Grothendieck functor

An imbedding functor (cf. Imbedding of categories) from a category $\mathcal{C}$ into the category $\hat{\mathcal{C}}$ of contravariant functors defined on $\mathcal{C}$ and taking values in the category of sets $\mathsf{Ens}$. Let $X$ be an object in a category $\mathcal{C}$; the mapping $Y \mapsto \mathrm{Hom}_{\mathcal{C}}(Y,X)$ defines a contravariant functor $h_X$ from $\mathcal{C}$ into the category of sets. For any object $F$ of $\hat{\mathcal{C}}$ there exists a natural bijection $F(X) \leftrightarrow \mathrm{Hom}_{\hat{\mathcal{C}}}(h_X,F)$ (Yoneda's lemma). Hence, in particular $$\mathrm{Hom}_{\hat{\mathcal{C}}}(h_X,h_Y) \leftrightarrow \mathrm{Hom}_{\mathcal{C}}(X,Y) \ .$$

Accordingly, the mapping $X \mapsto h_X$ defines a full imbedding $h : \mathcal{C} \rightarrow \hat{\mathcal{C}}$, 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