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#import "template/lib.typ": *
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#import "@preview/fletcher:0.5.8" as fletcher: diagram, node, edge
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#let scheme = dark
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#let edge = edge.with(stroke: scheme.fg, crossing-fill: scheme.bg)
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#show: paper.with(scheme: scheme, title: "SEMPL", subtitle: "Lesson 1")
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#let environment = environment.with(scheme: scheme)
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#let environment_ = environment_.with(scheme: scheme)
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#let theorem = environment.with(name: "Theorem")
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#let definition = environment.with(name: "Definition")
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#let exercise = environment.with(name: "Exercise")
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#let proof = environment_.with(name: "Proof of", follows: true)
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#outline()
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= Course Organization <nonumber>
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Paul-André Melliès
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Website : `https://www.irif.fr/~mellies/mpri/mpri-m2/mpri-mellies-slides-1.pdf`
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E-mail : `mellies@irif.fr`
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= Introduction : $lambda$-calculus <nonumber>
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There are two ways to work with $lambda$-calculus semantics:
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- Denotational Semantics
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- Operational Semantics
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With two syntax evaluations:
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- big step
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- small step
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And three strategies:
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- call-by-name (CBN)
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- call-by-value (CBV)
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- call-by-need
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And using Curry-Howard, we have a correspondance between proofs and programs.
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With C, pointers allows you access memory without knowing all the details
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behind it.
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The course will mainly speak about semantics and points of view about programs.
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The shift of C to Rust, and seperation logic were born from semantics.
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How do we have access to god ?
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= Domains and continuous functions
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It's intuitionistic logic.
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== Cartesian Closed Categories <nonumber>
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Intuitionistic (minimal) logic.
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It is deeply connected to the simply-typed $lambda$-calculus.
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Domains
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Can be seen as Domains = ordered set structure.
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$A times B$, $A => B$. A type is a Domain.
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== Introduction to domain theory
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=== Category Theory
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#let Hom = "Hom"
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#definition(subtitle: "Category")[
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A category is a _class_ of vertices (called objects) and a set of edges
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(called maps or morphisms) equiped with a family of functions
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$compose : Hom(B, C) times Hom(A, B) -> Hom(A, C)$ with:
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1. An identity map $id_A in Hom(A, A)$ such that $id_A compose f = f$ and $g = g compose id_A$
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2. Associativity $h compose (g compose f) = (h compose g) compose f$
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]
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#environment(name: "Ad", counter: none)[
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1-2-3 october at the Institut Henri Poincaré
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URL : https://www.ihp.fr/en/events/bridging-linguistic-gap-between-mathematician-and-machine ?
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]
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#definition[
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The inf of $a$ and $b$ is an element $a and b$ such that:
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1. $a and b <= a$
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2. $a and b <= b$
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3. $forall x, x <= a and x <= b => x <= a and b$
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#diagram(
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node((0, 0), $a$),
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node((2, 0), $b$),
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node((1, 1), $a and b$),
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node((1, 2), $x$),
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edge((1, 1), (0, 0), "->"),
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edge((1, 1), (2, 0), "->"),
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edge((1, 2), (1, 1), "->"),
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edge((1, 2), (0, 0), "-->", bend: 20deg),
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edge((1, 2), (2, 0), "-->", bend: -20deg),
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)
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]
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#definition[
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A cartesian product of $A$ and $B$ is a triple $(A times B, A times B arrow.r^(pi_1) A, A times B arrow.r^(pi_2) B)$
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#diagram(
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node((0, 0), $A$),
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node((2, 0), $B$),
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node((1, 1), $A times B$),
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node((1, 2), $X$),
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edge((1, 1), (0, 0), "->", $pi_1$),
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edge((1, 1), (2, 0), "->", $pi_2$),
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edge((1, 2), (1, 1), "->", $h$),
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edge((1, 2), (0, 0), "-->", bend: 20deg),
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edge((1, 2), (2, 0), "-->", bend: -20deg),
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)
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]
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#definition[
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Given two objects $A$, $B$ in a category $C$, we define the category $"Span"(A, B)$:
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- whose objects are triples $(X, X ->^f A, X ->^g B)$
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- whose maps are $(X, X ->^f A, X ->^g B) -> (Y, Y ->^f A, Y ->^g B)$ is a
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map $h: X -> Y$ in the category $C$ such that $f_X = f_Y compose h$ and $g_X = g_Y compose h$.
