Difference between revisions of "Skeleton of a category"
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| − | A minimal [[Full subcategory|full subcategory]] of a category that is equivalent to the category itself. In general, a category | + | {{TEX|done}} |
| + | A minimal [[Full subcategory|full subcategory]] of a category that is equivalent to the category itself. 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. | 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. | ||
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====Comments==== | ====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 | + | 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|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==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.J. Freyd, A. Scedrov, "Categories, allegories" , North-Holland (1990)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P.J. Freyd, A. Scedrov, "Categories, allegories" , North-Holland (1990)</TD></TR></table> | ||
Revision as of 16:19, 19 September 2014
A minimal full subcategory of a category that is equivalent to the category itself. 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, [a1].
References
| [a1] | P.J. Freyd, A. Scedrov, "Categories, allegories" , North-Holland (1990) |
Skeleton of a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skeleton_of_a_category&oldid=13897