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The contravariant powerset functor $P : C^{\text{op}} \rightarrow C$ is the canonical example of duality.[1]

There are several intereresting duality principles that amount to application of the contravariant powerset functor. The duality between products and coproducts in category theory can be seen as application of the contravariant powerset functor on the definition of the coproduct. $P(A+B) \cong PA \times PB$ and \begin{gather*} P([f,g] : A+B \rightarrow D) = \\ \langle Pf,Pg\rangle : PD \rightarrow P(A + B). \end{gather*} Similarly for injections to the coproduct: \begin{gather*} P(i_1 : A \rightarrow A+B) = \\ \pi_1 : P(A+B) \rightarrow PA. \end{gather*} By expanding $P(A+B) \cong PA \times PB$ and using the coproduct laws, these are seen to be the product laws, which justifies use of that notation for products above, where you often suppress explicit notation for the isomorphism. The other projection is similar. This kind of application of the contravariant powerset functor can be done for all finite colimits.

For exponentials, the situation is much different. Trying to find operation $B \setminus A$ (not the set subtraction, but close) such that the contravariant powerset functor produces an exponential, something along the lines of $P(B \setminus A) \cong PA \Rightarrow PB$ is a cause of much confusion about duality, since coexponentials are not very natural concept [2] and can cause havoc when combined with, say, a topos. Inverting the arrows in $C^{op}$, in essence $B \setminus A \cong B \Rightarrow A$, seems obvious solution, but there aren't necessarily exponentials in $C^{op}$, and (speculation:) the notion of exponential as internalization of hom-sets somehow inverts as well. There are several important notions of topology (open sets, bounded sets) that seem suitable for such definition in terms of exponential $PA \Rightarrow PB$. This also interacts with problems defining a left adjoint to a partially applied coproduct functor $F \dashv A + -$ (which might also be called subtraction if it exists). I can see two possible outcomes for such construct. Either the power set hierarchy expands without limit and iterating the exponential (which is similar to iterating the power set) produces ever larger categories. Or the possibilities for such expansion are limited, and some principle, e.g. Rice's theorem, limits the expressive power of repeated towers of exponentials.

[1] Lawvere, Rosebrugh: "Sets for mathematics"

[2] Crolard: Subtractive logic

The contravariant powerset functor, i.e. covariant functor $P : C^{\text{op}} \rightarrow C$ defined as $P(A)=2^A$ and $P(f)(g) = g \circ f$ is the canonical example of duality.[1] The category $C$ is usually $Set$, but generalizations to toposes exist.

There are several intereresting duality principles that amount to application of the contravariant powerset functor. The duality between products and coproducts in category theory can be seen as application of the contravariant powerset functor on the definition of the coproduct. $P(A+B) \cong PA \times PB$ and \begin{gather*} P([f,g] : A+B \rightarrow D) = \\ \langle Pf,Pg\rangle : PD \rightarrow P(A + B). \end{gather*} Similarly for injections to the coproduct: \begin{gather*} P(i_1 : A \rightarrow A+B) = \\ \pi_1 : P(A+B) \rightarrow PA. \end{gather*} By expanding $P(A+B) \cong PA \times PB$ and using the coproduct laws, these are seen to be the product laws, which justifies use of that notation for products above, where you often suppress explicit notation for the isomorphism. The other projection is similar. This kind of application of the contravariant powerset functor can be done for all finite colimits.

For exponentials, the situation is much different. Trying to find operation $B \setminus A$ (not the set subtraction, but close) such that the contravariant powerset functor produces an exponential, something along the lines of $P(B \setminus A) \cong PA \Rightarrow PB$ is a cause of much confusion about duality, since coexponentials are not very natural concept [2] and can cause havoc when combined with, say, a topos. Inverting the arrows in $C^{op}$, in essence $B \setminus A \cong B \Rightarrow A$, seems obvious solution, but there aren't necessarily exponentials in $C^{op}$, and (speculation:) the notion of exponential as internalization of hom-sets somehow inverts as well. There are several important notions of topology (open sets, bounded sets) that seem suitable for such definition in terms of exponential $PA \Rightarrow PB$. This also interacts with problems defining a left adjoint to a partially applied coproduct functor $F \dashv A + -$ (which might also be called subtraction if it exists). I can see two possible outcomes for such construct. Either the power set hierarchy expands without limit and iterating the exponential (which is similar to iterating the power set) produces ever larger categories. Or the possibilities for such expansion are limited, and some principle, e.g. Rice's theorem, limits the expressive power of repeated towers of exponentials.