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]
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#exercise[
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Show that $(A times B, pi_1, pi_2)$ is a cartesian product of $A$ and $B$ iff
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it is *terminal* in $bold("Span"(A, B))$.
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]
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= Coherence spaces and cliques
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It's linear logic (Girard late 80's).
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A type is a set of points (or states). M a program is among those points when
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seen as a type.
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Monoidal Closed Categories.
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$(A -o B)^bot ~= (A (x) B)^bot$
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Linear from A to B => One input produces one output.
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$
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n |-> n + 1 &: NN -o NN\
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n |-> n + n &: !NN -o NN\
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f |-> sum^(f(7))_(i=0)(f(i)) &: !(NN -o NN) -o NN
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$
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= Dialogue games and strategies
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heh
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= Categories ?
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#definition[
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A cartesian closed category (CCC) is a cartesian category $(cal(C), times, bb(1))$
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equipped for all $A, B$ objects of $cal(C)$ with an object $A => B$ and a map
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$
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"eval"_(A, B) : A times (A => B) --> B
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$
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Such that for all objects $C$ and for all maps
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$
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f : A times C --> B
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$
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there is a unique map
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$
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h : C --> (A => B)
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$
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such that the diagramm commute
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#align(center, diagram($
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A times (A => B) edge("->", label: "eval") &B\
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A times C edge("-->", "u", label: A times h) edge("->", "ur")
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$))
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]
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== Functors and natural transformations
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#definition[
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A functor $F : cal(A) -> cal(B)$ between categories $cal(A), cal(B)$ is the
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data of an object $F A$ of $cal(B)$ for all objects $A$ of $cal(A)$, and a
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family of functions:
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$
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Hom_cal(A)(A, A') &--> Hom_cal(B)(F A, F A')\
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f : A --> A' &mapsto.long F f : F A --> F A'
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$
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Such that the composition law and identities are preserved.
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]
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#definition[
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Given two functiors $F, G : cal(A) --> cal(B)$,
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a natural transformation $theta : F => G$ is a family of maps
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$theta_A : F A --> G A$ indexed by objects $A$ of $cal(A)$.
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]
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#definition[
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If $cal(A)$ and $cal(B)$ are categories, $cal(A) times cal(B)$ whose
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objects are pairs, whose maps are pairs.
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]
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Claim : every cartesian category $(cal(C), times, bb(1))$ comes equipped with a
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functor:
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$
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times : &cal(C) times cal(C) --> cal(C)\
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&(A, B) mapsto A times B
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$
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#exercise[
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1. Show that $x times x$ defines a functor.
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2. Show that the functor $times : cal(C) times cal(C) --> cal(C)$ comes
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with two natural transformations.
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$
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phi_(A, B) : A times B ==>^(pi_1) A = "fst"(A, B)\
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psi_(A, B) : A times B ==>^(pi_2) B = "snd"(A, B)
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$
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3. $Delta_A : A --> A times A$
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#align(center, diagram({
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node((-2, 0), [$cal(C)$])
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node((0, 0), [$cal(C)$])
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node((-3, 0), [$A$])
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node((1, 0), [$A times A$])
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node((-1, -2), [$(A, A)$])
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node((-1, -1), [$cal(C) times cal(C)$])
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edge((-2, 0), (0, 0), [$"Id"$], label-side: right, "->", bend: -36deg)
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edge((-2, 0), (-1, -1), [$"diag"$], label-side: left, "->", bend: 18deg)
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edge((-1, -1), (0, 0), [$times$], label-side: left, "->", bend: 18deg)
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edge((-3, 0), (-1, -2), "|->", bend: 36deg)
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edge((-1, -2), (1, 0), "|->", bend: 36deg)
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edge((-3, 0), (1, 0), "|->", bend: -50deg)
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edge((-1, 0.4), (-1, -1), [$Delta$], label-side: right, "=>")
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}))
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4. $gamma_(A, B) : A times B --> B times A$
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#align(center, diagram({
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node((-2, 0), [$cal(C)$])
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node((0, 0), [$cal(C)$])
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node((-3, 0), [$A$])
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node((1, 0), [$A times A$])
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node((-1, -2), [$(A, A)$])
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node((-1, -1), [$cal(C) times cal(C)$])
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edge((-2, 0), (0, 0), [$"Id"$], label-side: right, "->", bend: -36deg)
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edge((-2, 0), (-1, -1), [$"diag"$], label-side: left, "->", bend: 18deg)
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edge((-1, -1), (0, 0), [$times$], label-side: left, "->", bend: 18deg)
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edge((-3, 0), (-1, -2), "|->", bend: 36deg)
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edge((-1, -2), (1, 0), "|->", bend: 36deg)
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edge((-3, 0), (1, 0), "|->", bend: -50deg)
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edge((-1, 0.4), (-1, -1), [$gamma$], label-side: right, "=>")
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}))
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]
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== Adjunctions
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#environment(name: "Fact")[
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Every category $cal(C)$ comes equipped with a functor structure on the hom sets:
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$Hom_cal(C)(A, B)$.
