Tricocycloids, Effect Monoids & Effectuses

Supervised by:   Richard Garner

Sydney,   April - August 2019

Working Environment

Centre Of Australian Category Theory (CoACT)

Richard Garner
My Supervisor, Senior Lecturer

Steve Lack
Director, Associate Professor

Ross Street
Founding Director, Emeritus Professor

Dominic Verity
Associate Director, Professor

Centre Of Australian Category Theory (CoACT)

Tom Leinster calls them "ninja category theorists"

« One of my early Honours students at Macquarie University baffled his proposed Queensland graduate studies supervisor who asked whether the student knew the definition of a topological space. The aspiring researcher on dynamical systems answered positively:
"Yes, it is a relational $β$-module!"
I received quite a bit of flak from colleagues concerning that one; but the student Peter Kloeden went on to become a full professor of mathematics in Australia then Germany. »

— Ross Street (An Australian conspectus of higher categories)

I. The distribution monad $\small {\D M}$

Shown to be a linear exponential monad Garner's 2018 "Hypernormalisation" article.

In what follows: $M ≝ [0, 1] ⊆ ℝ$ and $\;\ovee ≝ +$

Discrete distribution monad $\; \D M：\Set → \Set$:

• \begin{align*} \D M(X) &≝ \Big\lbrace φ：X → M \;\mathrel{\Big|}\;\supp(φ) \text{ finite and } \bigovee\limits_{x ∈ X} φ(x) = 1\Big\rbrace\\ &= \bigg\lbrace {\color{orange} \sum\limits_{i = 1}^n r_i \Ket{x_i}} \;\mathrel{\Big|}\; x_i ∈ X, \, r_i ∈ M, \, \bigovee\limits_{i} r_i = 1\bigg\rbrace \end{align*}\\
• $$η_X(x) ≝ 1 \Ket x \qquad\qquad μ_X\Big(\sum\limits_{i = 1}^n r_i \Ket{φ_i}\Big) ≝ \sum\limits_{x ∈ X} \Big(\bigovee\limits_{i=1}^n r_i ⋅ φ_i(x)\Big) \Ket x$$

Garner: Linear exponential monads are the perfect setting for hypernormalisation.

What is so special about the unit interval?

which is related to... and leads to...
$[0,1]$ is an effect monoid quantum mechanics foundations $\D {[0, 1]}$ being a monad
$(0,1)$ is a tricocycloid quantum algebra $\D {[0, 1]}$ being $\underbrace{\text{linear exponential}}_{\llap{\text{related to linear logic} \, ⟶ \, \textbf{quantum }\text{logic}}}$
$\Kl(\D {[0, 1]})$ is an effectus quantum logic/computation $\EM(\D {[0, 1]}) ≅ \Conv {[0, 1]}$ in a state-and-effect triangle

⟹ How do these three notions relate?

Quick categorical recap

Kleisli category $\Kl(T)$ of a monad $T：𝒞 → 𝒞$

$\Kl(T)$:
• objects: $X ∈ 𝒞$
• morphisms: $X \xto f T(Y)$ for every $X \xto f Y ∈ 𝒞$

If $𝒞$ has coproducts $+$, so does $\Kl(T)$

Notation: $\; \KlN(T) ⊆ \Kl(T)$:
full subcategory whose objects are of the form $n ≝ 1 + ⋯ + 1 ∈ \Kl(T)$

Eilenberg-Moore category/category of algebras $\; \EM(T)$

$\EM(T)$:
• objects: $(X, T(X) \xto {ξ} X ∈ 𝒞)$ such that $ξ ∘ μ_X = ξ ∘ T ξ$ and $\; ξ ∘ η_X = \id[X]$
• morphisms $(X, ξ) \xto f (Y, ν)$: $X \xto f Y ∈ 𝒞$ such that $f ∘ ξ = ν ∘ T(f)$

II. What makes $\small {\D M}$ a monad?

$\D M$ being a monad:
we only use the fact that $M ≝ [0,1]$ is an effect monoid $M ∈ \EMon$, i.e. a monoid in the category $\EA$ of effect algebras.
Effect algebra $(E, \ovee, 0, (-)^⊥)$:
1. a Partial Commutative Monoid (PCM) where
• $\ovee ：E × E ⇀ E$ is a partial sum and $0 ∈ E$
• $\ovee\;$ is associative, commutative and satisfies the unit law (when defined)
2. equipped with an involution $(-)^⊥：E → E$ such that:
• $0$ is the unique element orthogonal to $1 ≝ 0^⊥$
• for all $x ∈ E$, $x^⊥$ is the unique element (orthocomplement) satisfying $x \,\ovee\, x^⊥ = 1$

$x_1, …, x_n ∈ M$ orthogonal: if their partial sum is defined. When $\; x, y ∈ M \text{ orthogonal}$: $x \,⊥\, y$.

Morphisms of effect algebras:
maps preserving the partial sum and $1$.

