Category theory: abstracting mathematical construction (essay)

Definition. A category ${\mathcal {C}}$ consists of the following data:

1. A class ${\text{ob}}({\mathcal {C}})$, called the "objects" of the category
2. For any two objects $X,Y\in {\text{ob}}({\mathcal {C}})$, a set ${\text{mor}}_{\mathcal {C}}(X,Y)$, called the "morphisms from $X$ to $Y$." We take $f:X\rightarrow Y$ to be a statement which means the exact same thing as the statement $f\in {\text{mor}}_{\mathcal {C}}(X,Y)$.
3. For any two morphisms $f:X\rightarrow Y$ and $g:Y\rightarrow Z$ , a function
   $$\circ :{\text{mor}}_{\mathcal {C}}(Y,Z)\times {\text{mor}}_{\mathcal {C}}(X,Y)\rightarrow {\text{mor}}_{\mathcal {C}}(X,Z),$$
   which we call "composition." That is, we can compose $f$ and $g$ to get an element $g\circ f:X\rightarrow Z$,

subject to the following conditions:

1. For any object $X\in {\text{ob}}({\mathcal {C}})$, there exists an "identity" morphism $1_{X}:X\rightarrow X$, satisfying $1_{X}\circ f=f$ for any $f:Y\rightarrow X$ and $g\circ 1_{X}=g$ for any $g:X\rightarrow Z$ .
2. Composition is associative: $(f\circ g)\circ h=f\circ (g\circ h)$.

♦

One example of a category---the example, in fact---is ${\text{Set}}$, the category of sets. The objects of ${\text{Set}}$ are sets, the morphisms of ${\text{Set}}$ are functions, and composition of morphisms of ${\text{Set}}$ is composition of functions: $(g\circ f)(x):=g(f(x)).$ The identity morphism of a set $X$ is the function $X\rightarrow X$ which does nothing.

Another example of a category is "the discrete category with 2 elements," which I shall call $\mathbb {I}$. The objects ${\text{ob}}(\mathbb {I} )$ are a set consisting of 2 elements (for concreteness we could take ${\text{ob}}(\mathbb {I} ):=\{a,b\}$ where $a=\emptyset$ and $b=\{\emptyset \}$). The morphisms of $\mathbb {I}$ are the identity elements, and nothing else. More formally, we have${\text{mor}}_{\mathbb {I} }(a,a)=\{*\}$, ${\text{mor}}_{\mathbb {I} }(a,b)=\emptyset$, and ${\text{mor}}_{\mathbb {I} }(b,b)=\{*\}$.

Another example of a category is the category ${\text{Grp}}$ of groups. Here, the objects are groups, and the morphisms are group homomorphisms.

Another example of a category is the (naive) category ${\text{Top}}$ of topological spaces. Here, the objects are topological spaces, and the morphisms are continuous maps.

We could go on like this ad infinitum. There are a huge variety of mathematical objects that form categories. Rings, graphs, vector spaces, representations, the points of the sphere, the open subsets of the plane, the natural numbers, and even categories themselves. In fact, every mathematical object forms a category, though perhaps not in an interesting sense. Part of modern mathematical folklore is that you should always "categorify" whatever mathematical objects it is that you are working with, i.e. you should find a way to define everything in a category-theoretic way.

Note that the definition I gave of categories is set theoretic. Shea has informed me that category theory can be taken as foundational, see e.g. [TODO]. I have never studied Lawvere's formalism, but I can provide a couple insights about it nonetheless. 1) This formalism is not mainstream at all; few of the mathematicians using category theory know it. 2) When it comes to actually proving things, set theory is much easier than category theory. It has been my experience that even the most routine, "obvious" things can become very difficult to prove when phrased categorically (there is a reason for that, which I will get to later). When mathematicians want to understand a category, they often embed it into the category of sets and try to understand it there (see "Yoneda embedding").

Functors

Let's suppose I'm doing some construction. Like