This article is about a general term. A list of important particular cases (instances) is available at Category:Types of group objects
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 be a category with a terminal object and which allows for taking finite products. A group object over is an object is endowed with the following additional structures:
|Structure name||Structure description||Interpretation when is the category of sets|
|multiplication map||a morphism where is the product in the category ( is sometimes denoted )||is a binary operation from the cartesian product to . In the concrete category case, the multiplication map may be denoted using an infix operator such as or even simply by concatenation, as with the usual notion of groups.|
|identity element||a morphism||a chosen element of (which plays the role of identity element). The element itself is denoted or >|
|inverse map||a morphism ( is sometimes denoted as ).||is a unary map from the set to itself. In the concrete category case, the inverse map may be denoted by the superscript , as with the usual notion of groups.|
satisfying the following compatibility conditions
|Condition name||Description of condition (verbal)||Description of condition (commutative diagram)||Interpretation when is the category of sets|
|Associativity||.Here, denotes the identity map from to itself.||PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]||In this case, it just becomes the associativity condition on elements: for all .|
|Identity element|| where is the canonical projection from to
where is the canonical projection from to
|PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]||for all elements .|
|Inverse||if is the diagonal map (see definition below) and is composed with the unique map from to then||PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]||for all elements .|
The diagonal map is the unique map which, when post-composed with the projection map to either factor , gives the identity map from to .
Group objects in a category form a new category
For any category that has products and a terminal object, the group objects in form a category with a forgetful functor to . The category is described explicitly as follows:
- The objects of this category are the group objects in .
- A morphism in this category between two group objects in is a morphism between them as objects of that commutes with the group operations.
The forgetful functor simply sends a group object in to its underlying object in , and views morphisms as -morphisms. It is a faithful functor.
Group objects and functors between categories
Suppose and are categories and is a functor. It is tempting to believe that induces a functor from the category of group objects of to the category of group objects of . This is true if the functor 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 ) 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.
|Category (described by name for its typical object)||Notion of group object|
|group||abelian group (this is a corollary of the Eckmann-Hilton argument)|
|topological space||topological group|
|algebraic variety over a fixed (usually, algebraically closed) field||algebraic group over the same field|
|manifold||real Lie group|
|differential manifold||real Lie group|
|simplicial set||simplicial group|
|cosimplicial set||cosimplicial group|