Effect Monoid $M ∈ \EMon ⊆ \EA$:
Effect algebra with an associative multiplication distributing over the partial sum and having $1$ as neutral element.

Effect Module $E ∈ \EMod M ≝ \EM(M ⊗ (-))$ over an effect monoid $M ∈ \EMon$:
Effect algebra $E$ with a scalar multiplication $⋅：M ⊗ E → E$ preserving $0$ and the partial sum coordinatewise and such that $1⋅e = e$ and $r⋅(s⋅e) = (r⋅s)⋅e$.

Effect algebras: arising in quantum logic

• Generalise both
• quantum effects: self-adjoint bounded linear operators on a Hilbert space (quantum observables) between $0$ and $\id$
• and orthomodular lattices
• abstraction of the lattice of closed subspaces of a Hilbert space (Von Neumann's experimental propositions about quantum observables)

« Dr. Von Neumann, I would like to know what is a Hilbert space? » — David Hilbert (at the end of a lecture given by Von Neumann in Göttingen in 1929)

Also generalise

• Boolean algebras (distributive complemented lattices)
• orthomodular lattices (quantum case): not distributive
• and $σ$-algebras
• Probability measures: morphisms in $\EA$

⟹ Can be thought of as generalising both propositions and probabilities

Can we generalise from $\D {[0,1]}$ to a general $\D M$, where $M$ is an arbitrary effect monoid?

⟶ If so, do we still have a linear exponential monad?

III. What makes $\small {\D M}$ linear exponential?

• Lifts a certain tensor product $\star_{(0,1)}$ making $(\Set, \star_{(0,1)})$ symmetric monoidal to a coproduct in $\EM(\D M)$

• in Linear Logic (LL): exponential modality $?$ turning additives into multiplicatives:
$$?(A ⊕ B) \; ≡ \; ?A ⅋ ?B$$
• One of the 3 ingredients to have a

Model of classical linear logic (MAELL):
1. M(ultiplicative fragment): Symmetric monoidal closed category (SMCC) which is $\ast$-autonomous
2. A(dditive fragment): Finite co/products
3. E(xponential fragment): Linear exponential co/monad

⟶ Linear logic: argued to be the "proper incarnation" of quantum logic (see Pratt's 1992 article)

« One of the wild hopes that this suggests is the possibility of a direct connection with quantum mechanics… but let’s not dream too much! » — Girard (Linear logic, 1987)

III.1. Because ${\small (\Set, \star_{(0,1)})}$ is monoidal

What makes the tensor $A \star_{(0,1)} B ≝ A + (0,1) × A × B + B$ endow $\Set$ with a symmetric monoidal structure ⟶ the fact that $(0,1)$ is a tricocycloid in $\Set$

A tricocycloid in a symmetric monoidal category $(𝒞, ⊗, α, λ, ρ, σ)$:

is an object $H ∈ 𝒞$ with an isomorphism $v：H ⊗ H → H ⊗ H$
satisfying the 3-cocycle condition:

$$(v ⊗ 1)(1 ⊗ σ)(v ⊗ 1) = (1 ⊗ v) (v ⊗ 1)(1 ⊗ v)$$

A symmetry $γ$ for a tricocycloid $H$:
is an involution $γ：H → H$ such that
$$(1 ⊗ γ) v (1 ⊗ γ) = v (γ ⊗ 1) v$$

A tricocycloid with a symmetry: symmetric tricocycloid.

Tricocycloids

• Quantum algebraic objects introduced by Street in 1998

• Generalisation of Hopf algebras
• generalising group algebras: bialgebra with an antipode
• ⟶ quantum groups
• 3-cocycle condition: $(v ⊗ 1)(1 ⊗ σ)(v ⊗ 1) = (1 ⊗ v) (v ⊗ 1)(1 ⊗ v)$

• non-abelian cohomology: geometrical nerve of the double suspension of the braided monoidal category seen as tricategory with one object and one morphism

Why such a cocycle condition? How is it related to the problem at hand?

Cocycles in non-abelian cohomology ⟷ coherence conditions in higher-dimensional categories (associativity for 3-cocycle).

Here: associativity/coherence condition: an axiom of abstract convex sets (objects of the category of algebras)

• Yang-Baxter equations: recurrent in quantum mechanics

Key lemma (Street and Garner): If $H ∈ 𝒞$ is a symmetric tricocycloid and $𝒞$ has finite coproducts $+$ over which $⊗$ distributes: we can co-universally make $(𝒞, ⋆_{H})$ a monoidal category (with $0$ as unit) by putting

$$A ⋆_{H} B ≝ A + H ⊗ A ⊗ B + B$$

The unit interval is a tricocycloid in ${\small (\Set, ×)}$

$(0,1) ∈ \Set$ is a symmetric tricocycloid:
• $v ≝ \begin{cases} (0,1)^2 &→ (0,1)^2\\ (r,s) &↦ \Big(rs, \frac {rs^\ast}{(rs)^\ast}\Big) \end{cases}$

