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Small category

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2020 Mathematics Subject Classification: Primary: 18A05 [MSN][ZBL]

A category 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.

How to Cite This Entry:
Small category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Small_category&oldid=42010
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article