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I tried dualizing the concept of exponentiation in a cartesian closed category. This lends a "coexponentiation". I know that exponentiation is like the internal hom or an abstract function space. I'm wondering if there is any similar intuition for what I called "coexponentiation".

Here's how I dualized things and what I ended up with. Call $C$ a dual cartesian closed category if: (i) the functor $!:C\to\mathbb{1}$ is such that $i\dashv !$. This lends an initial object $0=i(*)$.

(ii) the functor $\Delta:C\to C\times C$ is such that $(-)+(-)\dashv \Delta$. Let the unit and counit be $\iota : 1_{C\times C}\Rightarrow\Delta\circ(-+-)$ and $\gamma:(-+-)\circ\Delta\Rightarrow 1_C$. This lends coproducts with inclusion maps determined by $\iota$.

(iii) the functor $+$ is closed, with pointwise left adjoint $ce:C\times C\to C$. That is, for all $x\in C$, $ce(X)=ce_X\dashv X+(-)$. Let the unit and counit be $_X\eta:ce_X\circ(X+-)\Rightarrow 1_C$ and $_X\epsilon:1_C\Rightarrow(X+-)\circ ce_X$.

I worked out some basic properties of such a $C$. They are, as one would expect, just the dual properties of a cartesian closed category. Since we have a natural isomorphism $\iota_0:1_C\Rightarrow(-)+0$, we can take $_X\epsilon_0:ce_X(A)\to0$. It follows that we have a morphism $f:Y\to X$ if, and only if, we have a morphism $\bar{f}=\epsilon_0ce_X(f):ce_X(Y)\to 0$. And there are some easily provable "computation" rules such as $f=(X+\bar{f}_X)_X\eta_Y$.

So there's an interesting correspondence with $f:Y\to X$ and $\bar{f}:ce_X(Y)\to 0$. And we can "compute" the morphism $f$ via $(X+\bar{f})\eta_Y$. Is there any kind of structure this can be thought of? In many familiar categories there are no non-trival maps into the initial object. So maybe there just are no familiar categories that would non-trivially admit this structure. Are there any categories with interesting maps into the initial object? And are there any categories that have this "dual cartiesan closed structure"?

If not, I guess I curious as to why maps out of the terminal object are often very interesting and useful, but the same is not true of maps into the initial object.

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    $\begingroup$ Just a side note: if one takes the dual of the categories you consider when you say "maps out of the terminal object are often interesting", we find categories where maps to the initial object are interesting. It might look like a formal thing to do, but there are interesting examples. To define a "point" of a frame, one say it is a map to the initial frame. The same goes for Boolean algebras or distributive lattices (where maps from the terminal object are not interesting but maps to the initial object are, cf. Stone duality). $\endgroup$ Jun 6, 2022 at 4:46
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    $\begingroup$ Take any cartesian closed category, like Set for instance. Its dual (complete atomic Boolean algebras) has a "dual cartesian closed structure"... But I don't have much intuition of it. $\endgroup$ Jun 6, 2022 at 4:56

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