group functor

plural group functors

1. algebraic geometryA group object that is an object in a category of functors; a functor with certain properties that generalise the concept of group.

   In order to be able to efficiently state the main results of this chapter (in §4) we formulate here a group-functor version of the R-nilpotent tower lemma (Ch.III, 6.4).

   1972, A. K. Bousfield, D. M. Kan, Homotopy Limits, Completions and Localizations, Springer, page 106

   2 more examples

   A direct product of k–group functors is again a k–group functor; so is a fibre product if the morphisms used in its construction are homomorphisms of k–group functors.

   2003, Jens Carsten Jantzen, Representations of Algebraic Groups, 2nd edition, American Mathematical Society, page 20

   By a group functor we mean a functor from small k-algebras to groups. A homomorphism of group functors is a natural transformation. A subgroup functor of a group functor G is a subfunctor H such that H(R) is a subgroup of G(R) for all R; it is normal if H(R) is normal in G(R) for all R.

   2017, J. S. Milne, Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, page 118