nLab
functoriality of categories of presheaves
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Idea
Given a small category C C , one can consider the category of presheaves PSh ( C , D ) PSh(C, D) valued in some category D D . Given some assumptions on D D , any functor of small categories F : C → C ′ F : C \to C' induces two adjoint pairs
F ! : PSh ( C , D ) ⇄ PSh ( C ′ , D ) : F * F_! : PSh(C, D) \rightleftarrows PSh(C', D) : F^* F * : PSh ( C ′ , D ) ⇄ PSh ( C , D ) : F * F^* : PSh(C', D) \rightleftarrows PSh(C, D) : F_*
Definitions
Definition
Let F : C → C ′ F : C \to C' be a functor of small categories and D D some category. The restriction of scalars functor F * : PSh ( C ′ , D ) → PSh ( C , D ) F^* : PSh(C', D) \to PSh(C, D) is given by the formula H ↦ H ∘ f H \mapsto H \circ f , i.e. mapping a presheaf H : C ′ op → D H : C'^{op} \to D to the composite
C op → F C ′ op → H D . C^{op} \stackrel{F}{\to} C'^{op} \stackrel{H}{\to} D.
Properties
Lemma
Let F : C ⇄ C ′ : G F : C \rightleftarrows C' : G be an adjoint pair and consider the induced functors ( F ! , F * , F * ) (F_!, F^*, F_*) and ( G ! , G * , G * ) (G_!, G^*, G_*) . One has
F ! F_! is left adjoint to G ! G_! ,
F * F^* is left adjoint to G * G^* ,
F * F_* is left adjoint to G * G_* ,
F * = G * F_* = G^* ,
F * = G ! F^* = G_! .
Note that all these claims are in fact equivalent.
See also
For functoriality of sheaves , see
References
Last revised on July 24, 2014 at 01:22:04.
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