[1] Lawvere, Rosebrugh: "Sets for mathematics"

[2] Crolard: Subtractive logic

The contravariant powerset functor $P : C^{\text{op}} \rightarrow C$ is the canonical example of duality.[1]

There are several intereresting duality principles which amount to application of the contravariant powerset functor. The duality between products and coproducts in category theory can be seen as application of the contravariant powerset functor on the definition of the coproduct. $P(A+B) \cong PA \times PB$ and \begin{gather*} P([f,g] : A+B \rightarrow D) = \\ \langle Pf,Pg\rangle : PD \rightarrow P(A + B). \end{gather*} Similarly for injections to the coproduct: \begin{gather*} P(i_1 : A \rightarrow A+B) = \\ \pi_1 : P(A+B) \rightarrow PA. \end{gather*} By expanding $P(A+B) \cong PA \times PB$ and using the coproduct laws, these are seen to be the product laws, which justifies use of that notation for products above, where you often suppress explicit notation for the isomorphism. The other projection is similar. This kind of application of the contravariant powerset functor can be done for all finite colimits.

For exponentials, the situation is much different. Trying to find operation $B \setminus A$ (not the set subtraction, but close) such that the contravariant powerset functor produces an exponential, something along the lines of $P(B \setminus A) \cong PA \Rightarrow PB$ is a cause of much confusion about duality, since coexponentials are not very natural concept [2] and can cause havoc when combined with, say, a topos. Inverting the arrows in $C^{op}$, in essence $B \setminus A \cong B \Rightarrow A$, seems obvious solution, but there aren't necessarily exponentials in $C^{op}$, and (speculation:) the notion of exponential as internalization of hom-sets somehow inverts as well. There are several important notions of topology (open sets, bounded sets) that seem suitable for such definition in terms of exponential $PA \Rightarrow PB$. This also interacts with problems defining a left adjoint to a partially applied coproduct functor $F \dashv A + -$ (which might also be called subtraction if it exists). I can see two possible outcomes for such construct. Either the power set hierarchy expands without limit and iterating the exponential (which is similar to iterating the power set) produces ever larger categories. Or the possibilities for such expansion are limited, and some principle, e.g. Rice's theorem, limits the expressive power of repeated towers of exponentials.

[1] Lawvere, Rosebrugh: "Sets for mathematics"

[2] Crolard: Subtractive logic

The contravariant powerset functor $P : C^{\text{op}} \rightarrow C$ is the canonical example of duality.[1]

There are several intereresting duality principles that amount to application of the contravariant powerset functor. The duality between products and coproducts in category theory can be seen as application of the contravariant powerset functor on the definition of the coproduct. $P(A+B) \cong PA \times PB$ and \begin{gather*} P([f,g] : A+B \rightarrow D) = \\ \langle Pf,Pg\rangle : PD \rightarrow P(A + B). \end{gather*} Similarly for injections to the coproduct: \begin{gather*} P(i_1 : A \rightarrow A+B) = \\ \pi_1 : P(A+B) \rightarrow PA. \end{gather*} By expanding $P(A+B) \cong PA \times PB$ and using the coproduct laws, these are seen to be the product laws, which justifies use of that notation for products above, where you often suppress explicit notation for the isomorphism. The other projection is similar. This kind of application of the contravariant powerset functor can be done for all finite colimits.

For exponentials, the situation is much different. Trying to find operation $B \setminus A$ (not the set subtraction, but close) such that the contravariant powerset functor produces an exponential, something along the lines of $P(B \setminus A) \cong PA \Rightarrow PB$ is a cause of much confusion about duality, since coexponentials are not very natural concept [2] and can cause havoc when combined with, say, a topos. Inverting the arrows in $C^{op}$, in essence $B \setminus A \cong B \Rightarrow A$, seems obvious solution, but there aren't necessarily exponentials in $C^{op}$, and (speculation:) the notion of exponential as internalization of hom-sets somehow inverts as well. There are several important notions of topology (open sets, bounded sets) that seem suitable for such definition in terms of exponential $PA \Rightarrow PB$. This also interacts with problems defining a left adjoint to a partially applied coproduct functor $F \dashv A + -$ (which might also be called subtraction if it exists). I can see two possible outcomes for such construct. Either the power set hierarchy expands without limit and iterating the exponential (which is similar to iterating the power set) produces ever larger categories. Or the possibilities for such expansion are limited, and some principle, e.g. Rice's theorem, limits the expressive power of repeated towers of exponentials.

