A category consists of three things: a collection of objects, for each pair of objects a collection of morphisms (sometimes called "arrows") from one to another, and a binary operation defined on compatible pairs of morphisms called composition of morphisms. The category must satisfy an identity axiom and an associative axiom analogous to the monoid axioms.
The morphisms must obey the following laws (Stacks Project Authors 2026):
1. If 
and 
are morphisms, then there
is a morphism 
(commonly read "
composed with 
") from 
to 
.
2. Composition of morphisms is associative, so if 
, 
, and 
, then 
.
3. For each object 
, 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.
The morphisms are usually required 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, linear maps are chosen, and for differentiable manifolds, differentiable maps are chosen.
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.
