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Difference between revisions of "Grothendieck functor"

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An imbedding functor (cf. [[Imbedding of categories|Imbedding of categories]]) from a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g0451601.png" /> into the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g0451602.png" /> of contravariant functors defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g0451603.png" /> and taking values in the category of sets (Ens). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g0451604.png" /> be an object in a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g0451605.png" />; the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g0451606.png" /> defines a contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g0451607.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g0451608.png" /> into the category of sets. For any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g0451609.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g04516010.png" /> there exists a natural bijection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g04516011.png" /> (Yoneda's lemma). Hence, in particular
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An imbedding functor (cf. [[Imbedding of categories|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
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$$
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\mathrm{Hom}_{\hat{\mathcal{C}}}(h_X,h_Y) \leftrightarrow \mathrm{Hom}_{\mathcal{C}}(X,Y) \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g04516012.png" /></td> </tr></table>
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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]]).
 
 
Accordingly, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g04516013.png" /> defines a full imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045160/g04516014.png" />, 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 object]]; [[Group scheme|Group scheme]]).
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I. Bucur,  A. Deleanu,  "Introduction to the theory of categories and functors" , Wiley  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,  "Technique de descente et théorèmes d'existence en géométrie algébrique, II"  ''Sem. Bourbaki'' , '''Exp. 195'''  (1960)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  I. Bucur,  A. Deleanu,  "Introduction to the theory of categories and functors" , Wiley  (1968)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,  "Technique de descente et théorèmes d'existence en géométrie algébrique, II"  ''Sem. Bourbaki'' , '''Exp. 195'''  (1960)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Yoneda,  "On the homology theory of modules"  ''J. Fac. Sci. Tokyo. Sec. I'' , '''7'''  (1954)  pp. 193–227</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. MacLane,  "Categories for the working mathematician" , Springer  (1971)  pp. Chapt. IV, Sect. 6; Chapt. VII, Sect. 7</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Yoneda,  "On the homology theory of modules"  ''J. Fac. Sci. Tokyo. Sec. I'' , '''7'''  (1954)  pp. 193–227</TD></TR>
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</table>

Latest revision as of 19:18, 7 March 2017

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
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