Namespaces
Variants
Actions

Grothendieck functor

From Encyclopedia of Mathematics
Revision as of 16:56, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

An imbedding functor (cf. Imbedding of categories) from a category into the category of contravariant functors defined on and taking values in the category of sets (Ens). Let be an object in a category ; the mapping defines a contravariant functor from into the category of sets. For any object of there exists a natural bijection (Yoneda's lemma). Hence, in particular

Accordingly, the mapping defines a full imbedding , which is known as the Grothendieck functor. Using the Grothendieck functor it is possible to define algebraic structures on objects of a category (cf. Group object; Group scheme).

References

[1] I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)
[2] A. Grothendieck, "Technique de descente et théorèmes d'existence en géométrie algébrique, II" Sem. Bourbaki , Exp. 195 (1960)


Comments

In the English literature, the Grothendieck functor is commonly called the Yoneda embedding or the Yoneda–Grothendieck embedding.

References

[a1] S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7
[a2] N. Yoneda, "On the homology theory of modules" J. Fac. Sci. Tokyo. Sec. I , 7 (1954) pp. 193–227
How to Cite This Entry:
Grothendieck functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Grothendieck_functor&oldid=40223
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article