same starts becoming more important.\nWhen talking about fields, specially those that are finite, one can see that for any given order (cardinality) there is exactly 1 field (when the order is a prime power) or there simply isn't a field of that order.\nThis is in large contrast to the number of magmas for a given order.\nFor example, since 9 is 3 squared (a prime power) there is a single field with that order. The number of magmas with 9 elements is around 1.97 * 10^77.\n\n196627050475552913618075908526912116283103450944214766927315415537966391196809\n\nThen, one can create a sort of relationship between the number of algebraic structures of a given kind and order and the number of axioms imposed on that kind of algebraic structure.\nI like thinking of this relationship as a gradient.\nAt one end of this gradient, there are magmas (those that are partial) and at the opposite end there are fields.\nCategories sit somewhere in between these ends.\n\nIt is my perspective that taking an organic pattern and imposing structure onto it is the entire process of Mathematics.\nAlgebraic structures are abstract structures which we have imposed onto patterns we've seen all around Mathematics.\nSo the natural question now is to try and find what the best structure for imposing onto patterns is.\n\nCT\n\nCategory Theory (CT) is all about modeling things in terms of categories.\nArguably, CT is the field of Mathematics where you spend the most time trying to impose structure.\nThe organic pattern that has developed here is that categories are the de facto algebraic structure to impose onto other organic patterns.\nSo, it must sit in the \"sweet spot\" where the best structure for imposing patterns onto is.\nIt is because of this that I think that categories are the best algebraic structure.\n\nOther Structures\n\nYou might be asking yourself:\n\nWhy does this not happen with other structures?\n\nGood question, and I don't think I am qualified to answer it.\nBut 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.\nI have yet to have seen someone have a different perspective about CT.\nAnd I have yet to have seen someone have a perspective like this about some other field of Mathematics.\nI don't mean to say that there isn't anyone out there with a perspective like that out there.\nBut if there are, they are few and far between and definitely aren't as many as those who view CT this way.\n\nI 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.\nIn this case though, many things in ST can be reframed as basic concepts of CT.\nYou can read more about this on my post about ETCS.\n\nMotivation\n\nSo, why is this important?\n\nI have seen many Mathematicians turn down Category-Theoretical (CTic) constructions.\nMany of them with very valid arguments for doing so.\nIn fact, as an undergraduate student, there are way more people extremely more qualified than me to opine on this differently than me, that have seen more constructions than me and who probably know lots of counterexamples to this.\nThis is where I'll say:\n\nCTic constructions are not a panacea.\n\nI am suggesting that CTic constructions and thinking are beneficial for organizing patterns and giving structure.\nThese suggestions might not always apply, but, as it seems to me, most of the time they do.\n\nContinuum\n\nThe most natural question to ask now is:\n\nHas this been done before?\n\nBut before answering that question we can take a step back and contemplate the nature of asking a question like that.\nLook at us, trying to find a structure to impose onto the very thought of thinking about imposing structure.\nAnd this is where I want to divert the thesis of this blog post from categories as a structure to categories as a discipline.\nCT is able to link branches of Mathematics together, it lets us communicate in a universal language and it, personally my favorite, it is not exclusive to new ideas.\n\nNow, claiming that categories are the \"best algebraic structure\" or that CT is the \"best field of Mathematics\" screams naïveté.\nIf Maslow claims that once you have a hammer everything looks like a nail, the field you study in hands you a hammer and tells you the whole world is nails.\n\nClaiming that there is a \"best\" structure or field implies that there is a set of criteria with which to rate these and I am not convinced that there can be any meaningful set of criteria.\n\nSo, if I'm not going to claim what the \"best\" structure or field is, what was the point of writing this essay?\n\nThe point is that CT as a discipline is super useful and I personally wish more people felt this way about it.\n\nFood for thought, you know ?",
  "title": "The best Algebraic Structure"
}