• $γ ≝ (-)^\ast = \begin{cases} (0,1) &→ (0,1)\\ r &↦ r^\ast ≝ 1-r \end{cases}$

III.1. Because ${\small \EM(\D {[0,1]}) \,≅\, \Conv {[0,1]}}$

Very ad hoc to $\D {[0,1]}$. What about other probability monads? ⟶ Way harder to generalise

but... $\Kl(\D {[0,1]})$ (and the Kleisli category other probability monads as well) happens to be an effectus! (Jacobs 2018)

Effectus theory:
young branch of categorical logic, developed by Jacobs, Cho and the Westerbaan brothers. Effectuses = categorical models capturing the fundamentals of discrete/continuous/quantum logic and probability.

$$\frac{\text{Effectus theory}}{\text{Quantum logic}} \; ≃ \; \frac{\text{Topos theory}}{\text{Intuitionistic logic}}$$

Effectus $𝔹$:
a category with a terminal object $1$ and finite coproducts $+$ which satisfies mild pullback assumptions and a joint monicity requirement so that, for all $X ∈ 𝔹$
• the set $\M ≝ \Hom[𝔹]{1, 2}$ of scalars forms an effect monoid
• the set $\Stat(X) ≝ \Hom[𝔹]{1, X}$ of states of $X$ forms an abstract convex set over $\M$
• the set $\Pred(X) ≝ \Hom[𝔹]{X, 2}$ of predicates over $X$ forms an effect module over $\M$

Examples of effectuses:

• $\Set$ (modelling classical computation and logic)
• the opposite category $\opposite \Cpu$ of $C^\ast$-algebras and positive unital maps (modelling quantum computation/logic)
• the opposite categories $\opposite \DL$ of distributive lattices, $\opposite \BA$ of boolean algebras, and $\opposite \Rng$ of rings
• Octavio Zapata 2019: $\Kl(\D M)$ is an effectus for $M$ is an effect monoid.

An effectus $𝔹$ induces a state-and-effect triangle:

$\Kl(\D \M)$ is an effectus (Zapata 2019): how is it related to $𝔹$?

Further results

• If $𝔹$ has finite coproducts of $1$ as objects, with $q = [q^1, …, q^m]：m → n$:

$$F ≝ \begin{cases} 𝔹 & \overset{≅}{⟶} \; \KlN(\D \M) \\ n &⟼ \; n\\ m \xto q n &⟼ \begin{cases} m &⟶ \; \D \M (n) \\ k &⟼ \; \displaystyle \smash{\sum\limits_{1 ≤ i ≤ n}} ⊳_i q^k \Ket i \end{cases} \end{cases}$$

from which we get

and $\Stat \text{ is fully-faithful } ⟺ \Nerve i \text{ is fully-faithful } ⟺ i \text{ is dense}$

• When $M$ has normalisation:

• objects in $\Conv M$ given by binary convex sums (generalises Garner's situation).
• $\D \M$-algebras are idempotent commutative $⋆_H$-monoids ⟹ $\D \M$ is linear exponential
• Jacobs: $\Stat$ preserves $+$ when $\M = [0,1]$ ⟶ what about a $\M$ with normalisation?
• Models of LL: does $\D M$ live in a model of linear logic?

• $(\Set, ⋆_{\mathring M})$ is a SMC, but $0$ is a unit for $⋆_{\mathring M}$ ⟹ can't be closed.

Do we have a model of $\text{MELL}^{-}$ (without units)? (for Girard's proof nets)

Houston 2013: defines a good notion of model of $\text{MLL}^{-}$. Criterion: every arrow $A → B$ stems from a unique linear element of $A ⊸ B$ (ie a $γ_X：X → (A ⊸ B) ⊗ X$ such that $α_{A ⊸ B, X, Y} (γ_X ⊗ Y) = γ_{X ⊗ Y}：X ⊗ Y → (A ⊸ B) ⊗ (X ⊗ Y)$).

BUT condition not met in our case.

• To fix $0$ being the unit for the tensor: how about $⋆_M$, where $M$ is seen as a lax tricocycloid? ⟶ Fails: Street's construction only yields a lax semi-monoidal category.

• If $F：\catA → \catB$ is lax monoidal ⟹ the left Kan extension $\Lan F：\Psh\catA → \Psh\catB$ can be shown to be lax monoidal. So if $T：\catA → \catB$ is an opmonoidal monad, $\opposite T： \opposite\catA → \opposite\catB$ is lax monoidal, and $\Lan F：[\catA, \Set] → [\catB, \Set]$ is lax monoidal, thus $\opposite{\Lan F}：\opposite{[\catA, \Set]} → \opposite{[\catB, \Set]}$ is oplax monoidal. If $T$ is linear exponential, does $\opposite{\Lan F}$ remain linear exponential?