The subobject classifier is a set??
$begingroup$
I'm reading about subobject classifiers in presheaf categories, and got confused.
The book speaks of a subobject classifier $Omega$ of the presheaf category $[P^{op}, Set]$, where $P$ is a poset having a largest element $T$. So, as I understand, $Omega$ is an element of $[P^{op}, Set]$, meaning it is a contravariant functor from $P$ to $Set$.
But then a few lines later, the book says we can regard the truth values of the subobject classifier as a Heyting Algebra, and this induces a monoid structure on $Omega$. The book goes on to speak of actions of this monoid on a set, all the while using notation like $alpha in Omega$. In other words, it looks like they are saying $Omega$ as a set.
My question is, how can $Omega$ have any elements, if it is a functor? Are they writing $Omega$ when they actually mean $Omega$ applied to $T$? If this is the case then my confusion is (mostly) cleared.
category-theory
asked Dec 6 '18 at 2:09
SSFSSF
406110
$endgroup$
add a comment |
$begingroup$
I'm reading about subobject classifiers in presheaf categories, and got confused.
The book speaks of a subobject classifier $Omega$ of the presheaf category $[P^{op}, Set]$, where $P$ is a poset having a largest element $T$. So, as I understand, $Omega$ is an element of $[P^{op}, Set]$, meaning it is a contravariant functor from $P$ to $Set$.
But then a few lines later, the book says we can regard the truth values of the subobject classifier as a Heyting Algebra, and this induces a monoid structure on $Omega$. The book goes on to speak of actions of this monoid on a set, all the while using notation like $alpha in Omega$. In other words, it looks like they are saying $Omega$ as a set.
My question is, how can $Omega$ have any elements, if it is a functor? Are they writing $Omega$ when they actually mean $Omega$ applied to $T$? If this is the case then my confusion is (mostly) cleared.
category-theory
asked Dec 6 '18 at 2:09
SSFSSF
406110
$endgroup$
$begingroup$
Which book${}$?
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:11
$begingroup$
a handout from my lecturer, I believe it's based off on Sheaves in Geometry and Logic: A First Introduction to Topos Theory.
$endgroup$
– SSF
Dec 6 '18 at 2:13
$begingroup$
$Omega$ is an internal Heyting algebra, that is, it is naturally a presheaf of Heyting algebras on $P$. The Heyting algebra of truth values is indeed $Omega(T)$. It's hard to tell without further details, but your monoidal actions might indeed be of this monoid.
$endgroup$
– Kevin Carlson
Dec 6 '18 at 2:27
3
$begingroup$
In a topos, sometimes $ain A$ means that $a$ is an arrow from the terminal object $1$ to the object $A$.
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:38
add a comment |
$begingroup$
I'm reading about subobject classifiers in presheaf categories, and got confused.
The book speaks of a subobject classifier $Omega$ of the presheaf category $[P^{op}, Set]$, where $P$ is a poset having a largest element $T$. So, as I understand, $Omega$ is an element of $[P^{op}, Set]$, meaning it is a contravariant functor from $P$ to $Set$.
But then a few lines later, the book says we can regard the truth values of the subobject classifier as a Heyting Algebra, and this induces a monoid structure on $Omega$. The book goes on to speak of actions of this monoid on a set, all the while using notation like $alpha in Omega$. In other words, it looks like they are saying $Omega$ as a set.
My question is, how can $Omega$ have any elements, if it is a functor? Are they writing $Omega$ when they actually mean $Omega$ applied to $T$? If this is the case then my confusion is (mostly) cleared.
category-theory
asked Dec 6 '18 at 2:09
SSFSSF
406110
$endgroup$
I'm reading about subobject classifiers in presheaf categories, and got confused.
The book speaks of a subobject classifier $Omega$ of the presheaf category $[P^{op}, Set]$, where $P$ is a poset having a largest element $T$. So, as I understand, $Omega$ is an element of $[P^{op}, Set]$, meaning it is a contravariant functor from $P$ to $Set$.
But then a few lines later, the book says we can regard the truth values of the subobject classifier as a Heyting Algebra, and this induces a monoid structure on $Omega$. The book goes on to speak of actions of this monoid on a set, all the while using notation like $alpha in Omega$. In other words, it looks like they are saying $Omega$ as a set.
My question is, how can $Omega$ have any elements, if it is a functor? Are they writing $Omega$ when they actually mean $Omega$ applied to $T$? If this is the case then my confusion is (mostly) cleared.
category-theory
category-theory
asked Dec 6 '18 at 2:09
SSFSSF
406110
asked Dec 6 '18 at 2:09
SSFSSF
406110
asked Dec 6 '18 at 2:09
SSFSSF
406110
asked Dec 6 '18 at 2:09
SSFSSF
406110
asked Dec 6 '18 at 2:09
SSFSSF
406110
406110
$begingroup$
Which book${}$?
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:11
$begingroup$
a handout from my lecturer, I believe it's based off on Sheaves in Geometry and Logic: A First Introduction to Topos Theory.
