Working Environment
Department of Mathematics: Macquarie University, Sydney
Centre Of Australian Category Theory (CoACT)
Advisory Board | |
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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)
> « 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 $\;\ovee ≝ +$
- $$\begin{align*} \D M(X) &≝ \Big\lbrace φ:X → M \;\mathrel{\Big|}\;\supp(φ) \text{ finite and } \bigovee\limits_{x ∈ X} φ(x) = 1\Big\rbrace\\ &= \bigg\lbrace {\color{orange} \sum\limits_{i = 1}^n r_i \Ket{x_i}} \;\mathrel{\Big|}\; x_i ∈ X, \, r_i ∈ M, \, \bigovee\limits_{i} r_i = 1\bigg\rbrace \end{align*}\\$$ - $$ η_X(x) ≝ 1 \Ket x \qquad\qquad μ_X\Big(\sum\limits_{i = 1}^n r_i \Ket{φ_i}\Big) ≝ \sum\limits_{x ∈ X} \Big(\bigovee\limits_{i=1}^n r_i ⋅ φ_i(x)\Big) \Ket x $$
What is so special about the unit interval?
which is related to… | and leads to… | |
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$[0,1]$ 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 $T:𝒞 → 𝒞$
If $𝒞$ has coproducts $+$, so does $\Kl(T)$
Eilenberg-Moore category/category of algebras $\; \EM(T)$
II. What makes $\small {\D M}$ a monad?
- $\D M$ being a monad:
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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 Monoid $M ∈ \EMon ⊆ \EA$:
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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 $M ∈ \EMon$:
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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?
Linear exponential monad
- 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: \(?(A ⊕ B) \; ≡ \; ?A ⅋ ?B\)
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One of the 3 ingredients to have a
- Model of classical linear logic (MAELL):
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- M(ultiplicative fragment): Symmetric monoidal closed category (SMCC) which is $\ast$-autonomous
- A(dditive fragment): Finite co/products
- 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)
Two reasons why we have a linear exponential monad
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 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
- Generalisation of Hopf algebras
- 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:
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$v ≝ \begin{cases} (0,1)^2 &→ (0,1)^2
(r,s) &↦ \Big(rs, \frac {rs^\ast}{(rs)^\ast}\Big) \end{cases}$ -
$γ ≝ (-)^\ast = \begin{cases} (0,1) &→ (0,1)
r &↦ r^\ast ≝ 1-r \end{cases}$
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Geometric interpretation
Two reasons why we have a linear exponential monad
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:
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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.
- Effectus $𝔹$:
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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 Kleisli categories of the distribution/Giry/Radon/Kantorovitch probability monads
- 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 $𝔹$?
Effectuses v Effect Monoids v Tricocycloids
Further results
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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?
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$(\Set, ⋆{\mathring M})$ 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?
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