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Eric Wofsey
Eric Wofsey
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You can recover $F$ from $\mathrm{Res}_F$; this is an exercise in using the Yoneda lemma. Let $H_A$ denote the presheaf represented by an object $A$ of $\mathbf C$ or $\mathbf D$. Then for $A\in \mathbf{C}$ and $B\in\mathbf{D}$ we have natural isomorphisms $$\operatorname{Hom}_{\mathbf D}(FA,B)=H_B(FA)=\mathrm{Res}_FH_B(A)\cong\operatorname{Hom}_{\widehat{\mathbf C}}(H_A,\mathrm{Res}_FH_B),$$ $$\operatorname{Hom}_{\mathbf D}(FA,B)=H_B(FA)=\mathrm{Res}_FH_B(A).$$ the last isomorphism being Yoneda. ByBy Yoneda, the left hand side is enough to recover the functor $F$.

More generally, ifWe can also use Yoneda again to rewrite the left-hand side as $\operatorname{Hom}_{\widehat{\mathbf D}}(H_{FA},H_B)$ and the right-hand side as $\operatorname{Hom}_{\widehat{\mathbf C}}(H_A,\mathrm{Res}_FH_B)$. This suggests that $\mathrm{Res}_F$ has a left adjoint which is given by $H_A\mapsto H_{FA}$ on representable presheaves. When our categories are small then, this is true: $\mathrm{Res}_F$ has a left adjoint which is the left Kan extension of the composition $\mathbf{C}\to\mathbf{D}\to\widehat{\mathbf{D}}$ along $\mathbf{C}\to\widehat{\mathbf{C}}$. This adjoint coincides with $F$ on representable presheaves and in general is computed by writing any presheaf as a colimit of representable presheaves, applying $F$ to each of the representing objects, and taking the corresponding colimit in $\widehat{\mathbf{D}}$.

You can recover $F$ from $\mathrm{Res}_F$; this is an exercise in using the Yoneda lemma. Let $H_A$ denote the presheaf represented by an object $A$ of $\mathbf C$ or $\mathbf D$. Then for $A\in \mathbf{C}$ and $B\in\mathbf{D}$ we have natural isomorphisms $$\operatorname{Hom}_{\mathbf D}(FA,B)=H_B(FA)=\mathrm{Res}_FH_B(A)\cong\operatorname{Hom}_{\widehat{\mathbf C}}(H_A,\mathrm{Res}_FH_B),$$ the last isomorphism being Yoneda. By Yoneda, the left hand side is enough to recover the functor $F$.

More generally, if our categories are small then $\mathrm{Res}_F$ has a left adjoint which is the left Kan extension of the composition $\mathbf{C}\to\mathbf{D}\to\widehat{\mathbf{D}}$ along $\mathbf{C}\to\widehat{\mathbf{C}}$. This adjoint coincides with $F$ on representable presheaves and in general is computed by writing any presheaf as a colimit of representable presheaves, applying $F$ to each of the representing objects, and taking the corresponding colimit in $\widehat{\mathbf{D}}$.

You can recover $F$ from $\mathrm{Res}_F$; this is an exercise in using the Yoneda lemma. Let $H_A$ denote the presheaf represented by an object $A$ of $\mathbf C$ or $\mathbf D$. Then for $A\in \mathbf{C}$ and $B\in\mathbf{D}$ we have $$\operatorname{Hom}_{\mathbf D}(FA,B)=H_B(FA)=\mathrm{Res}_FH_B(A).$$ By Yoneda, the left hand side is enough to recover the functor $F$.

We can also use Yoneda again to rewrite the left-hand side as $\operatorname{Hom}_{\widehat{\mathbf D}}(H_{FA},H_B)$ and the right-hand side as $\operatorname{Hom}_{\widehat{\mathbf C}}(H_A,\mathrm{Res}_FH_B)$. This suggests that $\mathrm{Res}_F$ has a left adjoint which is given by $H_A\mapsto H_{FA}$ on representable presheaves. When our categories are small, this is true: $\mathrm{Res}_F$ has a left adjoint which is the left Kan extension of the composition $\mathbf{C}\to\mathbf{D}\to\widehat{\mathbf{D}}$ along $\mathbf{C}\to\widehat{\mathbf{C}}$. This adjoint coincides with $F$ on representable presheaves and in general is computed by writing any presheaf as a colimit of representable presheaves, applying $F$ to each of the representing objects, and taking the corresponding colimit in $\widehat{\mathbf{D}}$.

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Eric Wofsey
Eric Wofsey
  • 31.2k
  • 2
  • 115
  • 151

You can recover $F$ from $\mathrm{Res}_F$; this is an exercise in using the Yoneda lemma. Let $H_A$ denote the presheaf represented by an object $A$ of $\mathbf C$ or $\mathbf D$. Then for $A\in \mathbf{C}$ and $B\in\mathbf{D}$ we have natural isomorphisms $$\operatorname{Hom}_{\mathbf D}(FA,B)=H_B(FA)=\mathrm{Res}_FH_B(A)\cong\operatorname{Hom}_{\widehat{\mathbf C}}(H_A,\mathrm{Res}_FH_B),$$ the last isomorphism being Yoneda. By Yoneda, the left hand side is enough to recover the functor $F$.

More generally, if our categories are small then $\mathrm{Res}_F$ has a left adjoint which is the left Kan extension of the composition $\mathbf{C}\to\mathbf{D}\to\widehat{\mathbf{D}}$ along $\mathbf{C}\to\widehat{\mathbf{C}}$. This adjoint coincides with $F$ on representable presheaves and in general is computed by writing any presheaf as a colimit of representable presheaves, applying $F$ to each of the representing objects, and taking the corresponding colimit in $\widehat{\mathbf{D}}$.