Mixing
Inclusion of Reflective Subcategory Creates Limits
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(Full) subcategory $D$ of the category $C$ is called reflexive if there exists a localization functor $L : C to D$ and it is left adjoint to the inclusion functor $I : D hookrightarrow C$.
There is theorem:
If $D$ is a reflective subcategory of $C$, then $I$ creates all limits that $C$ admits.
But if this theorem is true, then every diagram $x: I toD$, which admits a limit $y = lim_i in I Ix_i $, will also have a limit in $D$. But as inclusion $I$ is right adjoint, it preserves limits, and so $y = IBig(lim_i in I x_i Big)$. Then, as $y$ is included into $C$, it must hold that $lim_i in I x_i = Ly cong y$. But, I don't know how to prove this fact, as $L$ only preserves colimits.
I can prove that $Ly$ indeed forms a cone over $x$ and for any other cone $c$
there exists a morphism $psi: c to Ly$, but I don't know how to prove uniqueness.
I construct $psi$ as $Lphi$, there $phi: Ic to y$, which exists by universal property of limit $y$.
What am I missing here? áan you help me prove uniqueness?
category-theory localization adjoint-functors
asked 14 hours ago
Nik Pronko
678617
add a comment |Â
up vote
2
down vote
favorite
$
newcommandCmathcalC
newcommandDmathcalD
newcommandImathcalI
$
(Full) subcategory $D$ of the category $C$ is called reflexive if there exists a localization functor $L : C to D$ and it is left adjoint to the inclusion functor $I : D hookrightarrow C$.
There is theorem:
If $D$ is a reflective subcategory of $C$, then $I$ creates all limits that $C$ admits.
But if this theorem is true, then every diagram $x: I toD$, which admits a limit $y = lim_i in I Ix_i $, will also have a limit in $D$. But as inclusion $I$ is right adjoint, it preserves limits, and so $y = IBig(lim_i in I x_i Big)$. Then, as $y$ is included into $C$, it must hold that $lim_i in I x_i = Ly cong y$. But, I don't know how to prove this fact, as $L$ only preserves colimits.
I can prove that $Ly$ indeed forms a cone over $x$ and for any other cone $c$
there exists a morphism $psi: c to Ly$, but I don't know how to prove uniqueness.
I construct $psi$ as $Lphi$, there $phi: Ic to y$, which exists by universal property of limit $y$.
What am I missing here? áan you help me prove uniqueness?
category-theory localization adjoint-functors
asked 14 hours ago
Nik Pronko
678617
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
$
newcommandCmathcalC
newcommandDmathcalD
newcommandImathcalI
$
(Full) subcategory $D$ of the category $C$ is called reflexive if there exists a localization functor $L : C to D$ and it is left adjoint to the inclusion functor $I : D hookrightarrow C$.
There is theorem:
If $D$ is a reflective subcategory of $C$, then $I$ creates all limits that $C$ admits.
But if this theorem is true, then every diagram $x: I toD$, which admits a limit $y = lim_i in I Ix_i $, will also have a limit in $D$. But as inclusion $I$ is right adjoint, it preserves limits, and so $y = IBig(lim_i in I x_i Big)$. Then, as $y$ is included into $C$, it must hold that $lim_i in I x_i = Ly cong y$. But, I don't know how to prove this fact, as $L$ only preserves colimits.
I can prove that $Ly$ indeed forms a cone over $x$ and for any other cone $c$
there exists a morphism $psi: c to Ly$, but I don't know how to prove uniqueness.
I construct $psi$ as $Lphi$, there $phi: Ic to y$, which exists by universal property of limit $y$.
What am I missing here? áan you help me prove uniqueness?
category-theory localization adjoint-functors
asked 14 hours ago
Nik Pronko
678617
$
newcommandCmathcalC
newcommandDmathcalD
newcommandImathcalI
$
(Full) subcategory $D$ of the category $C$ is called reflexive if there exists a localization functor $L : C to D$ and it is left adjoint to the inclusion functor $I : D hookrightarrow C$.
There is theorem:
If $D$ is a reflective subcategory of $C$, then $I$ creates all limits that $C$ admits.
