Cayley's Theorem (Category Theory)

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

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.

Proof Sketch
Define a functor $H : \mathbf C \rightarrow \mathbf{Sets}$ where


 * $H_A = \{f \in \mathbf C\ |\ f : \_ \rightarrow A\}$
 * $H_{f : A \rightarrow B} = g \mapsto f \circ g$

To check that $H$ is a functor we see that $H_{1_A} = 1_{H_A}$. Then, we see
 * $H_{f \circ g} = h \mapsto (f \circ g) \circ h = h \mapsto f \circ H_g(h) = H_f \circ H_g.$

Suppose now that $H_g = H_h$. Then for all arrows $f$, we have $g \circ f = h \circ f$, when these compositions make sense. This means $g = h$ (verify).

$H$ is faithful and injective on objects, so $\mathbf C$ is isomorphic to the subcategory of $\mathbf{Sets}$ represented by $H$.

Also see

 * Cayley's Theorem (Group Theory)

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

In fact, as Steve Awodey 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 Cayley as well.