Skeleton of a category

A minimal full subcategory of a category that is equivalent to the category itself. In general, a category $\mathfrak K$ contains many skeletons. Any skeleton can be built up as follows. One chooses a representative in every isomorphism class of objects of $\mathfrak K$. Then the full subcategory of $\mathfrak K$ generated by these objects is a skeleton of $\mathfrak K$.

Two categories are equivalent if and only if their skeletons are isomorphic. A skeleton of a category inherits many properties of the category itself: local smallness, existence of a bicategory structure, various forms of completeness, etc.

Comments

A category is said to be skeletal if it is a skeleton of itself, that is, if no two distinct objects are isomorphic. A skeleton of $\mathcal K$ may also be defined as a skeletal full subcategory which meets every isomorphism class of objects of $\mathcal K$. The construction of skeletons described above makes an obvious use of the axiom of choice; indeed, it can be shown that the assertions "every category has a skeleton" and "any two skeletons of a given category are isomorphic" are both equivalent to the axiom of choice, [a1].

References

[a1]	P.J. Freyd, A. Scedrov, "Categories, allegories" , North-Holland (1990)