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A [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s0857801.png" /> whose class of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s0857802.png" /> is a set. A small category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s0857803.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s0857805.png" />-category if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s0857806.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s0857807.png" /> is a universe. For a small category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s0857808.png" /> and an arbitrary locally small category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s0857809.png" /> the category of covariant (contravariant) functors (cf. [[Functor|Functor]]) from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s08578010.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s08578011.png" /> is locally small. In particular, the small categories form the [[Closed category|closed category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s08578012.png" /> of small categories, one of the basic categories of mathematics [[#References|[1]]].
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A [[category]] $\mathfrak{K}$ whose class of morphisms $\text{Mor}\,\mathfrak{K}$ is a set. A small category $\mathfrak{K}$ is called a $U$-category if $\text{Mor}\,\mathfrak{K} \subset U$, where $U$ is a universe. For a small category $\mathfrak{K}$ and an arbitrary locally small category $\mathfrak{C}$ the category of covariant (contravariant) functors (cf. [[Functor]]) from $\mathfrak{K}$ to $\mathfrak{C}$ is locally small. In particular, the small categories form the [[closed category]] $\textsf{Cat}$ of small categories, one of the basic categories of mathematics [[#References|[1]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F.W. Lawvere,  "The category of categories as a foundation for mathematics"  S. Eilenberg (ed.)  et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer  (1966)  pp. 1–20</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  F.W. Lawvere,  "The category of categories as a foundation for mathematics"  S. Eilenberg (ed.)  et al. (ed.) , ''Proc. conf. categorical algebra (La Jolla, 1965)'' , Springer  (1966)  pp. 1–20</TD></TR>
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====Comments====
 
====Comments====
A category is called locally small if, for any pair of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s08578013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s08578014.png" />, the class of morphisms from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s08578015.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085780/s08578016.png" /> is a set. (Some authors assume this condition as part of the definition of a category.) A locally small category is small if and only if its class of objects is a set.
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A category is called locally small if, for any pair of objects $A$ and $B$, the class of morphisms from $A$ to $B$ is a set. (Some authors assume this condition as part of the definition of a category.) A locally small category is small if and only if its class of objects is a set.
  
Cf. also [[Universe|Universe]].
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Cf. also [[Universe]].
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Revision as of 18:57, 28 October 2016

A category $\mathfrak{K}$ whose class of morphisms $\text{Mor}\,\mathfrak{K}$ is a set. A small category $\mathfrak{K}$ is called a $U$-category if $\text{Mor}\,\mathfrak{K} \subset U$, where $U$ is a universe. For a small category $\mathfrak{K}$ and an arbitrary locally small category $\mathfrak{C}$ the category of covariant (contravariant) functors (cf. Functor) from $\mathfrak{K}$ to $\mathfrak{C}$ is locally small. In particular, the small categories form the closed category $\textsf{Cat}$ of small categories, one of the basic categories of mathematics [1].

References

[1] F.W. Lawvere, "The category of categories as a foundation for mathematics" S. Eilenberg (ed.) et al. (ed.) , Proc. conf. categorical algebra (La Jolla, 1965) , Springer (1966) pp. 1–20


Comments

A category is called locally small if, for any pair of objects $A$ and $B$, the class of morphisms from $A$ to $B$ is a set. (Some authors assume this condition as part of the definition of a category.) A locally small category is small if and only if its class of objects is a set.

Cf. also Universe.

How to Cite This Entry:
Small category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Small_category&oldid=12738
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article