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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.

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  • $\begingroup$ Heyting algebra is bounded Lattice (Partial Order with all finite meets and Joins , Greatest and lowest element) with Exponentials and satisfying condition : (x ∧ a) ≤ b iff x ≤ (a ⇒ b) $\endgroup$
    – Pushpa
    Dec 7, 2014 at 16:08
  • $\begingroup$ Asked same question on stackexchange.cs but got no response in a week and so deleted it and reposted it here. $\endgroup$
    – Pushpa
    Dec 7, 2014 at 16:17

2 Answers 2

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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.

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  • $\begingroup$ Still not Completely got it ? Could You simplify your argument a little. I'm a Computer science guy . Might not be able to digest everything said above . $\endgroup$
    – Pushpa
    Dec 7, 2014 at 17:02
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    $\begingroup$ You need to understand first that posets can be viewed as special categories, namely those in which any two parallel morphisms are equal (sorry, had a typo here!) and where the only isomorphisms are the identities. Then, you need to look up the definition of coproduct and product in categories, and convince yourself that, in the case of categories arising from posets, these coincide with the notions of join and meet, respectively. Finally, view the defining property $a\wedge b\leq c\Leftrightarrow (a\leq (b\Rightarrow c))$ of the $\Rightarrow$-operator in a Heyting algebra as an adjunction... $\endgroup$
    – Hanno
    Dec 7, 2014 at 17:09
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    $\begingroup$ ... between $-\wedge b$ and $b\Rightarrow -$, and look up the fact that left adjoints preserve colimits. This is a classical result and you will probably find it in any book on category theory. $\endgroup$
    – Hanno
    Dec 7, 2014 at 17:11
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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.

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