Small set (category theory)
For other uses of the term, see Small set (disambiguation).
In category theory, a small set is one in a fixed universe of sets (as the word universe is used in mathematics in general). Thus, the category of small sets is the category of all sets one cares to consider. This is used when one does not wish to bother with set-theoretic concerns of what is and what is not considered a set, which concerns would arise if one tried to speak of the category of "all sets".
In this context, a large set is any set that is not small.
A small set is not to be confused with a small category, which is a category whose collection of arrows (and therefore of objects) forms a small set. For more on small categories, see Category theory.
See also
References
- S. Mac Lane, Ieke Moerdijk, Sheaves in geometry and logic: a first introduction to topos theory, ISBN 0-387-97710-4, ISBN 3-540-97710-4, the chapter on "Categorical preliminaries"
- Small set in nLab
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