Cayley's Theorem (Category Theory)

Theorem
Let $\mathbf C$ be a small category.

Denote with $\mathbf{Set}$ the category of sets.

Then there exists a category $\mathbf D$, subject to:


 * $(1): \quad $ The objects of $\mathbf D$ are sets.
 * $(2): \quad $ The morphisms of $\mathbf D$ are mappings.
 * $(3): \quad \mathbf C \cong \mathbf D$, i.e. $\mathbf C$ and $\mathbf D$ are isomorphic.

That is, $\mathbf C$ is isomorphic to a subcategory of $\mathbf{Set}$.

Proof
Define a functor $H: \mathbf C \to \mathbf{Set}$ by:


 * $H C := \left\{{f \in \operatorname{mor} \mathbf C: \operatorname{cod} f = C}\right\}$
 * $H f: H A \to H B, g \mapsto f \circ g$

for $f: A \to B$ a morphism of $\mathbf C$.

It is immediate by the definition of identity morphism that:


 * $H \left({\operatorname{id}_A}\right) = \operatorname{id}_{H A}$

For $f: A \to B$ and $g: B \to C$, observe:

Thus by Equality of Mappings, $H \left({g \circ f}\right) = H g \circ H f$.

It follows that $H$ is a functor.

It is clear that $H$ is injective on objects.

Suppose now that $H$ were not faithful.

Then there would be morphisms $g, h: A \to B$ of $\mathbf C$ such that $H g = H h$.

Since $\operatorname{id}_A \in H A$, this means in particular that:


 * $g \circ \operatorname{id}_A = h \circ \operatorname{id}_A$

by Equality of Mappings.

But the definition of identity morphism then reduces this to $g = h$.

Hence, $H$ is faithful.

By Functor is Embedding iff Faithful and Injective on Objects, it follows that $H$ is an embedding.

Thus $\mathbf C$ is isomorphic to a subcategory of $\mathbf{Set}$.

Also see

 * Cayley's Theorem (Group Theory)

Although did not prove this result, it is very similar in both statement and proof to Cayley's Theorem in group theory.

In fact, as states it in :


 *  [ Cayley's ] theorem may be generalized to show that any category that is not "too big" can be represented as a [...] category of sets and functions.

The contributor Lord_Farin subsequently was as audacious as to name the general result after as well.