Endomorphismes

Endomorphismes are morphisms from an object to itself within a given mathematical structure. In category theory, an endomorphism of an object X is any morphism f: X → X; the set of all endomorphisms is denoted End(X). When the ambient structure is merely a set, End(X) consists of all functions X → X.

In algebraic contexts, endomorphisms are structure-preserving maps from the object to itself: group endomorphisms, ring endomorphisms, module endomorphisms, and linear endomorphisms of vector spaces. The collection End(A) carries a natural ring structure: addition is defined pointwise and composition serves as multiplication. For a vector space V over a field F, End_F(V) is the set of all linear maps V → V. If V has dimension n, End_F(V) is isomorphic to the matrix algebra M_n(F).

Automorphisms are invertible endomorphisms. The group Aut(X) consists of endomorphisms that are bijective and hence invertible. In linear algebra, Aut(V) is the general linear group GL(V).

Examples include projections P with P^2 = P, scalar multiplications aI, rotations in the plane as linear endomorphisms, and nilpotent endomorphisms N with N^k = 0 for some k.

In category theory, End(X) = Hom(X, X) forms a monoid under composition, and Aut(X) is its group of units. Endomorphisms play a central role in many areas of mathematics, providing invariants and structures through eigenvalues, minimal polynomials, invariant subspaces, and representations.