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$
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Hom_cal(C) : &cal(C)^"op" times cal(C) --> &"Set"\
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&(A, B) mapsto.long &Hom_cal(C)(A, B)
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$
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]
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== Cartesian closed categories
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#definition[
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1. Based on $"eval" : A times (A => B) --> B$
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2. Based on adjunctions :
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A cartesian closed category $cal(C)$ is closed when for all objects $A$ in $cal(C)$,
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the functor $A times \_ : &cal(C) --> cal(C)\ &B mapsto.long A times B$ has a right
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adjoint $A_(=>) : &cal(C) --> cal(C)\ &B mapsto.long A => B$.
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The two definitions are equivalent.
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Given definition 2, we want to recover the family of maps $"eval" : A times (A => B) --> B$.
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]
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#environment_(name: "Geneal facts about adjunctions", counter: none)[
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// #diagram($
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// "Set" edge(->, label: "Free") edge(<-, label: "Forgetful") &"Mon"
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// $)
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Example 1 :
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#align(center, diagram({
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node((0, 0), [$"Set"$])
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node((1, 0), [$"Mon"$])
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node((.5, 0), [$bot$])
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edge((0, 0), (1, 0), [Free], label-side: left, shift: 0.2, "->")
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edge((1, 0), (0, 0), [Forgetful], label-side: left, shift: 0.2, "->")
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}))
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$
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"Set"(A, M)\
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tilde.equiv\
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"Mon"(A^*, (M, eta, mu))
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$
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Example 2 :
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#align(center, diagram({
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node((0, 0), [$"Set"$])
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node((1, 0), [$"Set"$])
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node((.5, 0), [$bot$])
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edge((0, 0), (1, 0), [$A times \_$], label-side: left, shift: 0.2, "->")
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edge((1, 0), (0, 0), [$A => \_$], label-side: left, shift: 0.2, "->")
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}))
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$
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"Set"(A times B, C)\
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tilde.equiv\
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"Mon"(A, B => C)
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$
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It's currification.
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General case :
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#align(center, diagram({
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node((0, 0), [$A$])
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node((1, 0), [$B$])
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node((.5, 0), [$bot$])
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edge((0, 0), (1, 0), [$L$], label-side: left, shift: 0.2, "->")
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edge((1, 0), (0, 0), [$R$], label-side: left, shift: 0.2, "->")
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||||
}))
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Suppose $L : A --> B$ left adjoint to $R : B --> A$
|
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|
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In that case, there exists families of maps / arrows
|
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$eta : A --> R L A$ (unit of the adjonction) and $epsilon : L R B --> B$
|
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(co unit of the adjunction).
