Previously on the blog, we've discussed a recurring theme throughout mathematics: making new things from old things. Today, I'd like to focus on a particular way to build a new vector space from old vector spaces: the tensor product. This construction often come across as scary and mysterious, but I hope to shine a little light and dispel a little fear. In particular, we won't talk about axioms, universal properties, or commuting diagrams. Instead, we'll take an elementary, concrete look: Given two vectors $\mathbf{v}$ and $\mathbf{w}$, we can build a new vector, called the tensor product $\mathbf{v}\otimes \mathbf{w}$. But what is that vector, really? Likewise, given two vector spaces $V$ and $W$, we can build a new vector space, also called their tensor product $V\otimes W$. But what is that vector space, really?

You see, it's very different than other branches of math. Rather than being another sibling lined up in the family photograph, it's more like a common gene that unites them in the first place. With this vantage point, it becomes evident that different areas of math share common patterns/trends/structures. This becomes extraordinarily useful when you want to solve a problem in one realm (say, topology) but don't have the right tools at your disposal. By transporting the problem to a different realm (say, algebra), you can see the problem in a different light and perhaps discover new tools, and the solution may become much easier. a "template" for all of mathematics: depending on what you feed into the template, you'll recover one of the mathematical realms. Naturally, then, you're prompted to also ask about relationships between categories. These are called functors. But why stop there? What about the relationships between those relationships? These are called natural transformations. (And yes, you can ask a hierarchy of questions: "What about the relationships between the relationships between the relationships between the...?" This leads to infinity categories. [And a possible brain freeze.]