Category
A category is an algebraic structure consisting of
- Objects:
- Morphisms (arrows):
such that
- Each morphism has specified domain and codomain objects:
signifies that is a morphism with domain and codomain . - Each object has a designated identity morphism
. - For any pair of morphisms
, with the codomain of equal to the domain of , there exists a specified composite morphism whose domain is equal to the domain of and whose codomain is equal to the codomain of . The composition law is subject to the following axioms: - Unital with identity morphisms: For any
, the composites and are both equal to . - Associativity:
.
e.g.
- The real numbers
and continuous functions . - Topological spaces and continuous mappings.
- Smooth manifolds and smooth mappings.
Commutative Diagram
A commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. A commutative diagram often consists of three parts:
- objects (also known as vertices)
- morphisms (also known as arrows or edges)
- paths or composites
Subcategory
A subcategory
of a category is defined by restricting to a subcollection of objects and subcollection of morphisms subject to the requirements that the subcategory contains the domain and codomain of any morphism in , the identity morphism of any object in , and the composite of any composable pair of morphisms in .
Foundations, Large and Small
Small and Large
A category
is called small if both the collection of objects of and the collection of arrows of are sets. Otherwise, is called large.
e.g. All finite categories are clearly small, the category
Locally-Small
A category
is called locally small if for any objects in , the collection is a set (called a hom-set).
e.g. Many of the large categories we want to consider are in fact locally small.
Proposition
Any small category is locally small.