|
||||
|
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*Adjunction* $L tack.l R : cal(B)(L -, =) tilde.equiv cal(A)(-, R =)$
|
||||
|
||||
$Phi_(A, B) : cal(B)(L A, B) tilde.equiv cal(A)(A, R B)$ (natural in A and B)
|
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|
||||
$B = L A : cal(B)(L A, L A) tilde.equiv cal(A)(A, R L A)$ ($eta_A = Phi_(A, L A)(id_(L A))$)
|
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|
||||
$A = R B : cal(B)(L R B, B) tilde.equiv cal(A)(R B, R B)$ ($epsilon_B = Phi^(-1)_(R B, B)(id_(R B))$)
|
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#align(center, diagram({
|
||||
node((0, -1), [$cal(A)^(op) times cal(B)$])
|
||||
node((0, 1), [$"Set"$])
|
||||
node((-1, 0), [$cal(A)^op times cal(A)$])
|
||||
node((1, 0), [$cal(B)^op times cal(B)$])
|
||||
node((2, 0), [$(L A, B)$])
|
||||
node((-2, 0), [$(A, R B)$])
|
||||
edge((0, -1), (-1, 0), [$\_ times R$], label-side: right, "->")
|
||||
edge((0, -1), (1, 0), [$L^op times \_$], label-side: left, "->")
|
||||
edge((-1, 0), (0, 1), [$"Hom"_cal(A)$], label-side: right, "->")
|
||||
edge((1, 0), (0, 1), [$"Hom"_cal(B)$], label-side: left, "->")
|
||||
edge((1, 0), (-1, 0), "=>")
|
||||
}))
|
||||
|
||||
$eta$ is a natural transformation :
|
||||
|
||||
#align(center, diagram({
|
||||
node((-1, -1), [$A$])
|
||||
node((0, -1), [$R L A$])
|
||||
node((-1, 0), [$A'$])
|
||||
node((0, 0), [$R L A'$])
|
||||
edge((-1, -1), (0, -1), [$eta_A$], label-side: left, "->")
|
||||
edge((-1, -1), (-1, 0), [$h$], label-side: right, "->")
|
||||
edge((0, -1), (0, 0), [$R L h$], label-side: left, "->")
|
||||
edge((-1, 0), (0, 0), [$epsilon_A'$], "->")
|
||||
}))
|
||||
|
||||
Which commutes for all $h : A --> A'$.
|
||||
|
||||
#align(center, diagram({
|
||||
node((-1, -1), [$L R B$])
|
||||
node((0, -1), [$B$])
|
||||
node((-1, 0), [$L R B'$])
|
||||
node((0, 0), [$B'$])
|
||||
edge((-1, -1), (0, -1), [$epsilon_B$], label-side: left, "->")
|
||||
edge((-1, -1), (-1, 0), [$L R h$], label-side: right, "->")
|
||||
edge((0, -1), (0, 0), [$h$], label-side: left, "->")
|
||||
edge((-1, 0), (0, 0), [$epsilon_B'$], label-side: right, "->")
|
||||
}))
|
||||
|
||||
Which commutes for all $h : B --> B'$.
|
||||
|
||||
We have $eta : "Id"_cal(A) => R compose L$ and $epsilon : L compose R => "Id"_cal(B)$.
|
||||
|
||||
#align(center, diagram({
|
||||
node((-1, -1), [$cal(A)$])
|
||||
node((1, -1), [$cal(A)$])
|
||||
node((0, 0), [$cal(B)$])
|
||||
node((0, -2))
|
||||
edge((-1, -1), (1, -1), [$"Id"_cal(A)$], label-side: left, "->", bend: 54deg)
|
||||
edge((-1, -1), (0, 0), [$L$], label-side: right, "->", bend: -18deg)
|
||||
edge((0, 0), (1, -1), [$R$], label-side: right, "->", bend: -18deg)
|
||||
edge((0, -1.5), (0, 0), [$eta$], label-side: left, "=>")
|
||||
}))
|
||||
|
||||
Back to example 1 :
|
||||
|
||||
A a set,
|
||||
|
||||
$eta_A : A &--> A^*\ a &mapsto.long [a]$
|
||||
|
||||
M a monoid,
|
||||
|
||||
$"eval"_M = epsilon_M ; M^* &--> M\ [m_1, ..., m_k] &mapsto.long m_A ⋅ ... ⋅ m_k$
|
||||
|
||||
Back to example 2 :
|
||||
|
||||
unit of $A times \_ tack.l A => \_$ is a CCC.
|
||||
|
||||
$eta^A_B : B &--> (A => (A times B))\ b &mapsto.long (a mapsto (A, B))$ (it's coeval)
|
||||
|
||||
counit of $A times \_ tack.l A => \_$ is a CCC.
|
||||
|
||||
$epsilon^A_B ; (A times (A => B)) &--> B\ (a, f) &mapsto.long f(A)$ (it's eval)
|
||||
]
|
||||
|
||||
== Coherence Categories
|
||||
|
||||
Category *Coh* :
|
||||
$
|
||||
f : A --> B
|
||||
$
|
||||
is a relation
|
||||
$
|
||||
f subset.eq |A| times |B|
|
||||
$
|
||||
such that :
|
||||
$
|
||||
forall (a, b) in f, (a', b') in f, a "coh"_A a' => b "coh"_B b' => a "ncoh"_A a'
|
||||
$
|
||||
i.e., $f$ is a clique of $A -> B$.
