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Math and science::Algebra::Aluffi

Group. Definition.

Groups. Categorical definition.

A group is a groupoid with a single object.

More specifically, a group is the set of morphisms of a single object groupoid.

Recall that a groupoid is a category where every morphism is an isomorphism.

Now the usual approach.

Groups. Standard definition.

A group \( (G, \bullet) \) is a set \( G \) and a function \( \bullet : G \times G \to G \) (called a binary operation) where the following three properties are satisfied:


\( \bullet \) is associative,

\[ \forall g, h, k \in G, \quad (g \bullet h) \bullet k = g \bullet ( h \bullet k). \]

There exists an identity element denoted \( e_G \) for \( \bullet \),

\[ \exists e_G \in G, \forall g \in G, \quad g \bullet e_G = g = e_G \bullet g . \]
Every element of \( G \) has an inverse with respect to \( \bullet \),

\[ \forall g \in G, \quad \exists h \in G, g \bullet h = h \bullet g . \]

Categorical perspective of a group

From the categorical perspective, consider a category \( C \) that has a single object, \( a \in \catobj{C} \). This is our sort of dummy category that holds our group. Call \( G \) to be a group, and define it to be the set of morphisms of \( C \), \( G = \cathom{C} \). Elements \( g, h \in G \) are morphisms. The operation \( \bullet \) of the group is the function that exists by definition of a category: \( \forall a, b, c \in \cathom{C}, \exists f: \cathom{C}(a, b) \times \cathom{C}(b, c) \to \cathom{C}(a, c) \). As there is just one object in the category, there is only one of these functions, and that is our group \( G \). So the function, denoted \( \bullet \) is: \( \bullet: G \times G \to G \). While morphisms typically carry the connotations of "morphing one object to another", as there is only one object, they lose this connotation.

Take the integers and addition as an example of a group. The elements of \( \mathbb{Z} \) can be thought of as morphisms that "act" on \( \mathbb{Z} \), but don't do anything to \( \mathbb{Z} \). The elements of \( \mathbb{Z} \) compose to equal other elements of \( \mathbb{Z} \), and they do so associatively.

The great generality of categories

With the above perspective of groups, these isn't really anything that can be considered to be "acted upon" or changed. The fact that such a rich sense of change and morphing still emerges when conceptualizing groups is testament to the even greater power of the framework of categories which can contain all of these group dynamics despite being restricted to a single object with all morphisms being isomorphisms.