But if this theorem is true, then every diagram $x: I toD$, which admits a limit $y = lim_i in I Ix_i $, will also have a limit in $D$. But as inclusion $I$ is right adjoint, it preserves limits, and so $y = IBig(lim_i in I x_i Big)$. Then, as $y$ is included into $C$, it must hold that $lim_i in I x_i = Ly cong y$. But, I don't know how to prove this fact, as $L$ only preserves colimits.
I can prove that $Ly$ indeed forms a cone over $x$ and for any other cone $c$
there exists a morphism $psi: c to Ly$, but I don't know how to prove uniqueness.
I construct $psi$ as $Lphi$, there $phi: Ic to y$, which exists by universal property of limit $y$.
What am I missing here? áan you help me prove uniqueness?
category-theory localization adjoint-functors
asked 14 hours ago
Nik Pronko
678617
edited 14 hours ago
asked 14 hours ago
Nik Pronko
678617
asked 14 hours ago
Nik Pronko
678617
asked 14 hours ago
Nik Pronko
678617
678617
add a comment |Â
add a comment |Â
1 Answer
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Let the adjunction take the form of a natural isomorphism $Psi: textHom_D(L(-),-)cong textHom_D(-,I(-))$ between functors mapping $C^textoptimes DtotextbfSet$. Let $x:mathcal Ito D$ be a functor, and suppose $Ix:mathcal Ito C$ has a limit given by the cone $mathcal K=lim(Ix)oversetpi_ilongrightarrow IX(i), iintextOb(mathcal I)$.
Lemma: $forall dintextOb(D), exists eta_d: LI(d)cong d$, and this correspondence is natural in $d$. In other words, $LI$ and $textid_D$ are naturally isomorphic as functors.
Fullness of the inclusion functor $I$ gives canonical isomorphisms $$textHom_D(LI(d_1),d_2)cong textHom_C(LI(d_1),I(d_2))oversetPsicong textHom_C(I(d_1),d_2) cong textHom_D(d_1,d_2)$$ Natural in pairs $left<d_1,d_2right>intextOb(D^textoptimes D)$. In particular, there is a natural mapping $$Psi_:textHom_D(LI(-),-)cong textHom_D(-,-)$$ Where $textHom_D(LI(-),-)$ and $textHom_D(-,-)$ are functors $D^textoptimes Dto textbfSet$. The existence of $eta$ follows by the Yoneda Lemma (or Yoneda Lemma-style arguments) $Box$
Conceptually, we want to show that $Lmathcal K$ "behaves" in $D$ as $mathcalK$ does in $D$, and in particular, that $L(lim(Ix))$ is itself the limit of $x$. After all, "pre-filling" the the left argument with $lim(Ix)intextOb(C)$ and discarding the "irrelevant" components of $Psi$ gives natural isomorphism $$Psi_lim(Ix):textHom_D(L(lim(Ix)),-)cong textHom_C(lim(Ix),I(-))$$ So the two should "see $D$ in the same way" in some sense.
Consider some arbitrary cone $K$ in $D$ over $x$ of the form $K=koversetp_ilongrightarrow X(i), iintextOb(mathcal I)$.
Passing this cone through $I$ gives another cone $IK=I(k)oversetIp_ilongrightarrow IX(i), iintextOb(mathcal I)$—this time in $C$—over $Ix$. Definitionally there exists some (unique!) $C$-morphism $kappa:kto lim(IX)$ commuting with $IK$, $mathcal K$ and $Ix$. Passing everything "back into $D$" through $L$ gives $D$-morphism $Lkappa:LI(k)to L(lim(IX))$ commuting with $LIK$, $Lmathcal K$ and $LIx$.
Finally, applying $eta$ to $LIK$ and $LIX$ gives an arrow $kappacirc eta_k^-1:kto L(lim(IX))$ making cones $K$, $Lmathcal K$ and diagram $x$ commute. Uniqueness of $kappacirc eta_k^-1$ in this regard follows from fullness of $LI:Dto D$ and isomorphism of $eta^-1$.
So $Lmathcal K$ is in fact the limiting cone of $x:mathcal Ito C$, as desired $blacksquare$
answered 6 hours ago
Rafay A.
