Skeleton (category theory)

In category theory , the skeleton of a category is a sub-category that does not contain superfluous isomorphisms . In a sense, the skeleton of a category is the smallest equivalent category that retains all of its categorical properties. Indeed, two categories are equivalent if and only if they have isomorphic skeletons.

definition

A skeleton for a category is a full, dense sub-category in which two (different) objects may not be isomorphic. That means in detail: A skeleton of is a category , so that: ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$

• Every object of is an object of . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$
• For each object of , the identity of is also the identity of . ${\displaystyle d}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle d}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle d}$
• The composition in is the restriction of the composition in to the morphisms of . ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$
• If , are arbitrary objects of , then the morphisms from to are exactly the morphisms from to , in formulas: ${\displaystyle d_{1}}$ ${\displaystyle d_{2}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle d_{1}}$ ${\displaystyle d_{2}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle d_{1}}$ ${\displaystyle d_{2}}$

${\displaystyle \operatorname {Hom} _{D}(d_{1},d_{2})=\operatorname {Hom} _{C}(d_{1},d_{2})}$

• Every object is isomorphic to an object. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$
• Any two different objects are not isomorphic. ${\displaystyle {\mathcal {D}}}$

Existence and uniqueness

It is fundamental that each category has a skeleton. (This statement is equivalent to the axiom of choice for classes, such as the Neumann-Bernays-Gödel set theory provides.) Even if a category can have several different skeletons, they are isomorphic as categories . So, apart from isomorphism, each category has a unique skeleton.

The meaning of skeletons comes from the fact that they (except for isomorphism) are canonical representatives of the equivalence classes with regard to the equivalence of categories. This follows from the fact that every category is equivalent to a skeleton, and that two categories are equivalent if and only if they have isomorphic skeletons.

Examples

• The category Set , consisting of all sets and figures , has the subcategory of the cardinal numbers as a skeleton.
• The category Vekt _{${\displaystyle K}$} consisting of all - vector spaces and - linear mappings for a fixed body , has the sub-category as a skeleton consisting of the , there being a cardinal number is. ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K^{n}}$ ${\displaystyle n}$
• The category of well-orders has the sub-category of ordinal numbers as a skeleton.

literature

• J. Adámek, H. Herrlich, GE Strecker: Abstract and concrete categories (PDF; 4.4 MB), John Wiley (1990)