Mathematical Structure to Define a General Kind of Basis

Earlier, on August 6, 2021, I had tried to formalize a mathematical concept that would generalize the idea of a “basis,” as seen in the theory of vector spaces or the Fundamental Theorem of Arithmetic. I discuss in this post what I tried and what I learned.

In order to generalize beyond vector spaces and free modules, I considered replacing the underlying field of scalars with an arbitrary set . Then, a “basis space” would be a pair of sets with two operations, “vector” combination and “scalar multiplication” , such that there exists an expansion of any element in in terms of scalars and a “smaller” subset of called the basis.

I later realized that this concept may not totally capture what I wanted with the idea of a basis, since trivially the entire set is a basis (and other similar trivial examples would probably be possible too.) One way to deal with this is to require the expansion to be unique in some way. However, this should be qualified properly, since in many practical cases of basis-like constructions, expansions are considered unique up to a certain level. (For example, for the Fundamental Theorem of Arithmetic in say a unique factorization domain or UFD, multiplication by units is explicitly stated to be ignored in the definition of uniqueness.)

Then, on April 28, 2022, I found a beautiful, elegant, and general answer to this question, and that made me all the more excited to explore further. This is the free object, used in universal algebra and even more generally in category theory. It is a massively general construction that plays on the idea of generating all the possible expansions from a certain set of primitives subject to any set of conditions for considering expansions to be equivalent; then, for any structure, any model isomorphic to this would necessarily be the proper generalization of the concept of basis to that structure.

I don’t know if I would have appreciated the power of the free object construction as much if I hadn’t tried this exploration on my own — it gave me massive appreciation and awe for the elegance of the free object. This is a construction that I will probably write an expository post about at some point.