$endgroup$
– SSF
Dec 6 '18 at 2:13
$begingroup$
$Omega$ is an internal Heyting algebra, that is, it is naturally a presheaf of Heyting algebras on $P$. The Heyting algebra of truth values is indeed $Omega(T)$. It's hard to tell without further details, but your monoidal actions might indeed be of this monoid.
$endgroup$
– Kevin Carlson
Dec 6 '18 at 2:27
3
$begingroup$
In a topos, sometimes $ain A$ means that $a$ is an arrow from the terminal object $1$ to the object $A$.
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:38
add a comment |
$begingroup$
Which book${}$?
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:11
$begingroup$
a handout from my lecturer, I believe it's based off on Sheaves in Geometry and Logic: A First Introduction to Topos Theory.
$endgroup$
– SSF
Dec 6 '18 at 2:13
$begingroup$
$Omega$ is an internal Heyting algebra, that is, it is naturally a presheaf of Heyting algebras on $P$. The Heyting algebra of truth values is indeed $Omega(T)$. It's hard to tell without further details, but your monoidal actions might indeed be of this monoid.
$endgroup$
– Kevin Carlson
Dec 6 '18 at 2:27
3
$begingroup$
In a topos, sometimes $ain A$ means that $a$ is an arrow from the terminal object $1$ to the object $A$.
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:38
$begingroup$
Which book${}$?
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:11
$begingroup$
Which book${}$?
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:11
$begingroup$
a handout from my lecturer, I believe it's based off on Sheaves in Geometry and Logic: A First Introduction to Topos Theory.
$endgroup$
– SSF
Dec 6 '18 at 2:13
$begingroup$
a handout from my lecturer, I believe it's based off on Sheaves in Geometry and Logic: A First Introduction to Topos Theory.
$endgroup$
– SSF
Dec 6 '18 at 2:13
$begingroup$
$Omega$ is an internal Heyting algebra, that is, it is naturally a presheaf of Heyting algebras on $P$. The Heyting algebra of truth values is indeed $Omega(T)$. It's hard to tell without further details, but your monoidal actions might indeed be of this monoid.
$endgroup$
– Kevin Carlson
Dec 6 '18 at 2:27
$begingroup$
$Omega$ is an internal Heyting algebra, that is, it is naturally a presheaf of Heyting algebras on $P$. The Heyting algebra of truth values is indeed $Omega(T)$. It's hard to tell without further details, but your monoidal actions might indeed be of this monoid.
$endgroup$
– Kevin Carlson
Dec 6 '18 at 2:27
3
3
$begingroup$
In a topos, sometimes $ain A$ means that $a$ is an arrow from the terminal object $1$ to the object $A$.
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:38
$begingroup$
In a topos, sometimes $ain A$ means that $a$ is an arrow from the terminal object $1$ to the object $A$.
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:38
add a comment |
1 Answer
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If by "element" they mean "global element", then a global element (of $A$) is an arrow from $1to A$ where $1$ is the terminal object. Since $T$ is the top element of the poset, it is the terminal object of that poset when viewed as a category. Since $mathsf{Hom}$ is continuous in its second argument, we have that $mathsf{Hom}(-,T)cong 1$ where $1$ is the terminal object of the category of presheaves. A global element of the presheaf $P$ is an element of $$mathsf{Nat}(1,P)congmathsf{Nat}(mathsf{Hom}(-,T),P)cong P(T)$$ where $mathsf{Nat}(F,G)$ is the set of natural transformations between $F$ and $G$, i.e. the hom-set of the category of presheaves. The second isomorphism is Yoneda. So the global elements of the presheaf $Omega$ are in correspondence with the elements of the set $Omega(T)$.
More naturally, we can use the notion of the internal language of an elementary topos which can be presented as the Mitchell-Bénabou language. This lets us write what looks like "normal" mathematics but interpret it into any elementary topos. In this context, global elements correspond to closed terms. In other words, the elements of $Omega(T)$, in this case, correspond to the closed terms of type $Omega$ in the internal language.
answered Dec 6 '18 at 4:00
Derek ElkinsDerek Elkins
16.7k11437
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$begingroup$
If by "element" they mean "global element", then a global element (of $A$) is an arrow from $1to A$ where $1$ is the terminal object. Since $T$ is the top element of the poset, it is the terminal object of that poset when viewed as a category. Since $mathsf{Hom}$ is continuous in its second argument, we have that $mathsf{Hom}(-,T)cong 1$ where $1$ is the terminal object of the category of presheaves. A global element of the presheaf $P$ is an element of $$mathsf{Nat}(1,P)congmathsf{Nat}(mathsf{Hom}(-,T),P)cong P(T)$$ where $mathsf{Nat}(F,G)$ is the set of natural transformations between $F$ and $G$, i.e. the hom-set of the category of presheaves. The second isomorphism is Yoneda. So the global elements of the presheaf $Omega$ are in correspondence with the elements of the set $Omega(T)$.
