Category
A category consists of three things: a collection of objects, for each pair of objects a collection of morphisms (sometimes call "arrows") from one to another, and a binary operation defined on compatible pairs of morphisms called composition. The category must satisfy an identity axiom and an associative axiom which is analogous to the monoid axioms.
The morphisms must obey the following laws:
1. If 
is a morphism from
to 
(in short, 
), and 
, then
there is a morphism 
(commonly
read "
composed with 
") from 
to 
.
2. Composition of morphisms, where defined, is associative, so if 
, 
, and 
, then
.
3. For each object a, there is an identity morphism 
, such that for
any 
, 
and
.
In most concrete categories over sets, an object is some mathematical structure (e.g., a group, vector space, or smooth manifold) and a morphism is a map between two objects. The identity map between any object and itself is then the identity morphism, and the composition of morphisms is just function composition.
One usually requires the morphisms to preserve the mathematical structure of the objects. So if the objects are all groups, a good choice for a morphism would be a group homomorphism. Similarly, for vector spaces, one would choose linear maps, and for differentiable manifolds, one would choose differentiable maps.
In the category of topological spaces, morphisms are usually continuous maps between topological spaces. However, there are also other category structures having topological spaces as objects, but they are not nearly as important as the "standard" category of topological spaces and continuous maps.
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