A group object in a category with finite products is an object 
equipped with morphisms 
, 
, and 
, where 1 is a terminal
object, satisfying the associativity, identity,
and inverse axioms as commutative diagrams.
The morphism 
gives multiplication, 
gives inversion, and 
gives the identity (Snowden 2013).
In the category of sets, this definition recovers an ordinary group. A group object in
the category of schemes over
a fixed base scheme 
is a group scheme over 
.
