Categorical Proofs of the Commutative Law, Distributive Law, and Exponential Law

27 Dec 2025

This post was originally written in Korean, and has been machine translated into English. It may contain minor errors or unnatural expressions. Proofreading will be done in the near future.

Theorem.

1. Distributive law: $(A + B) \times C = A \times C + B \times C$
2. Exponential law: $(A \times B)^C = A^C \times B^C$

In a previous post, we explored categorical sums and products. Sums are a type of colimit, while products are a type of limit. From this observation, the distributive and exponential laws can be generalised into the following theorem.

Theorem. Left adjoints preserve colimits, and right adjoints preserve limits.

The precise meaning of this theorem is as follows. Let $F: \mathcal{A} \to \mathcal{B}$ and $G: \mathcal{B} \to \mathcal{A}$, and suppose $F \dashv G$. The statement that $G$ preserves limits means that for any small category $\mathbf{I}$ and functor $D : \mathbf{I} \to \mathcal{B}$, the following holds:

If $\big(B \xrightarrow{q_I} D(I)\big)_{I \in \mathbf{I}}$ is a limit cone in $\mathcal{B}$, then $\big(G(B) \xrightarrow{G(q_I)} G(D(I))\big)_{I \in \mathbf{I}}$ is a limit cone in $\mathcal{A}$.

From this, we can deduce the following when $G$ preserves limits:

\[G\left(\lim_{\leftarrow \mathbf{I}} D\right) \cong \lim_{\leftarrow \mathbf{I}} G \circ D\]

For colimits, the definition is analogous, replacing cones with cocones.

On the other hand, for any $X, Y, Z \in \mathbf{Set}$, the following holds: $\hom(X \times Y, Z) \cong \hom(X, Z^Y)$ (this is a common technique in functional programming known as currying). From this, the following adjoint relationship arises:

\[(-) \times Y \dashv (-)^Y\]

Thus, the functor $(-) \times Y$ preserves colimits, and the functor $(-)^Y$ preserves limits. From this, we can derive the generalised distributive and exponential laws.

Theorem. For sets $A, B, Y$, if $A$ and $B$ are disjoint, the following holds:

1. $(A + B) \times Y \cong (A \times Y) + (B \times Y)$
2. $(A \times B)^Y \cong A^Y \times B^Y$

Here, $+$ denotes the disjoint union of sets, which is equivalent to $\sqcup$, but the notation $+$ is used to emphasise its similarity to the distributive law. The left-hand side corresponds to $G\left(\lim_{\leftarrow \mathbf{I}} D\right)$, while the right-hand side corresponds to $\lim_{\leftarrow \mathbf{I}} G \circ D$. By applying cardinality operations to the sets in the above theorem, we recover the distributive and exponential laws for natural numbers.

Additionally, the following property of limits is also known (the $\bullet, -$ notation is explained in this post):

Theorem. Let $\mathbf{I}$ and $\mathbf{J}$ be small categories, and let $\mathcal{S}$ be a locally small category that admits limits of shape $\mathbf{I}$ and $\mathbf{J}$. For $D: \mathbf{I} \times \mathbf{J} \to \mathcal{S}$, the following holds:

\[\lim_{\leftarrow \mathbf{J}}\lim_{\leftarrow \mathbf{I}} D(\bullet, -) \cong \lim_{\leftarrow \mathbf{I} \times \mathbf{J}} D \cong \lim_{\leftarrow \mathbf{I}}\lim_{\leftarrow \mathbf{J}} D(-, \bullet)\]

In other words, limits satisfy the commutative law. This can be thought of as a generalisation of the commutative laws for addition and multiplication.