Jamie Vicary
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View article: Parameter-free approximate equivariance for tasks with finite group symmetry
Parameter-free approximate equivariance for tasks with finite group symmetry Open
Equivariant neural networks incorporate symmetries through group actions, embedding them as an inductive bias to improve performance on a wide variety of tasks. However, existing equivariant methods can be computationally intensive, with h…
View article: Beyond Eckmann-Hilton: Commutativity in Higher Categories
Beyond Eckmann-Hilton: Commutativity in Higher Categories Open
We show that in a weak globular $ω$-category, all composition operations are equivalent and commutative for cells with sufficiently degenerate boundary, which can be considered a higher-dimensional generalisation of the Eckmann-Hilton argu…
View article: Naturality for higher-dimensional path types
Naturality for higher-dimensional path types Open
We define a naturality construction for the operations of weak omega-categories, as a meta-operation in a dependent type theory. Our construction has a geometrical motivation as a local tensor product with a directed interval, and behaves …
View article: From dual-unitary to biunitary: a 2-categorical model for exactly-solvable many-body quantum dynamics
From dual-unitary to biunitary: a 2-categorical model for exactly-solvable many-body quantum dynamics Open
Dual-unitary brickwork circuits are an exactly-solvable model for many-body chaotic quantum systems, based on 2-site gates which are unitary in both the time and space directions. Prosen has recently described an alternative model called d…
View article: A Syntax for Strictly Associative and Unital ∞-Categories
A Syntax for Strictly Associative and Unital ∞-Categories Open
We present the first definition of strictly associative and unital ∞-category. Our proposal takes the form of a type theory whose terms describe the operations of such structures, and whose definitional equality relation enforces desired s…
View article: The theory and applications of anticolimits
The theory and applications of anticolimits Open
Colimits are a fundamental construction in category theory. They provide a way to construct new objects by gluing together existing objects that are related in some way. We introduce a complementary notion of anticolimits, which provide a …
View article: homotopy.io: A Proof Assistant for Finitely-Presented Globular n-Categories
homotopy.io: A Proof Assistant for Finitely-Presented Globular n-Categories Open
We present the proof assistant homotopy.io for working with finitely-presented semistrict higher categories. The tool runs in the browser with a point-and-click interface, allowing direct manipulation of proof objects via a graphical repre…
View article: Posetal Diagrams for Logically-Structured Semistrict Higher Categories
Posetal Diagrams for Logically-Structured Semistrict Higher Categories Open
We now have a wide range of proof assistants available for compositional\nreasoning in monoidal or higher categories which are free on some generating\nsignature. However, none of these allow us to represent categorical operations\nsuch as…
View article: A Categorical Model for Classical and Quantum Block Designs
A Categorical Model for Classical and Quantum Block Designs Open
Classical block designs are important combinatorial structures with a wide\nrange of applications in Computer Science and Statistics. Here we give a new\nabstract description of block designs based on the arrow category construction.\nWe s…
View article: On Structures in Arrow Categories
On Structures in Arrow Categories Open
In this article we investigate which categorical structures of a category C are inherited by its arrow category. In particular, we show that a monoidal equivalence between two categories gives rise to a monoidal equivalence between their a…
View article: A layout algorithm for higher-dimensional string diagrams
A layout algorithm for higher-dimensional string diagrams Open
The algebraic zigzag construction has recently been introduced as a combinatorial foundation for a higher dimensional notion of string diagram. For use in a proof assistant, a layout algorithm is required to determine the optimal rendering…
View article: From dual-unitary to biunitary: a 2-categorical model for exactly-solvable many-body quantum dynamics
From dual-unitary to biunitary: a 2-categorical model for exactly-solvable many-body quantum dynamics Open
Dual-unitary brickwork circuits are an exactly-solvable model for many-body chaotic quantum systems, based on 2-site gates which are unitary in both the time and space directions. Prosen has recently described an alternative model called '…
View article: A Syntax for Strictly Associative and Unital $\infty$-Categories
A Syntax for Strictly Associative and Unital $\infty$-Categories Open
We present the first definition of strictly associative and unital $\infty$-category. Our proposal takes the form of a type theory whose terms describe the operations of such structures, and whose definitional equality relation enforces de…
View article: Exact dynamics in dual-unitary quantum circuits with projective measurements
Exact dynamics in dual-unitary quantum circuits with projective measurements Open
Dual-unitary circuits have emerged as a minimal model for chaotic quantum many-body dynamics in which the dynamics of correlations and entanglement remains tractable. Simultaneously, there has been intense interest in the effect of measure…
View article: The Word Problem for Braided Monoidal Categories is Unknot-Hard
The Word Problem for Braided Monoidal Categories is Unknot-Hard Open
We show that the word problem for braided monoidal categories is at least as hard as the unknotting problem. As a corollary, so is the word problem for Gray categories. We conjecture that the word problem for Gray categories is decidable.
