group object

Noun

 * 1)  Given a category C, any object X ∈ C on which morphisms are defined corresponding to the group theoretic concepts of a binary operation (called multiplication), identity and inverse, such that multiplication is associative and properties are satisfied that correspond to the existence of inverse elements and the identity element.
 * 2) * 2005, Angelo Vistoli, Part 1: Grothendieck typologies, fibered categories, and descent theory, Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, Fundamental Algebraic Geometry: Grothendieck's FGA Explained,, page 20,
 * The identity is obviously a homomorphism from a group object to itself. Furthermore, the composite of homomorphisms of group objects is still a homomorphism; thus, group objects in a fixed category form a category, which we denote by $$\text{Grp}(C)$$.
 * The identity is obviously a homomorphism from a group object to itself. Furthermore, the composite of homomorphisms of group objects is still a homomorphism; thus, group objects in a fixed category form a category, which we denote by $$\text{Grp}(C)$$.

Usage notes
Alternatively, and more concisely, an object $$X\in C$$ such that for any $$Y\in C$$, the set of morphisms $$\text{Hom}_C(Y,X)$$ is a group and the correspondence $$Y\rightarrow\text{Hom}_C(Y,X)$$ is a functor from $$C$$ into the category of groups $$\text{Gr}$$.

Group objects generalise the concept of group to objects of greater complexity than mere sets. In the process, attention is withdrawn from individual elements and placed more strongly on operations. A typical example of a group object might be a topological group where the object is a topological space on which the group operations are differentiable.