[1] Lawvere, Rosebrugh: "Sets for mathematics"

[2] Crolard: Subtractive logic

The contravariant powerset functor $P : C^{\text{op}} \rightarrow C$ is the canonical example of duality.[1]

There are several intereresting duality principles which amount to application of the contravariant powerset functor. The duality between products and coproducts in category theory can be seen as application of the contravariant powerset functor on the definition of the coproduct. $P(A+B) \cong PA \times PB$ and \begin{gather*} P([f,g] : A+B \rightarrow D) = \\ \langle Pf,Pg\rangle : PD \rightarrow P(A + B). \end{gather*} Similarly for injections to the coproduct: \begin{gather*} P(i_1 : A \rightarrow A+B) = \\ \pi_1 : P(A+B) \rightarrow PA. \end{gather*} By expanding $P(A+B) \cong PA \times PB$ and using the coproduct laws, these are seen to be the product laws, which justifies use of that notation for products above, where you often suppress explicit notation for the isomorphism. The other projection is similar. This kind of application of the contravariant powerset functor can be done for all finite colimits.

For exponentials, the situation is much different. Trying to find operation $B \setminus A$ (not the set subtraction, but close) such that the contravariant powerset functor produces an exponential, something along the lines of $P(B \setminus A) \cong PA \Rightarrow PB$ is a cause of much confusion about duality, since coexponentials are not very natural concept [2] and can cause havoc when combined with, say, a topos. Inverting the arrows $B \setminus A \cong B \Rightarrow A$ in $C^{op}$ seems obvious solution, but there aren't necessarily exponentials in $C^{op}$, and (speculation:) the notion of exponential as internalization of hom-sets somehow inverts as well. There are several important notions of topology (open sets, bounded sets) that seem suitable for such definition in terms of exponential $PA \Rightarrow PB$. This also interacts with problems defining a left adjoint to a partially applied coproduct functor $F \dashv A + -$ (which might also be called subtraction if it exists). I can see two possible outcomes for such construct. Either the power set hierarchy expands without limit and iterating the exponential (which is similar to iterating the power set) produces ever larger categories. Or the possibilities for such expansion are limited, and some principle, e.g. Rice's theorem, limits the expressive power of repeated towers of exponentials.

[1] Lawvere, Rosebrugh: "Sets for mathematics"

[2] Crolard: Subtractive logic

The contravariant powerset functor $P : C^{\text{op}} \rightarrow C$ is the canonical example of duality.[1]

There are several intereresting duality principles which amount to application of the contravariant powerset functor. The duality between products and coproducts in category theory can be seen as application of the contravariant powerset functor on the definition of the coproduct. $P(A+B) \cong PA \times PB$ and \begin{gather*} P([f,g] : A+B \rightarrow D) = \\ \langle Pf,Pg\rangle : PD \rightarrow P(A + B). \end{gather*} Similarly for injections to the coproduct: \begin{gather*} P(i_1 : A \rightarrow A+B) = \\ \pi_1 : P(A+B) \rightarrow PA. \end{gather*} By expanding $P(A+B) \cong PA \times PB$ and using the coproduct laws, these are seen to be the product laws, which justifies use of that notation for products above, where you often suppress explicit notation for the isomorphism. The other projection is similar. This kind of application of the contravariant powerset functor can be done for all finite colimits.

For exponentials, the situation is much different. Trying to find operation $B \setminus A$ (not the set subtraction, but close) such that the contravariant powerset functor produces an exponential, something along the lines of $P(B \setminus A) \cong PA \Rightarrow PB$ is a cause of much confusion about duality, since coexponentials are not very natural concept [2] and can cause havoc when combined with, say, a topos. Inverting the arrows in $C^{op}$, in essence $B \setminus A \cong B \Rightarrow A$, seems obvious solution, but there aren't necessarily exponentials in $C^{op}$, and (speculation:) the notion of exponential as internalization of hom-sets somehow inverts as well. There are several important notions of topology (open sets, bounded sets) that seem suitable for such definition in terms of exponential $PA \Rightarrow PB$. This also interacts with problems defining a left adjoint to a partially applied coproduct functor $F \dashv A + -$ (which might also be called subtraction if it exists). I can see two possible outcomes for such construct. Either the power set hierarchy expands without limit and iterating the exponential (which is similar to iterating the power set) produces ever larger categories. Or the possibilities for such expansion are limited, and some principle, e.g. Rice's theorem, limits the expressive power of repeated towers of exponentials.

[1] Lawvere, Rosebrugh: "Sets for mathematics"

[2] Crolard: Subtractive logic