Skeleton (category theory)

In mathematics, a skeleton of a is a 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 equivalent if and only if they have isomorphic skeletons. A category is called skeletal if isomorphic objects are necessarily identical. A skeleton of a category C is an 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 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 , 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 has a skeleton. (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 of all sets has the subcategory of all cardinal numbers as a skeleton. The category 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 × m matrices with entries in K.