Small set (category theory)
- Small set (category theory)
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.
References
*S. Mac Lane, I. 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"
ee also
*Category of sets
Wikimedia Foundation.
2010.
Look at other dictionaries:
Small set — In mathematics, the term small set may refer to: *Small set (category theory) *Small set (combinatorics)ee also*Ideal (set theory) *Natural density *Large set (Ramsey theory) … Wikipedia
Category theory — In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects and morphisms . Categories now appear in most branches of mathematics and in… … Wikipedia
Glossary of category theory — This is a glossary of properties and concepts in category theory in mathematics.CategoriesA category A is said to be: * small provided that the class of all morphisms is a set (i.e., not a proper class); otherwise large. * locally small provided… … Wikipedia
Limit (category theory) — In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint… … Wikipedia
Nerve (category theory) — In category theory, the nerve N(C) of a small category C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space, called the classifying space of the category C … Wikipedia
Diagram (category theory) — In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms. An indexed family of sets is a collection of sets … Wikipedia
Cone (category theory) — In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances in category theory as well. Contents 1 Definition 2 Equivalent formulations 3 Category … Wikipedia
Span (category theory) — A span, in category theory, is a generalization of the notion of relation between two objects of a category. When the category has all pullbacks (and satisfies a small number of other conditions), spans can be considered as morphisms in a… … Wikipedia
Skeleton (category theory) — In mathematics, a skeleton of a category is a subcategory which, roughly speaking, does not contain any extraneous isomorphisms. In a certain sense, the skeleton of a category is the smallest equivalent category which captures all categorical… … Wikipedia
Category (mathematics) — In mathematics, a category is an algebraic structure that comprises objects that are linked by arrows . A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A … Wikipedia