564
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1 Answer
1
active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Let the adjunction take the form of a natural isomorphism $Psi: textHom_D(L(-),-)cong textHom_D(-,I(-))$ between functors mapping $C^textoptimes DtotextbfSet$. Let $x:mathcal Ito D$ be a functor, and suppose $Ix:mathcal Ito C$ has a limit given by the cone $mathcal K=lim(Ix)oversetpi_ilongrightarrow IX(i), iintextOb(mathcal I)$.
Lemma: $forall dintextOb(D), exists eta_d: LI(d)cong d$, and this correspondence is natural in $d$. In other words, $LI$ and $textid_D$ are naturally isomorphic as functors.
Fullness of the inclusion functor $I$ gives canonical isomorphisms $$textHom_D(LI(d_1),d_2)cong textHom_C(LI(d_1),I(d_2))oversetPsicong textHom_C(I(d_1),d_2) cong textHom_D(d_1,d_2)$$ Natural in pairs $left<d_1,d_2right>intextOb(D^textoptimes D)$. In particular, there is a natural mapping $$Psi_:textHom_D(LI(-),-)cong textHom_D(-,-)$$ Where $textHom_D(LI(-),-)$ and $textHom_D(-,-)$ are functors $D^textoptimes Dto textbfSet$. The existence of $eta$ follows by the Yoneda Lemma (or Yoneda Lemma-style arguments) $Box$
Conceptually, we want to show that $Lmathcal K$ "behaves" in $D$ as $mathcalK$ does in $D$, and in particular, that $L(lim(Ix))$ is itself the limit of $x$. After all, "pre-filling" the the left argument with $lim(Ix)intextOb(C)$ and discarding the "irrelevant" components of $Psi$ gives natural isomorphism $$Psi_lim(Ix):textHom_D(L(lim(Ix)),-)cong textHom_C(lim(Ix),I(-))$$ So the two should "see $D$ in the same way" in some sense.
Consider some arbitrary cone $K$ in $D$ over $x$ of the form $K=koversetp_ilongrightarrow X(i), iintextOb(mathcal I)$.
Passing this cone through $I$ gives another cone $IK=I(k)oversetIp_ilongrightarrow IX(i), iintextOb(mathcal I)$—this time in $C$—over $Ix$. Definitionally there exists some (unique!) $C$-morphism $kappa:kto lim(IX)$ commuting with $IK$, $mathcal K$ and $Ix$. Passing everything "back into $D$" through $L$ gives $D$-morphism $Lkappa:LI(k)to L(lim(IX))$ commuting with $LIK$, $Lmathcal K$ and $LIx$.
Finally, applying $eta$ to $LIK$ and $LIX$ gives an arrow $kappacirc eta_k^-1:kto L(lim(IX))$ making cones $K$, $Lmathcal K$ and diagram $x$ commute. Uniqueness of $kappacirc eta_k^-1$ in this regard follows from fullness of $LI:Dto D$ and isomorphism of $eta^-1$.
So $Lmathcal K$ is in fact the limiting cone of $x:mathcal Ito C$, as desired $blacksquare$
answered 6 hours ago
Rafay A.
564
add a comment |Â
up vote
0
down vote
Let the adjunction take the form of a natural isomorphism $Psi: textHom_D(L(-),-)cong textHom_D(-,I(-))$ between functors mapping $C^textoptimes DtotextbfSet$. Let $x:mathcal Ito D$ be a functor, and suppose $Ix:mathcal Ito C$ has a limit given by the cone $mathcal K=lim(Ix)oversetpi_ilongrightarrow IX(i), iintextOb(mathcal I)$.
Lemma: $forall dintextOb(D), exists eta_d: LI(d)cong d$, and this correspondence is natural in $d$. In other words, $LI$ and $textid_D$ are naturally isomorphic as functors.