|
||||
|
||||
$f : 1 -> A$ is a clique of A.
|
||||
|
||||
$g : A -> bot tilde.eq 1$ an anticlique of $A$ <=> a clique of $A multimap bot tilde.eq A^bot$.
|
||||
|
||||
#align(center, diagram({
|
||||
node((-3, -1), [$A$])
|
||||
node((-5, -1), [$1$])
|
||||
node((-1, -1), [$bot tilde.equiv 1$])
|
||||
node((-6, -1), [$*$])
|
||||
node((0, -1), [$*$])
|
||||
edge((-5, -1), (-3, -1), [$f "(clique)"$], label-side: left, "->")
|
||||
edge((-3, -1), (-1, -1), [$g "(anti clique)"$], label-side: left, "->")
|
||||
edge((-5, -1), (-1, -1), "->", bend: -30deg)
|
||||
edge((-6, -1), (-5, -1), [$in$], label-side: center, " ")
|
||||
edge((0, -1), (-1, -1), [$in.rev$], label-side: center, " ")
|
||||
}))
|
||||
|
||||
=== Categorical structure of *Coh*
|
||||
|
||||
==== The cartesian product of A and B
|
||||
|
||||
#align(center, diagram({
|
||||
node((-1, -1), [$A \& B$])
|
||||
node((-2, -2), [$A$])
|
||||
node((0, -2), [$B$])
|
||||
node((-1, 1), [$X$])
|
||||
edge((-1, 1), (-1, -1), [$exists!h$], label-side: center, "-->")
|
||||
edge((-1, -1), (-2, -2), [$pi_1$], label-side: center, "->")
|
||||
edge((-1, -1), (0, -2), [$pi_2$], label-side: center, "->")
|
||||
edge((-1, 1), (-2, -2), [$f$], label-side: center, "->", bend: 18deg)
|
||||
edge((-1, 1), (0, -2), [$g$], label-side: center, "->", bend: -18deg)
|
||||
}))
|
||||
|
||||
$
|
||||
&pi_1 = {("inl" a, a) | a in |A|}\
|
||||
&pi_2 = {("inr" b, b) | b in |B|}\
|
||||
&|A \& B| = {"inl" a | a in |A|} union.plus {"inr" b | b in |B|}\
|
||||
&"given" f : X -> A "and" g : X -> B\
|
||||
&h = {(x, "inl" a) | (x, a) in f} union.plus {(x, "inr" b) | (x, b) in f}
|
||||
$
|
||||
|
||||
Why is $h$ a clique ?
|
||||
|
||||
terminal object : empty clique
|
||||
|
||||
==== The category of Coh is cartesian, but not cartesian closed
|
||||
|
||||
#import "@preview/pinit:0.2.2": *
|
||||
|
||||
$
|
||||
pin("1") A \& B --> C pin("2")
|
||||
$
|
||||
|
||||
#pinit-highlight("1", "2", fill: blue.transparentize(75%))
|
||||
#pinit-point-from("2", "No way to currify", offset-dy: 10pt, body-dy: 0pt, fill: white)
|
||||
|
||||
==== On the other hand, there is a bijection
|
||||
|
||||
$
|
||||
"Coh"(A (times) B, C) tilde.eq "Coh"(B, A -o C)
|
||||
$
|
||||
|
||||
=== Exponentials in coherence spaces and linear logic
|
||||
|
||||
#let coh = "⁐"
|
||||
#let ncoh = "ncoh"
|
||||
|
||||
$!A "has web" |!A|$ defined as the set of finite clique
|
||||
|
||||
$coh_(A!)$ when
|
||||
1. $exists w$ a finite clique $u, v subset.eq w$.
|
||||
2. (equiv) $forall a in u, forall tau' in v$, $a coh_(A!) a'$
|
||||
|
||||
==== Two main structures
|
||||
|
||||
$!A$ defines a _commutative comonoid_ in Coh.