View article: Computads for weak $ω$-categories as an inductive type
Computads for weak $ω$-categories as an inductive type Open
We give a new description of computads for weak globular $ω$-categories by giving an explicit inductive definition of the free words. This yields a new understanding of computads, and allows a new definition of $ω$-category that avoids the…
View article: Zigzag normalisation for associative n-categories
Zigzag normalisation for associative n-categories Open
The theory of associative 𝑛-categories has recently been proposed as a strictly associative and unital approach to higher category theory. As a foundation for a proof assistant, this is potentially attractive, since it has the potential to…
View article: Exact dynamics in dual-unitary quantum circuits with projective measurements
Exact dynamics in dual-unitary quantum circuits with projective measurements Open
Dual-unitary circuits have emerged as a minimal model for chaotic quantum many-body dynamics in which the dynamics of correlations and entanglement remains tractable. Simultaneously, there has been intense interest in the effect of measure…
View article: Normalization for planar string diagrams and a quadratic equivalence algorithm
Normalization for planar string diagrams and a quadratic equivalence algorithm Open
In the graphical calculus of planar string diagrams, equality is generated by exchange moves, which swap the heights of adjacent vertices. We show that left- and right-handed exchanges each give strongly normalizing rewrite strategies for …
View article: Traced Monoidal Categories as Algebraic Structures in Prof
Traced Monoidal Categories as Algebraic Structures in Prof Open
We define a traced pseudomonoid as a pseudomonoid in a monoidal bicategory\nequipped with extra structure, giving a new characterisation of Cauchy complete\ntraced monoidal categories as algebraic structures in Prof, the monoidal\nbicatego…
View article: A Type Theory for Strictly Associative Infinity Categories
A Type Theory for Strictly Associative Infinity Categories Open
Many definitions of weak and strict $\infty$-categories have been proposed. In this paper we present a definition for $\infty$-categories with strict associators, but which is otherwise fully weak. Our approach is based on the existing typ…
View article: Traced monoidal categories as algebraic structures in $\mathbf{Prof}$
Traced monoidal categories as algebraic structures in $\mathbf{Prof}$ Open
We define a traced pseudomonoid as a pseudomonoid in a monoidal bicategory equipped with extra structure, giving a new characterisation of Cauchy complete traced monoidal categories as algebraic structures in $\mathbf{Prof}$, the monoidal …
View article: The Word Problem for Braided Monoidal Categories is Unknot-Hard
The Word Problem for Braided Monoidal Categories is Unknot-Hard Open
We show that the word problem for braided monoidal categories is at least as hard as the unknotting problem. As a corollary, so is the word problem for Gray categories. We conjecture that the word problem for Gray categories is decidable.
View article: Proceedings of the 3rd Annual International Applied Category Theory\n Conference 2020
Proceedings of the 3rd Annual International Applied Category Theory\n Conference 2020 Open
The third annual International Applied Category Theory Conference (ACT2020)\nwas planned to take place at MIT in Cambridge, Massachusetts USA. However, the\nglobal COVID-19 pandemic made the prospect of holding a large in-person meeting\ni…
View article: A Type Theory for Strictly Unital $\infty$-Categories
A Type Theory for Strictly Unital $\infty$-Categories Open
We use type-theoretic techniques to present an algebraic theory of $\infty$-categories with strict units. Starting with a known type-theoretic presentation of fully weak $\infty$-categories, in which terms denote valid operations, we exten…
View article: Quantum teleportation with infinite reference-frame uncertainty
Quantum teleportation with infinite reference-frame uncertainty Open
We present two new schemes for quantum teleportation between parties whose reference frames are misaligned by the action of a compact Lie group. The first scheme produces a channel with increased purity compared to a standard protocol, wit…
View article: Coherence for Frobenius pseudomonoids and the geometry of linear proofs
Coherence for Frobenius pseudomonoids and the geometry of linear proofs Open
We prove coherence theorems for Frobenius pseudomonoids and snakeorators in monoidal bicategories. As a consequence we obtain a 3d notation for proofs in nonsymmetric multiplicative linear logic, with a geometrical notion of equivalence, a…
View article: Coherence for Frobenius pseudomonoids and the geometry of linear proofs
Coherence for Frobenius pseudomonoids and the geometry of linear proofs Open
We prove coherence theorems for Frobenius pseudomonoids and snakeorators in monoidal bicategories. As a consequence we obtain a 3d notation for proofs in nonsymmetric multiplicative linear logic, with a geometrical notion of equivalence, a…
View article: High-level methods for homotopy construction in associative n-categories
High-level methods for homotopy construction in associative n-categories Open
A combinatorial theory of associative $n$-categories has recently been proposed, with strictly associative and unital composition in all dimensions, and the weak structure arising as a combinatorial notion of homotopy with a natural geomet…