Proving distributivity of Heyting algebras with the Yoneda lemma.

Question

How can one prove distributivity of a Heyting Algebra via the Yoneda lemma?

I'm able to prove it using the Heyting algebra property $(x \wedge a) \leq b$ if and only if $x \leq (a \Rightarrow b)$. But using Yoneda Lemma, I'm unable to get that. Help Greatly appreciated . Thank You Guys.

Answer 1

For a Heyting algebra $H$, the category ${\mathcal H}$ associated to the poset underlying $H$ is a small, skeletal, Cartesian closed category with finite coproducts in which any two parallel morphisms are equal - and in fact this assignment is a bijection. Now if $x\in H$ then $x\wedge -: {\mathcal H}\to{\mathcal H}$ is a left adjoint with right adjoint $x\Rightarrow -$, hence preserves colimits up to canonical isomorphism; in particular, for any $y_1,...,y_n\in{\mathcal H}$, the canonical morphism $\varphi: (x\wedge y_1)\vee ...\vee (x\wedge y_n)\to x\wedge (y_1\vee ...\vee y_n)$ is an isomorphism in ${\mathcal H}$. By the above-mentioned properties of ${\mathcal H}$, this means that $(x\wedge y_1)\vee ...\vee (x\wedge y_n)=x\wedge (y_1\vee ...\vee y_n)$ and that $\varphi$ is the identity.

The Yoneda lemma is hidden here in the proof of the fact that left adjoints preserve colimits.

Answer 2

Use full and faithfulness of Yoneda embedding to prove every cartesian closed category C with finite coproducts must be distributive, i.e. $ a × (b + c)$ and $(a × b) + (a × c)$ are isomorphic objects in C.