|
||||
|
||||
#definition[
|
||||
A monoid $(M, m, e)$ in a monoidal category $(cal(C), (times), bb(1))$ is an object
|
||||
$M$ of $cal(C)$ equipped with two maps :
|
||||
|
||||
$m : M times M --> M$ $e : bb(1) --> M$
|
||||
|
||||
#align(center, diagram({
|
||||
node((0, 0), [$M times M times M$])
|
||||
node((1, 0), [$M times M$])
|
||||
node((0, 1), [$M times M$])
|
||||
node((1, 1), [$M$])
|
||||
node((3, 0), [$M$])
|
||||
node((4, 0), [$bb(1) times M$])
|
||||
node((4, 1), [$M times M$])
|
||||
node((6, 0), [$M$])
|
||||
node((7, 0), [$M times bb(1)$])
|
||||
node((7, 1), [$M times M$])
|
||||
edge((0, 0), (1, 0), [$M times m$], label-side: left, "->")
|
||||
edge((0, 0), (0, 1), [$m times M$], label-side: right, "->")
|
||||
edge((0, 1), (1, 1), [$m$], label-side: right, "->")
|
||||
edge((1, 0), (1, 1), [$m$], label-side: left, "->")
|
||||
edge((3, 0), (4, 0), [$lambda^(-1)_M$], "->")
|
||||
edge((4, 0), (4, 1), [$e times M$], label-side: left, "->")
|
||||
edge((4, 1), (3, 0), [$m$], label-side: left, "->")
|
||||
edge((7, 1), (6, 0), [$m$], label-side: left, "->")
|
||||
edge((7, 0), (7, 1), [$M times e$], label-side: left, "->")
|
||||
edge((6, 0), (7, 0), [$lambda^(-1)_M$], "->")
|
||||
}))
|
||||
|
||||
#align(center, diagram({
|
||||
node((3, 0), [$M$])
|
||||
node((3, 1), [$M$])
|
||||
edge((3, 0), (3, 1), [$id$], label-side: right, "->", bend: -36deg)
|
||||
edge((3, 0), (3, 1), [$m compose (e times M) compose lambda^(-1)_M$], label-side: left, "->", bend: 36deg)
|
||||
}))
|
||||
]
|
||||
|
||||
#definition[
|
||||
A comonoid in $(cal(C), (times), bb(A))$ is a monoid in the opposite monoidal category
|
||||
$(cal(C)"op", (times), bb(1))$.
|
||||
|
||||
$d : M --> M (times) M$ $u : M --> bb(1)$
|
||||
|
||||
#align(center, diagram({
|
||||
node((1, 0), [$M$])
|
||||
node((2, 0), [$M times.o M$])
|
||||
node((2, 1), [$M times.o M times.o M$])
|
||||
node((1, 1), [$M times.o M$])
|
||||
node((4, 0), [$M times.o M$])
|
||||
node((4, 1), [$bb(1) times.o M$])
|
||||
node((5, 0), [$M$])
|
||||
node((6, 0), [$M times.o M$])
|
||||
node((6, 1), [$M times.o bb(1)$])
|
||||
edge((1, 0), (2, 0), [$d$], label-side: left, "->")
|
||||
edge((1, 0), (1, 1), [$d$], label-side: right, "->")
|
||||
edge((1, 1), (2, 1), [$M times.o d$], label-side: right, "->")
|
||||
edge((2, 0), (2, 1), [$M times.o d$], label-side: left, "->")
|
||||
edge((5, 0), (4, 0), [$d$], label-side: right, "->")
|
||||
edge((5, 0), (6, 0), [$d$], label-side: left, "->")
|
||||
edge((4, 0), (4, 1), [$u times.o M$], label-side: right, "->")
|
||||
edge((6, 0), (6, 1), [$M times.o u$], label-side: left, "->")
|
||||
edge((4, 1), (5, 0), [$lambda_M$], label-side: right, "->")
|
||||
edge((6, 1), (5, 0), [$Gamma_M$], label-side: left, "->")
|
||||
}))
|
||||
|
||||
#align(center, diagram({
|
||||
node((1, 0), [$!A$])
|
||||
node((2, 0), [$!A times.o !A$])
|
||||
node((1, 1), [$!A times.o !A$])
|
||||
node((2, 1), [$!A times.o !A times.o !A$])
|
||||
edge((1, 0), (2, 0), [$d_A$], label-side: left, "->")
|
||||
