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4 types of categories to know about

Thursday February 29 2024.

If you wish to know more about categories (in the sense of category theory) then the following 4 types may become of importance to you:

  1. Concrete categories are ones where, roughly speaking, the objects are sets with some additional structure. Many objects in mathematics, like fields, vector spaces, groups, rings, and topological spaces, can be represented this way. For a detailed list of examples check out Example 1.1.3 in Category Theory in Context (free pdf). For example, if X is a topological space, then X would be an object in the category of topological spaces. Now, every topological space has an underlying set of points. This is represented by the forgetful functor from the category of topological spaces to the category of sets, sometimes denoted \(U:\mathrm{Top} \to \mathrm{Set}\). In that case \(U(X)\) would be the underlying set of points of \(X\).

  2. Monoidal categories. Many categories have a way to combine two objects, like the Cartesian product \(A \times B\) of sets, the direct product of two groups, the tensor product of two vector spaces, etc, that constitute a binary operation. Furthermore in each case there is a unit object which serves as an identity for the binary operation: the singleton set (with one element), the trivial vector space (with one point), etc. This is the premise of a monoidal category. Insert segue into Rosetta stone paper here.

  3. Cartesian closed categories. This type of category is commonly used to represent the type systems within some programming languages. Given two objects \(A\) and \(B\) in a “CCC”, you can form the product \(A \times B\) as well as the exponential \(B^A\), also written more suggestively as \(A \to B\). When \(A\) and \(B\) are sets (or types) then \(B^A\) can be understood as the set (type) of functions (or “programs”) from \(A\) to \(B\). If you combine this with point 2, so that the product \(\times\) obeys the monoidal laws, then you get a closed monoidal category. I need a better reference on this.

  4. Abelian categories. If you ever try to study algebraic topology you will come across the concept of homology (or cohomology) and the notion of a chain complex. Let \(Ab\) be the category of abelian groups (which is a concrete category as well as a monoidal category). A chain complex of abelian groups is a sequence \(A_0, A_1, A_2, \cdots\) of abelian groups, along with homomorphisms \(d_n: A_{n+1} \to A_n\) for each \(n\), such that \(d_{n} \circ d_{n+1} = 0\) for all \(n\), where on the left is the composition of \(d_{n+1}\) with \(d_n\) and on the right is the zero morphism from \(A_{n+2}\) to \(A_n\), which sends every element to zero. If one asks, “what is the most general category I can replace \(Ab\) with?” the answer would be one where you have zero morphisms, and a few other things… anyway, people already figured this out, and they are called abelian categories!