In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, a small set is one in a fixed
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
of
sets (as the word ''universe'' is used in mathematics in general). Thus, the category of small sets is the
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
of all sets one cares to consider. This is used when one does not wish to bother with
set-theoretic
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
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".
A category C is called small if both the collection of objects and arrows are sets. Otherwise the category is called large.
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
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
.
In other choices of foundations, such as Grothendieck universes, there exist both sets that belong to the universe, called “small sets” and sets that do not, such as the universe itself, “large sets”. We gain an intermediate notion of moderate set: a subset of the universe, which may be small or large. Every small set is moderate, but not conversely.
Since in many cases the choice of foundations is irrelevant, it makes sense to always say “small set” for emphasis even if one has in mind a foundation where all sets are small.
Similarly, a small family is a family indexed by a small set; the axiom of replacement (if it applies in the foundation in question) then says that the image of the family is also small.
See also
*
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...
References
*S. Mac Lane,
Ieke Moerdijk
Izak (Ieke) Moerdijk (; born 23 January 1958) is a Dutch mathematician, currently working at Utrecht University, who in 2012 won the Spinoza prize.
Education and career
Moerdijk studied mathematics, philosophy and general linguistics at the Uni ...
, ''Sheaves in geometry and logic: a first introduction to topos theory'', , , the chapter on "Categorical preliminaries"
* {{nlab, id=small+set, title=Small set
Categories in category theory