In this work, we propose a new architecture-agnostic method for training idempotent neural networks. An idempotent operator satisfies $f(x) = f(f(x))$, meaning it can be applied iteratively with no effect beyond the first application. Some neural networks used in data transformation tasks, such as image generation and augmentation, can represent non-linear idempotent projections. Using methods from perturbation theory we derive the recurrence relation ${\mathbf{K}’ \leftarrow 3\mathbf{K}^2 - 2\mathbf{K}^3}$ for iteratively projecting a real-valued matrix $\mathbf{K}$ onto the manifold of idempotent matrices. Our analysis shows that for linear, single-layer MLP networks this projection 1) has idempotent fixed points, and 2) is attracting only around idempotent points. We give an extension to non-linear networks by considering our approach as a substitution of the gradient for the canonical loss function, achieving an architecture-agnostic training scheme. We provide experimental results for MLP- and CNN-based architectures with significant improvement in idempotent error over the canonical gradient-based approach. Finally, we demonstrate practical applications of the method as we train a generative network successfully using only a simple reconstruction loss paired with our method.

We define a naturality construction for the operations of weak ω-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 logically as a globular analogue of Reynolds parametricity. Our construction operates as a "power tool" to support construction of terms with geometrical structure, and we use it to define composition operations for cylinders and cones in ω-categories. The machinery can generate terms of high complexity, and we have implemented our construction in a proof assistant, which verifies that the generated terms have the correct type. All our results can be exported to homotopy type theory, allowing the explicit computation of complex path type inhabitants.

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 representation. We describe the user interface and explain how the tool can be used in practice. We also describe the essential subsystems of the tool, including collapse, contraction, expansion, typechecking, and layout, as well as key implementation details including data structure encoding, memoisation, and rendering. These technical innovations have been essential for achieving good performance in a resource-constrained setting.

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 strictness conditions. The key technical device is a new computation rule in the definitional equality of the theory, which we call insertion, defined in terms of a universal property. On terms for which it is defined, this operation "inserts" one of the arguments of a substituted coherence into the coherence itself, appropriately modifying the pasting diagram and result type, and simplifying the syntax in the process. We generate an equational theory from this reduction relation and we study its properties in detail, showing that it yields a decision procedure for equality.

We use type-theoretic techniques to present an algebraic theory of ∞-categories with strict units. Starting with a known type-theoretic presentation of fully weak ∞-categories, in which terms denote valid operations, we extend the theory with a non-trivial definitional equality. This forces some operations to coincide strictly in any model, yielding the strict unit behaviour.

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 allow simple formal proofs of complex high-dimensional algebraic phenomena. However, the theory relies on an implicit term normalisation procedure to recognize correct composites, with no recursive method available for computing it.

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 connected string diagrams. We use this result to give a linear-time solution to the equivalence problem in the connected case, and a quadratic solution in the general case. We also give a stronger proof of the Joyal-Street coherence theorem, settling Selinger's conjecture on recumbent isotopy.

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 notion of `homotopy' with a natural geometrical interpretation. Such a theory has the potential to serve as an attractive foundation for a computer proof assistant for higher category theory, since it allows composites to be uniquely described, and relieves proofs from the bureaucracy of associators, unitors and their coherence. However, this basic theory lacks a high-level way to construct homotopies, which would be intractable to build directly in complex situations; it is not therefore immediately amenable to implementation. We tackle this problem by describing a `contraction' operation, which algorithmically constructs complex homotopies that reduce the lengths of composite terms. This contraction procedure allows building of nontrivial proofs by repeatedly contracting subterms, and also allows the contraction of those proofs themselves, yielding in some cases single-step witnesses for complex homotopies. We prove correctness of this procedure by showing that it lifts connected colimits from a base category to a category of zigzags, a procedure which is then iterated to yield a contraction mechanism in any dimension. We also present homotopy.io, an online proof assistant that implements the theory of associative n-categories, and use it to construct a range of examples that illustrate this new contraction mechanism.

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, and without the need for a global correctness criterion or thinning links. We argue that traditional proof nets are the 2d projections of these 3d diagrams.

This article introduces Globular, an online proof assistant for the formalization and verification of proofs in higher-dimensional category theory. The tool produces graphical visualizations of higher-dimensional proofs, assists in their construction with a point-and- click interface, and performs type checking to prevent incorrect rewrites. Hosted on the web, it has a low barrier to use, and allows hyperlinking of formalized proofs directly from research papers. It allows the formalization of proofs from logic, topology and algebra which are not formalizable by other methods, and we give several examples.

We give a mathematical analysis of a new type of classical computer network architecture, intended as a model of a new technology that has recently been proposed in industry. Our approach is based on groubits, generalizations of classical bits based on groupoids. This network architecture allows the direct execution of a number of protocols that are usually associated with quantum networks, including teleportation, dense coding and secure key distribution.

To any complex Hadamard matrix we associate a quantum permutation group. The correspondence is not one-to-one, but the quantum group encapsulates a number of subtle properties of the matrix. We investigate various aspects of the construction: compatibility to product operations, characterization of matrices which give usual groups, explicit computations for small matrices.

We present new data structures for quasistrict higher categories, in which associativity and unit laws hold strictly. Our approach has low axiomatic complexity compared to traditional algebraic approaches, and gives a practical method for performing calculations in quasistrict 4-categories. It is amenable to computer implementation, and we exploit this to give a machine-verified algebraic proof that every adjunction of 1-cells in a quasistrict 4-category can be promoted to a coherent adjunction satisfying the butterfly equations.

We use a categorical topological semantics to examine the Deutsch-Jozsa, hidden subgroup and single-shot Grover algorithms. This reveals important structures hidden by conventional algebraic presentations, and allows novel proofs of correctness via local topological operations, giving for the first time a satisfying high-level explanation for why these procedures work. We also investigate generalizations of these algorithms, providing improved analyses of those already in the literature, and a new generalization of the single-shot Grover algorithm.

We show that an orthogonal basis for a finite-dimensional Hilbert space can be equivalently characterised as a commutative †-Frobenius monoid in the category FdHilb, which has finite-dimensional Hilbert spaces as objects and continuous linear maps as morphisms, and tensor product for the monoidal structure. The basis is normalised exactly when the corresponding commutative †-Frobenius monoid is special. Hence, both orthogonal and orthonormal bases are characterised without mentioning vectors, but just in terms of the categorical structure: composition of operations, tensor product and the †-functor. Moreover, this characterisation can be interpreted operationally, since the †-Frobenius structure allows the cloning and deletion of basis vectors. That is, we capture the basis vectors by relying on their ability to be cloned and deleted. Since this ability distinguishes classical data from quantum data, our result has important implications for categorical quantum mechanics.

We propose a higher semantics for the description of quantum protocols, which deals with quantum and classical information in a unified way. Central to our approach is the modelling of classical data by information transfer to the environment, and the use of 2-category theory to formalize the resulting framework. This 2-categorical semantics has a graphical calculus, the diagrams of which correspond exactly to physically-implementable quantum procedures. Quantum teleportation in its most general sense is reformulated as the ability to remove correlations between a quantum system and its environment, and is represented by an elegant graphical identity. We use this new formalism to describe two new families of quantum protocols.