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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085670/s0856701.png" /> contains many skeletons. Any skeleton can be built up as follows. One chooses a representative in every isomorphism class of objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085670/s0856702.png" />. Then the full subcategory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085670/s0856703.png" /> generated by these objects is a skeleton of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085670/s0856704.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085670/s0856705.png" /> may also be defined as a skeletal full subcategory which meets every isomorphism class of objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085670/s0856706.png" />. 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]]].
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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)
How to Cite This Entry:
Skeleton of a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Skeleton_of_a_category&oldid=33321
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article