skeleton

Definition

A skeleton of a category $\mathcal{C}$ is a full subcategory

$$\mathcal{S}\hookrightarrow \mathcal{C}$$

such that:

• every object of $\mathcal{C}$ is isomorphic to an object of $\mathcal{S}$
• if $A,B\in \mathcal{S}$ and $A\cong B$, then $A=B$

Equivalently, a skeleton is a full subcategory containing exactly one representative from each isomorphism class of objects.

Remarks

A skeleton $\mathcal{S}$ is always equivalent to $\mathcal{C}$.

Indeed, since $\mathcal{S}$ is full, the inclusion

$$\mathcal{S}\hookrightarrow \mathcal{C}$$

is full and failthful, and since every object of $\mathcal{C}$ is isomorphic to one in $\mathcal{S}$, it is essentially surjective.

Thus every category is equivalent to any of its skeletons.

In general, constructing a skeleton requires choosing one representative from each isomorphism class of objects, so the existence of a skeleton may depend on some form of choice.