Fullness of the inclusion functor $I$ gives canonical isomorphisms $$textHom_D(LI(d_1),d_2)cong textHom_C(LI(d_1),I(d_2))oversetPsicong textHom_C(I(d_1),d_2) cong textHom_D(d_1,d_2)$$ Natural in pairs $left<d_1,d_2right>intextOb(D^textoptimes D)$. In particular, there is a natural mapping $$Psi_:textHom_D(LI(-),-)cong textHom_D(-,-)$$ Where $textHom_D(LI(-),-)$ and $textHom_D(-,-)$ are functors $D^textoptimes Dto textbfSet$. The existence of $eta$ follows by the Yoneda Lemma (or Yoneda Lemma-style arguments) $Box$
Conceptually, we want to show that $Lmathcal K$ "behaves" in $D$ as $mathcalK$ does in $D$, and in particular, that $L(lim(Ix))$ is itself the limit of $x$. After all, "pre-filling" the the left argument with $lim(Ix)intextOb(C)$ and discarding the "irrelevant" components of $Psi$ gives natural isomorphism $$Psi_lim(Ix):textHom_D(L(lim(Ix)),-)cong textHom_C(lim(Ix),I(-))$$ So the two should "see $D$ in the same way" in some sense.
Consider some arbitrary cone $K$ in $D$ over $x$ of the form $K=koversetp_ilongrightarrow X(i), iintextOb(mathcal I)$.
Passing this cone through $I$ gives another cone $IK=I(k)oversetIp_ilongrightarrow IX(i), iintextOb(mathcal I)$—this time in $C$—over $Ix$. Definitionally there exists some (unique!) $C$-morphism $kappa:kto lim(IX)$ commuting with $IK$, $mathcal K$ and $Ix$. Passing everything "back into $D$" through $L$ gives $D$-morphism $Lkappa:LI(k)to L(lim(IX))$ commuting with $LIK$, $Lmathcal K$ and $LIx$.
Finally, applying $eta$ to $LIK$ and $LIX$ gives an arrow $kappacirc eta_k^-1:kto L(lim(IX))$ making cones $K$, $Lmathcal K$ and diagram $x$ commute. Uniqueness of $kappacirc eta_k^-1$ in this regard follows from fullness of $LI:Dto D$ and isomorphism of $eta^-1$.
So $Lmathcal K$ is in fact the limiting cone of $x:mathcal Ito C$, as desired $blacksquare$
answered 6 hours ago
Rafay A.
564
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Let the adjunction take the form of a natural isomorphism $Psi: textHom_D(L(-),-)cong textHom_D(-,I(-))$ between functors mapping $C^textoptimes DtotextbfSet$. Let $x:mathcal Ito D$ be a functor, and suppose $Ix:mathcal Ito C$ has a limit given by the cone $mathcal K=lim(Ix)oversetpi_ilongrightarrow IX(i), iintextOb(mathcal I)$.
Lemma: $forall dintextOb(D), exists eta_d: LI(d)cong d$, and this correspondence is natural in $d$. In other words, $LI$ and $textid_D$ are naturally isomorphic as functors.
Fullness of the inclusion functor $I$ gives canonical isomorphisms $$textHom_D(LI(d_1),d_2)cong textHom_C(LI(d_1),I(d_2))oversetPsicong textHom_C(I(d_1),d_2) cong textHom_D(d_1,d_2)$$ Natural in pairs $left<d_1,d_2right>intextOb(D^textoptimes D)$. In particular, there is a natural mapping $$Psi_:textHom_D(LI(-),-)cong textHom_D(-,-)$$ Where $textHom_D(LI(-),-)$ and $textHom_D(-,-)$ are functors $D^textoptimes Dto textbfSet$. The existence of $eta$ follows by the Yoneda Lemma (or Yoneda Lemma-style arguments) $Box$
Conceptually, we want to show that $Lmathcal K$ "behaves" in $D$ as $mathcalK$ does in $D$, and in particular, that $L(lim(Ix))$ is itself the limit of $x$. After all, "pre-filling" the the left argument with $lim(Ix)intextOb(C)$ and discarding the "irrelevant" components of $Psi$ gives natural isomorphism $$Psi_lim(Ix):textHom_D(L(lim(Ix)),-)cong textHom_C(lim(Ix),I(-))$$ So the two should "see $D$ in the same way" in some sense.
Consider some arbitrary cone $K$ in $D$ over $x$ of the form $K=koversetp_ilongrightarrow X(i), iintextOb(mathcal I)$.
