group functor

English

Noun

group functor (plural group functors)

1. (category theory, algebraic geometry) A group object that is an object in a category of functors; a functor with certain properties that generalise the concept of group.
   • 1972, A. K. Bousfield, D. M. Kan, Homotopy Limits, Completions and Localizations, Springer, page 106:

     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).

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

     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.

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

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

Related terms

• subgroup functor

Further reading

• Functor category on Wikipedia.Wikipedia
• Group object on Wikipedia.Wikipedia
• group functor on nLab on Wikipedia.Wikipedia