Difference between revisions of "Small category"
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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]]]. | + | 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|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]]]. |
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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. | 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. | ||
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Latest revision as of 17:09, 5 October 2017
2020 Mathematics Subject Classification: Primary: 18A05 [MSN][ZBL]
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.
Small category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Small_category&oldid=42010