Skeleton (category theory)
In category theory, the skeleton of a category is a category that basically contains no superfluous isomorphisms. In a sense, the skeleton of a category is the "smallest " equivalent category that retains all " categorical properties ". In fact, two categories are well then equivalent if they have isomorphic skeletons.
Definition
A skeleton of a category C is a full, dense subcategory D in which two (different ) objects may not be isomorphic. That is, as follows: A skeleton of C is a class D, so that:
- Every object of D is an object of C.
- For each object d of D, the D- identity of d is also the identity of C d
- The composition in D is the restriction of the composition in C to morphisms of D.
- If d1, d2 arbitrary objects of D, then the C- morphisms are of d1 by d2 exactly the D- morphisms to the d1 d2 after, in formulas:
- Each object C is isomorphic to a D object.
- Any two distinct D objects are not isomorphic.
Existence and uniqueness
It is fundamental that each category has a skeleton. ( This statement is the axiom of choice for classes equivalent, as it provides about the Neumann - Bernays - Gödel set theory. ) However, if there is one category may have several different skeletons, they are isomorphic as categories. So, each category up to isomorphism a unique skeleton.
The importance of skeletons is because they are canonical ( up to isomorphism ) representatives of the equivalence classes with respect to the equivalence of categories. The results from the fact that each category is equivalent to a skeleton, and that two categories are well then equivalent if they have isomorphic skeletons.
Examples
- The category set, consisting of all sets and figures, has the sub-category of cardinal numbers as a skeleton.
- The category VektK consisting of all K- vector spaces and K- linear maps for a fixed field K, has the subcategory as a skeleton, consisting of the K (n ), where n is a cardinal number.
- The class of well-orderings, the sub-category of ordinals as a skeleton.