M Theory Lesson 224

Without worrying about drawing all paths through the graph defining a small category, we can start to discuss the Yoneda lemma, the first fundamental theorem of category theory.

A diagram (such as the cube) internal to a category C (such as Set) may be thought of as a functor from the small category formed from the diagram, which embeds the diagram in the category C. We will call this functor $F$. That is, instead of a vertex labelled $C_i$, we imagine that $C_i = F(A_i)$ where $A_i$ labels an object on the diagram.

Now consider a second functor $H$ from the diagram into the category C, defined as follows for the cube in Set. The object $H(C_i)$ is the set of all paths from $C_i$ into the target of the cube. The number of elements is defined by the numbers on the diagram where 0 includes the empty path from the target to itself. For $t$ the target, a category theorist would call such a (actually contravariant) functor Hom$(-,t)$, where the dash stands for the argument and Hom is short for homomorphism.

Observe that the target object on the new cube is the one element set. The correct way to introduce two dimensional structures to categories is to describe natural transformations between functors. Such a 2-arrow $\eta: H \Rightarrow F$ is described by a whole family of commuting squares made using arrows $\eta_{X}: H(X) \rightarrow F(X)$ for any object $X$. So what are these $\eta_{X}$ for the functors in question?

Since $H(t)$ is the one element set, $\eta_{t}$ picks out an element of the original target set $F(t)$, such as the set $C_0 \cup C_1 \cup C_2$ of the last lesson. In fact, the Yoneda lemma tells us that the set of all possible natural transformations $\eta$ is isomorphic to the set $F(t)$. Here $\eta$ sends a basic projection (of a square onto a 1-arrow) onto a representative of the projection in the 1-arrow of the cube. The message is that the higher dimensional arrows can deconstruct higher dimensional spaces into simple one dimensional paths, much as in the case of space filling curves for complicated geometries based on the real or complex numbers.