In
category theoryIn mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from sets and functions to objects linked in diagrams by morphisms or arrows....
, a branch of
mathematicsMathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....
, a
section is a right inverse of a morphism. Dually, a
retraction is a left inverse. In other words, if and are morphisms whose composition is the
identity morphismIn mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
on
Y, then
g is a section of
f, and
f is a retraction of
g.
If
section of a morphism exists, it is called
sectionable. Dually, if
retraction of a morphism exists, it is called
retractable.
The categorical concept of a section is important in
homological algebraHomological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
, and is also closely related to the notion of a
section of a fiber bundleIn the mathematical field of topology, a section of a fiber bundle, π: E → B, over a topological space, B, is a continuous map, s : B → E, such that π=x for all x in B.A section is a certain generalization of the notion of the graph of a function...
in topology: in the latter case, a section of a fiber bundle is a section of the bundle projection map of the
fiber bundleIn mathematics, and particularly topology, a fiber bundle is intuitively a space E which locally "looks" like a product space B × F, but globally may have a different topological structure...
.
Every section is a
monomorphismIn the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation ....
, and every retraction is an
epimorphismIn category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...
;
they are called respectively a
split monomorphism and a
split epimorphism (the inverse is the splitting).
Examples
Given a
quotient spaceIn topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
with quotient map , a section of is called a
transversalIn combinatorial mathematics, given a collection C of sets, a transversal is a set containing exactly one element from each member of the collection: it is a section of the quotient map induced by the collection. If the original sets are not disjoint, there are several different definitions...
.