Cohomology and Extensions
A group object in a category is a group internal to .
In terms of internal group objects
A group object or internal group in a category with binary products and a terminal object is an object in and arrows
(the unit map)
(the inverse map) and
(the multiplication map), such that the following diagrams commute:
(expressing the fact multiplication is associative),
(telling us that the unit map picks out an element that is a left and right identity), and
(telling us that the inverse map really does take an inverse), where we have let denote the composite and is a diagonal morphism.
Even if doesn't have all binary products, as long as products with (and the terminal object ) exist, then one can still speak of a group object in .
In terms of presheaves of groups
This is a special case of the general theory of structures in presheaf toposes.
In other words, the forgetful functor from to (obtained by composing with the forgetful functor Grp Set) creates representable group objects from representable objects.
An object in with an internal group structure is a diagram
This equips each object with an ordinary group structure, so in particular a product operation
Moreover, since morphisms in are group homomorphisms, it follows that for every morphism we get a commuting diagram
Taken together this means that there is a morphism
of representable presheaves. By the Yoneda lemma, this uniquely comes from a morphism , which is the product of the group structure on the object that we are after.
The basic results of elementary group theory apply to group objects in any category with finite products. (Arguably, it is precisely the elementary results that apply in any such category.)
The theory of group objects is an example of a Lawvere theory.
monoid, monoid object, monoid object in an (∞,1)-category
group, group object, group object in an (∞,1)-category
groupoid, groupoid object, groupoid object in an (∞,1)-category
infinity-groupoid, infinity-groupoid object, groupoid object in an (∞,1)-category
ring, ring object