Passing this cone through $I$ gives another cone $IK=I(k)oversetIp_ilongrightarrow IX(i), iintextOb(mathcal I)$—this time in $C$—over $Ix$. Definitionally there exists some (unique!) $C$-morphism $kappa:kto lim(IX)$ commuting with $IK$, $mathcal K$ and $Ix$. Passing everything "back into $D$" through $L$ gives $D$-morphism $Lkappa:LI(k)to L(lim(IX))$ commuting with $LIK$, $Lmathcal K$ and $LIx$.
Finally, applying $eta$ to $LIK$ and $LIX$ gives an arrow $kappacirc eta_k^-1:kto L(lim(IX))$ making cones $K$, $Lmathcal K$ and diagram $x$ commute. Uniqueness of $kappacirc eta_k^-1$ in this regard follows from fullness of $LI:Dto D$ and isomorphism of $eta^-1$.
So $Lmathcal K$ is in fact the limiting cone of $x:mathcal Ito C$, as desired $blacksquare$
answered 6 hours ago
Rafay A.
564
Let the adjunction take the form of a natural isomorphism $Psi: textHom_D(L(-),-)cong textHom_D(-,I(-))$ between functors mapping $C^textoptimes DtotextbfSet$. Let $x:mathcal Ito D$ be a functor, and suppose $Ix:mathcal Ito C$ has a limit given by the cone $mathcal K=lim(Ix)oversetpi_ilongrightarrow IX(i), iintextOb(mathcal I)$.
Lemma: $forall dintextOb(D), exists eta_d: LI(d)cong d$, and this correspondence is natural in $d$. In other words, $LI$ and $textid_D$ are naturally isomorphic as functors.
Fullness of the inclusion functor $I$ gives canonical isomorphisms $$textHom_D(LI(d_1),d_2)cong textHom_C(LI(d_1),I(d_2))oversetPsicong textHom_C(I(d_1),d_2) cong textHom_D(d_1,d_2)$$ Natural in pairs $left<d_1,d_2right>intextOb(D^textoptimes D)$. In particular, there is a natural mapping $$Psi_:textHom_D(LI(-),-)cong textHom_D(-,-)$$ Where $textHom_D(LI(-),-)$ and $textHom_D(-,-)$ are functors $D^textoptimes Dto textbfSet$. The existence of $eta$ follows by the Yoneda Lemma (or Yoneda Lemma-style arguments) $Box$
Conceptually, we want to show that $Lmathcal K$ "behaves" in $D$ as $mathcalK$ does in $D$, and in particular, that $L(lim(Ix))$ is itself the limit of $x$. After all, "pre-filling" the the left argument with $lim(Ix)intextOb(C)$ and discarding the "irrelevant" components of $Psi$ gives natural isomorphism $$Psi_lim(Ix):textHom_D(L(lim(Ix)),-)cong textHom_C(lim(Ix),I(-))$$ So the two should "see $D$ in the same way" in some sense.
Consider some arbitrary cone $K$ in $D$ over $x$ of the form $K=koversetp_ilongrightarrow X(i), iintextOb(mathcal I)$.
Passing this cone through $I$ gives another cone $IK=I(k)oversetIp_ilongrightarrow IX(i), iintextOb(mathcal I)$—this time in $C$—over $Ix$. Definitionally there exists some (unique!) $C$-morphism $kappa:kto lim(IX)$ commuting with $IK$, $mathcal K$ and $Ix$. Passing everything "back into $D$" through $L$ gives $D$-morphism $Lkappa:LI(k)to L(lim(IX))$ commuting with $LIK$, $Lmathcal K$ and $LIx$.
Finally, applying $eta$ to $LIK$ and $LIX$ gives an arrow $kappacirc eta_k^-1:kto L(lim(IX))$ making cones $K$, $Lmathcal K$ and diagram $x$ commute. Uniqueness of $kappacirc eta_k^-1$ in this regard follows from fullness of $LI:Dto D$ and isomorphism of $eta^-1$.
So $Lmathcal K$ is in fact the limiting cone of $x:mathcal Ito C$, as desired $blacksquare$
answered 6 hours ago
Rafay A.
564
edited 2 hours ago
answered 6 hours ago
Rafay A.
564
answered 6 hours ago
Rafay A.
564
answered 6 hours ago
Rafay A.
564
564
add a comment |Â
add a comment |Â
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