endomaps

Endomaps, also known as endomorphisms, are structure-preserving maps from a mathematical object to itself. In the broadest sense, an endomap is a morphism f: A → A where A is an object in a given category; when A is a set, endomaps are simply functions from the set to itself.

In algebra, endomorphisms preserve the underlying structure: for a group G, a map f: G → G satisfies f(xy) = f(x)f(y); for a vector space V, a linear endomorphism is a linear map V → V; for a ring or algebra, f preserves addition and multiplication; for a topological space X, a topological endomorphism is a continuous map X → X.

The set (or class) End(A) of endomorphisms of A forms a monoid under composition, with the identity map as its unit. In many contexts, the subset of invertible endomorphisms comprises the automorphisms; these form a group Aut(A) of structure-preserving isomorphisms.

In finite contexts, counting helps illustrate: there are n^n endomaps on a finite set with n elements. For a finite-dimensional vector space V of dimension n over a finite field F_q, the set of linear endomorphisms End(V) has size q^{n^2} and is isomorphic to the matrix ring M_n(F_q).

Endomaps are central in category theory, algebra, and topology, often studied for their algebraic and geometric properties. The term “endomap” is somewhat less common than “endomorphism” but used as a synonym in many texts.