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 categoryD in which isomorphic objects are equal. Typically, a skeleton is taken to be a
subcategoryD of C such that:
D is skeletal: any two isomorphic objects of D are equal.
Existence and uniqueness
It is a basic fact that every small category has a skeleton; more generally, every
accessible category has a skeleton.[citation needed] (This is equivalent to the
axiom of choice.) 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 (up to isomorphism of categories), 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.
The category K-Vect of all
vector spaces over a fixed
field has the subcategory consisting of all powers , where α is any cardinal number, as a skeleton; for any finite m and n, the maps are exactly the n × mmatrices with entries in K.
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.
Robert Goldblatt (1984). Topoi, the Categorial Analysis of Logic (Studies in logic and the foundations of mathematics, 98). North-Holland. Reprinted 2006 by Dover Publications.