Small set (category theory)
Encyclopedia
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 objects forms a set. For more on small categories, see Category theory
.
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
, a small set is one in a fixed universe
Universe (mathematics)
In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation...
of sets (as the word universe is used in mathematics in general). Thus, the category of small sets is the category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
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 objects forms a set. For more on small categories, see Category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
.