edge((1, 0), (1, 1), [$d_A$], label-side: right, "->")
|
||||
edge((1, 1), (2, 1), [$!A times.o d_A$], label-side: right, "->")
|
||||
edge((2, 0), (2, 1), [$d_A times.o !A$], label-side: left, "->")
|
||||
edge((1, 0), (2, 1), [$d_a times.o d_a$], label-side: center, "->")
|
||||
}))
|
||||
]
|
||||
|
||||
#definition[
|
||||
A commutative monoid in a symmetric monoidal category is a monoid $(M, m, e)$
|
||||
such that this commutes :
|
||||
#align(center, diagram({
|
||||
node((1, 0), [$M times.o M$])
|
||||
node((2, .5), [$M$])
|
||||
node((1, 1), [$M times.o M$])
|
||||
edge((1, 0), (2, .5), [$m$], label-side: left, "->")
|
||||
edge((1, 1), (2, .5), [$m$], label-side: right, "->")
|
||||
edge((1, 0), (1, 1), [$gamma_(M,M)$], label-side: right, "->")
|
||||
}))
|
||||
|
||||
By duality, a commutative comonoid :
|
||||
#align(center, diagram({
|
||||
node((2, 0), [$M times.o M$])
|
||||
node((1, .5), [$M$])
|
||||
node((2, 1), [$M times.o M$])
|
||||
edge((1, .5), (2, 0), [$d_m$], label-side: left, "->")
|
||||
edge((1, .5), (2, 1), [$d_m$], label-side: right, "->")
|
||||
edge((2, 0), (2, 1), [$gamma_(M,M)$], label-side: right, "->")
|
||||
}))
|
||||
]
|
||||
|
||||
#definition[
|
||||
A monad is a monoid in the category of endofunctors.
|
||||
|
||||
A monad $T : cal(C) --> cal(C)$ is a functor equipped with a pair of natural
|
||||
transformations.
|
||||
|
||||
$
|
||||
eta_A : A --> A quad mu_A : T T A --> T A
|
||||
$
|
||||
]
|
||||
|
||||
#definition[
|
||||
A comonad $K: cal(C) --> cal(C)$ on $cal(C)$ is defined as a monad on $cal(C)op$.
|
||||
|
||||
$
|
||||
delta_A : A --> A quad epsilon_A : K A --> A
|
||||
$
|
||||
]
|
||||
|
||||
|
||||
The exponential modality $!$ as a comonad $! : "Coh" --> "Coh"$ defines a functor.
|
||||
|
||||
$
|
||||
[f : A --> B] mapsto.long [! f : !A --> !B]
|
||||
$
|
||||
|
||||
$f$ a clique of $A -o B$.
|
||||
|
||||
$
|
||||
!f = { (u, v) cases(forall a in u\, exists b in v \, (a\, b) in f, forall b in v\, exists a in u \, (a\, b) in f, delim: "|") }
|
||||
$
|
||||
|
||||
$
|
||||
delta_A = {(u, {v_1, ..., v_n}) | u = v_1 union ... union v_n} "(each "v_i" is a finite clique of "A")"
|
||||
$
|
||||
|
||||
$
|
||||
epsilon_A = {({a}, a)}
|
||||
$
|
||||
|
||||
Intuitionistic linear logic :
|
||||
$
|
||||
A_1, ..., A_n tack B arrow.r.squiggly.long [A_1] (times) ... (times) [A_n] --> [B]
|
||||
$
|
||||
|
||||
Contraction : $(Gamma, !A, !A tack B)/(Gamma, !A tack B)$ which is $!A -->^(d_A) !A (times) !A$
|
||||
|
||||
Weakening : $(Gamma tack B)/(Gamma, !A tack B)$ which is $!A -->^(u_A) bb(1)$
|
||||
|
||||
Dereliction : $(Gamma, A tack B)/(Gamma, !A tack B)$ which is $!A -->^(epsilon_A) A$
|
||||
|
||||
Digging : $(!A_1, ..., !A_n tack B)/(!A_1, ..., !A_n tack !B)$
|
||||
|
||||
Which is $!A -->^(delta_A) !!A$ and $!A (times) !B tilde.equiv !(A \& B)$
|
||||
|
||||
$
|
||||
!(A \& A') tilde.equiv !A (times) !A' &-->f B\
|
||||
!!(A \& A') tilde.equiv !A (times) !A' &-->^(!f) !B
|
||||
$
|
||||
Loading…
Add table
Add a link
Reference in a new issue