# Group Theory/Group objects and torsors

**Definition (group object)**:

Let be a category that has a terminal object, which we shall denote by . A **group object** of consists of

- an object of such that the categorical products , and exist
- a morphism , thought of as the multiplication morphism,
- a morphism , thought of as the constant identity map (even though may not even be a set)
- and a morphism , thought of as the inversion morphism

which satisfy the following equations:

- (identity law)
- , where shall denote the diagonal of and is the unique morphism given by the definition of a terminal object (inverse law)
- , where is the canonical isomorphism (associative law)

Here, we wrote for the morphism that is induced by two maps and and the universal property of the product, by specifying that the map from to shall be given by , and that the map from to shall be given by .

**Definition (action of a group object)**:

Let be a category which possesses a terminal object, and let be a group object of . Let be another object of . An **action** of