Lecture 7: Step functions and exponential modalities

Teacher: Paul-André Melliès

\[\newcommand\xto\xrightarrow \newcommand\xfrom\xleftarrow \newcommand{\Tr}{\mathop{\mathrm{Tr}}}\]

Step functions

\[D_{A ⊸ B} = \underbrace{D_A ⊸ D_B}_{\text{domain of linear functions}}\]

NB: this theory is heavily inspired by functional analysis, and in particular step functions.

\[A ⊸ B = (A ⊗ B^⊥)^⊥ = A^⊥ ⅋ B\]

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Every pair $(a,b) ∈ \vert A \vert × \vert B \vert$ defines a linear function:

\[δ_{(a,b)} = u ⟼ \begin{cases} \lbrace b \rbrace &&\text{ if } a ∈ u \\ ∅ &&\text{ else} \end{cases}\] \[\Tr δ_{(a,b)} = \lbrace (\lbrace a \rbrace, b)\rbrace\]

Think of $δ_{(a,b)}$ as a “step function”.

Analogously, for $u ∈ !A, \, b ∈ B$:

\[δ_{(u,b)} = v ⟼ \begin{cases} \lbrace b \rbrace &&\text{ if } v ⊆ u \\ ∅ &&\text{ else} \end{cases}\]

Exercise: Show that $δ_{(u,b)}: D_A ⟶ D_B$ is a stable function.

\[\Tr δ_{(u,b)} = \lbrace (u, b)\rbrace\]

Fact: $f$ is the union of $δ_{(u,b)}$ for $(u,b) ∈ \Tr f$ whenever $f: D_A ⟶ D_B$ is a stable function:

\[f: x ⟼ \bigcup \lbrace δ_{(u,b)} \; \mid \; (u,b) ∈ \Tr f\rbrace (x) \\ = \bigcup \lbrace δ_{(u,b)}(x) \; \mid \; (u,b) ∈ \Tr f\rbrace\\ = \lbrace b \; \mid \; ∃ u ∈ !A, u ⊆ x; (u,b) ∈ \Tr f\rbrace\]

In the case of linear functions $D_A ⟶ D_B$: when can we define the union of $δ_{(a,b)}$ and $δ_{(a’,b’)}$? You want to map $\lbrace a, a’\rbrace$ to $\lbrace b, b’\rbrace$, if $a \sim_A b$ and $b \sim_B b’$: this precisely defines $A ⊸ B$!

Exponential modalities

Linear decomposition of the intuitionistic implication:

\[A ⇒ B \; = \; !A ⊸ B\] \[u \sim_{!A} v ⟺ \text{the union } u ∪ v \text{ is a finite clique }\\ ⟺ u ↑ v \text{ (compatible)}\] \[?A = (!A^⊥)^⊥\]

If $f: D_A ⟶ Σ = D_1 = D_⊥$ is a stable function determines and is determined by a set of pairwise incompatible elements of $!A$

So a stable function $D_A ⟶ Σ = D_1 = D_⊥$ is the same thing as an anticlique of $!A$, hence the same as a clique of $!A ⊸ ⊥$

More generally, a stable function $D_A ⟶ D_B$ is the same thing as a clique of $A ⊸ B$ (every stable function is the union of step functions $δ_{(u,b)}$ compatible with one another!)

\[!(A \& B) \; ≅ \; !A ⊗ !B \qquad !⊤ = 1\]

Categorical properties of the exponential:

\[Δ_A : A ⊸ A \& A\\ Δ_A = \lbrace (a, \texttt{inl } a) \; \mid \; a ∈ \vert A \vert\rbrace ∪ \lbrace (a, \texttt{inr } a) \; \mid \; a ∈ \vert A \vert\rbrace\] \[\underbrace{!A ⟶ !(A \& A) \overset{≅}{⟶} !A ⊗ !A}_{\text{comonoid structure → contraction}}\]

Using the $!$, we can turn the cartesian product (which is not closed!) into some kind of diagonal for the tensor

Monoids (category theory)

A monoid in $Set$:

\[M × M \overset{m}{⟶} M\]

that is associative (the evident rectangle commutes)