Teacher: Paul-André Melliès

Reminders

An adjoint pair is a pair of functors

equipped with a family of bijections:

natural in $a$ and $b$.

This operation defines functors:

NB: in Homotopy Type Theory, we don’t currently know how to deal with these opposite categories.

Naturality here amounts to every commutative square

being turned into a commutative square:

Mac Lane’s Parameter theorem

In a cartesian category, we already know that the cartesian product defines a bifunctor.

Similarly, in a CCC: (Mac Lane’s Parameter theorem) the family of cartesian exponentiations

defines a unique bifunctor:

such that the bijections ${\varphi }_{A,B,C}$ are natural in $A,B,C$.

Every CCC is enriched over itself

In a cartesian closed category, the functor $\mathrm{H}\mathrm{o}\mathrm{m}$ functors as:

NB: the terminal object $1$ exists, as $𝒞$ is cartesian (in the category of types, $1$ is the empty context: closed terms: $1\stackrel{t}{\to }A$)

NB: the converse is not true: a category may be enriched over itself but not cartesian closed.

Example: Topological spaces: not cartesian closed. But a sub-category thereof is:

the category of compactly generated topological spaces (in which every set is closed iff its intersection with every compact is closed) and continuous functios is a CCC.

Simply typed $\lambda$-calculus as CCC

Inference rules/typing judgement:

Logical rules

Structural rules

will be interpreted as

For every type variable $\alpha$, we’re given an object $\xi (\alpha )$

Interpretation of a type $A$: $⟦A⟧$

Then, by structural induction, we define $⟦\bullet ⟧$ for every type.

Every sequent $x_1:A_1,\dots ,x_k:A_k⊢P:B$ is translated into $⟦A_1⟧\times ⟦A_k⟧⟶⟦B⟧$

by induction on the derivation tree:

• Variable: $⟦A⟧\stackrel{i{d}_{⟦A⟧}}{\to }⟦A⟧$

• Lambda/Curryfication: $A\times \Gamma \stackrel{f}{\to }B$ becomes, by adjunction: $\Gamma \stackrel{{\varphi }_{A,\Gamma ,B}(f)}{\to }A\Rightarrow B$

• Application: $\Gamma \stackrel{f}{\to }A\text{ and }\Delta \stackrel{g}{\to }A\Rightarrow B$

  $$\Gamma \times \Delta \stackrel{f\times g}{\to }A\times (A\Rightarrow B)\stackrel{eva{l}_{A,B}}{\to }B$$

• Contraction $\Gamma \times A\times A\stackrel{f}{\to }B$ becomes $\Gamma \times A\stackrel{\Gamma \times {\delta }_A}{\to }\Gamma \times A\times A\stackrel{f}{\to }B$

• Weakening $\Gamma \times A\stackrel{{\pi }_1}{\to }\Gamma \stackrel{f}{\to }B$

  which is isomorphic to

  $$\Gamma \times A\stackrel{\Gamma \times wk}{\to }\Gamma \times 1\simeq \Gamma \stackrel{f}{\to }B$$

• Exchange: analogous

Soundness theorem

In every CCC $𝒞$, the interpretation $⟦\bullet ⟧$ is an invariant modulo $\beta ,\eta$.

If $\Gamma ⊢(\lambda x.M):A\Rightarrow B$ and $\Delta ⊢N:A$:

If $\Gamma ⊢M:A\Rightarrow B$:

Proof: Similar to the one of subject reduction (preservation of type by $\beta$-reduction and $\eta$-expansion): by induction on the typing judgment.

Linear Logic: an intermediate notion