More naturally, we can use the notion of the internal language of an elementary topos which can be presented as the Mitchell-Bénabou language. This lets us write what looks like "normal" mathematics but interpret it into any elementary topos. In this context, global elements correspond to closed terms. In other words, the elements of $Omega(T)$, in this case, correspond to the closed terms of type $Omega$ in the internal language.
answered Dec 6 '18 at 4:00
Derek ElkinsDerek Elkins
16.7k11437
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add a comment |
$begingroup$
If by "element" they mean "global element", then a global element (of $A$) is an arrow from $1to A$ where $1$ is the terminal object. Since $T$ is the top element of the poset, it is the terminal object of that poset when viewed as a category. Since $mathsf{Hom}$ is continuous in its second argument, we have that $mathsf{Hom}(-,T)cong 1$ where $1$ is the terminal object of the category of presheaves. A global element of the presheaf $P$ is an element of $$mathsf{Nat}(1,P)congmathsf{Nat}(mathsf{Hom}(-,T),P)cong P(T)$$ where $mathsf{Nat}(F,G)$ is the set of natural transformations between $F$ and $G$, i.e. the hom-set of the category of presheaves. The second isomorphism is Yoneda. So the global elements of the presheaf $Omega$ are in correspondence with the elements of the set $Omega(T)$.
More naturally, we can use the notion of the internal language of an elementary topos which can be presented as the Mitchell-Bénabou language. This lets us write what looks like "normal" mathematics but interpret it into any elementary topos. In this context, global elements correspond to closed terms. In other words, the elements of $Omega(T)$, in this case, correspond to the closed terms of type $Omega$ in the internal language.
answered Dec 6 '18 at 4:00
Derek ElkinsDerek Elkins
16.7k11437
$endgroup$
add a comment |
$begingroup$
If by "element" they mean "global element", then a global element (of $A$) is an arrow from $1to A$ where $1$ is the terminal object. Since $T$ is the top element of the poset, it is the terminal object of that poset when viewed as a category. Since $mathsf{Hom}$ is continuous in its second argument, we have that $mathsf{Hom}(-,T)cong 1$ where $1$ is the terminal object of the category of presheaves. A global element of the presheaf $P$ is an element of $$mathsf{Nat}(1,P)congmathsf{Nat}(mathsf{Hom}(-,T),P)cong P(T)$$ where $mathsf{Nat}(F,G)$ is the set of natural transformations between $F$ and $G$, i.e. the hom-set of the category of presheaves. The second isomorphism is Yoneda. So the global elements of the presheaf $Omega$ are in correspondence with the elements of the set $Omega(T)$.
More naturally, we can use the notion of the internal language of an elementary topos which can be presented as the Mitchell-Bénabou language. This lets us write what looks like "normal" mathematics but interpret it into any elementary topos. In this context, global elements correspond to closed terms. In other words, the elements of $Omega(T)$, in this case, correspond to the closed terms of type $Omega$ in the internal language.
answered Dec 6 '18 at 4:00
Derek ElkinsDerek Elkins
16.7k11437
$endgroup$
If by "element" they mean "global element", then a global element (of $A$) is an arrow from $1to A$ where $1$ is the terminal object. Since $T$ is the top element of the poset, it is the terminal object of that poset when viewed as a category. Since $mathsf{Hom}$ is continuous in its second argument, we have that $mathsf{Hom}(-,T)cong 1$ where $1$ is the terminal object of the category of presheaves. A global element of the presheaf $P$ is an element of $$mathsf{Nat}(1,P)congmathsf{Nat}(mathsf{Hom}(-,T),P)cong P(T)$$ where $mathsf{Nat}(F,G)$ is the set of natural transformations between $F$ and $G$, i.e. the hom-set of the category of presheaves. The second isomorphism is Yoneda. So the global elements of the presheaf $Omega$ are in correspondence with the elements of the set $Omega(T)$.
More naturally, we can use the notion of the internal language of an elementary topos which can be presented as the Mitchell-Bénabou language. This lets us write what looks like "normal" mathematics but interpret it into any elementary topos. In this context, global elements correspond to closed terms. In other words, the elements of $Omega(T)$, in this case, correspond to the closed terms of type $Omega$ in the internal language.
answered Dec 6 '18 at 4:00
Derek ElkinsDerek Elkins
16.7k11437
answered Dec 6 '18 at 4:00
Derek ElkinsDerek Elkins
16.7k11437
answered Dec 6 '18 at 4:00
Derek ElkinsDerek Elkins
16.7k11437
answered Dec 6 '18 at 4:00
Derek ElkinsDerek Elkins
16.7k11437
16.7k11437
add a comment |
add a comment |
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$begingroup$
Which book${}$?
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:11
$begingroup$
a handout from my lecturer, I believe it's based off on Sheaves in Geometry and Logic: A First Introduction to Topos Theory.
$endgroup$
– SSF
Dec 6 '18 at 2:13
$begingroup$
$Omega$ is an internal Heyting algebra, that is, it is naturally a presheaf of Heyting algebras on $P$. The Heyting algebra of truth values is indeed $Omega(T)$. It's hard to tell without further details, but your monoidal actions might indeed be of this monoid.
$endgroup$
– Kevin Carlson
Dec 6 '18 at 2:27
3
$begingroup$
In a topos, sometimes $ain A$ means that $a$ is an arrow from the terminal object $1$ to the object $A$.
$endgroup$
– Lord Shark the Unknown
Dec 6 '18 at 2:38