Group object

Definition
The notion of group object is a category-theoretic generalization of the concept of group. The definition is designed so that a group object in the category of sets is just a group.

Let $$C$$ be a category with a terminal object $$1$$ and which allows for taking finite products. A group object over $$C$$ is an object $$G$$ is $$C$$ endowed with the following additional structures:

satisfying the following compatibility conditions

The diagonal map $$\Delta$$ is the unique map $$G \to G \times G$$ which, when post-composed with the projection map to either factor $$G$$, gives the identity map from $$G$$ to $$G$$.

Group objects in a category form a new category
For any category $$C$$ that has products and a terminal object, the group objects in $$C$$ form a category with a forgetful functor to $$C$$. The category is described explicitly as follows:


 * The objects of this category are the group objects in $$C$$.
 * A morphism in this category between two group objects in $$C$$ is a morphism between them as objects of $$C$$ that commutes with the group operations.

The forgetful functor simply sends a group object in $$C$$ to its underlying object in $$C$$, and views morphisms as $$C$$-morphisms. It is a faithful functor.

Group objects and functors between categories
Suppose $$C_1$$ and $$C_2$$ are categories and $$F:C_1 \to C_2$$ is a functor. It is tempting to believe that $$F$$ induces a functor from the category of group objects of $$C_1$$ to the category of group objects of $$C_2$$. This is true if the functor $$F$$ sends the terminal object to the terminal object and sends products to products (in other words, it is a monoidal functor between the categories equipped with the Cartesian monoidal structure).

An example of the failure of this is that algebraic groups are not topological groups. The functor from algebraic varieties over a field (such as $$\mathbb{C}$$) to topological spaces that sends any algebraic variety to its underlying set with the Zariski topology. This functor does not preserve products, i.e., the Zariski topology on a product variety is not the product topology of the Zariski topologies on the varieties.