View source for Skeleton of a category

{{TEX|done}}{{MSC|18A05}}

A minimal [[full subcategory]] of a category that is equivalent to the category itself (cf. [[Equivalence of categories]]). In general, a category $\mathfrak K$ contains many skeletons. Any skeleton can be built up as follows. One chooses a representative in every isomorphism class of objects of $\mathfrak K$. Then the full subcategory of $\mathfrak K$ generated by these objects is a skeleton of $\mathfrak K$.

Two categories are equivalent if and only if their skeletons are isomorphic. A skeleton of a category inherits many properties of the category itself: local smallness, existence of a bicategory structure, various forms of completeness, etc.



====Comments====
A category is said to be skeletal if it is a skeleton of itself, that is, if no two distinct objects are isomorphic. A skeleton of $\mathcal K$ may also be defined as a skeletal full subcategory which meets every isomorphism class of objects of $\mathcal K$. The construction of skeletons described above makes an obvious use of the [[axiom of choice]]; indeed, it can be shown that the assertions  "every category has a skeleton"  and  "any two skeletons of a given category are isomorphic"  are both equivalent to the axiom of choice, [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Freyd,   A. Scedrov,   "Categories, allegories" , North-Holland  (1990)</TD></TR></table>