In mathematics, a skeleton of a category is a subcategory that, roughly speaking, does not contain any extraneous isomorphisms. In a certain sense, the skeleton of a category is the "smallest" equivalent category, which captures all "categorical properties" of the original. In fact, two categories are equivalentif and only if they have isomorphic skeletons. A category is called skeletal if isomorphic objects are necessarily identical.
Definition
A skeleton of a category C is an equivalent category D in which no two distinct objects are isomorphic. It is generally considered to be a subcategory. In detail, a skeleton of C is a category D such that:
D is a subcategory of C: every object of D is an object of C
for every pair of objects d1 and d2 of D, the morphisms in D are morphisms in C, i.e. and the identities and compositions in D are the restrictions of those in C.
The inclusion of D in C is full, meaning that for every pair of objects d1 and d2 of D we strengthen the above subset relation to an equality:
The inclusion of D in C is essentially surjective: Every C-object is isomorphic to some D-object.
D is skeletal: No two distinct D-objects are isomorphic.
It is a basic fact that every small category has a skeleton; more generally, every accessible category has a skeleton. Also, although a category may have many distinct skeletons, any two skeletons are isomorphic as categories, so up to isomorphism of categories, the skeleton of a category is unique. The importance of skeletons comes from the fact that they are, canonical representatives of the equivalence classes of categories under the equivalence relation of equivalence of categories. This follows from the fact that any skeleton of a category C is equivalent to C, and that two categories are equivalent if and only if they have isomorphic skeletons.
Examples
The category Set of all sets has the subcategory of all cardinal numbers as a skeleton.
A preorder, i.e. a small category such that for every pair of objects, the set either has one element or is empty, has a partially ordered set as a skeleton.
