Small category
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.
Small category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Small_category&oldid=42010