Symmetric monoidal equivalences of quantum field theories in dimension two and Frobenius algebras

We show that the canonical equivalences of categories between 2-dimensional (unoriented) topological quantum field theories valued in a symmetric monoidal category and (extended) commutative Frobenius algebras in that symmetric monoidal category are symmetric monoidal equivalences. As an application, we recover that the invariant of 2-dimensional manifolds given by the product of (extended) commutative Frobenius algebras in a symmetric tensor category is the multiplication of the invariants given by each of the algebras.Comment: 4 page

Universal support for triangulated categories

We revisit a result of Gratz and Stevenson on the universal space that carries supports for objects of a triangulated category, in the absence of a tensor product.Comment: 2 page

Homotopy liftings and Hochschild cohomology of some twisted tensor products

The Hochschild cohomology of a tensor product of algebras is isomorphic to a graded tensor product of Hochschild cohomology algebras, as a Gerstenhaber algebra. A similar result holds when the tensor product is twisted by a bicharacter. We present new proofs of these isomorphisms, using Volkov's homotopy liftings that were introduced for handling Gerstenhaber brackets expressed on arbitrary bimodule resolutions. Our results illustrate the utility of homotopy liftings for theoretical purposes.Comment: 14 pages, minor reference corrections, added Section 4 with example

A Primer on Twists in the Noncommutative Realm Focusing on Algebra, Representation Theory, and Geometry

We review several techniques that twist an algebra's multiplicative structure. We first consider twists by an automorphism, also known as Zhang twists, and we relate them to 2-cocycle twists of certain bialgebras. We then outline the classification and properties of twisted tensor products, and we examine twisted Segre products. Our exposition emphasizes clarity over generality, providing a wealth of interconnecting examples.Comment: 16 pages, expanded acknowledgements, added referee correction

Gerstenhaber brackets on Hochschild cohomology of general twisted tensor products

We present techniques for computing Gerstenhaber brackets on Hochschild cohomology of general twisted tensor product algebras. These techniques involve twisted tensor product resolutions and are based on recent results on Gerstenhaber brackets expressed on arbitrary bimodule resolutions.Comment: 14 pages, small changes in the presentation, minor corrections, additional referee corrections, to appear in J. Pure Appl. Algebr

A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks

Regulatory networks depict promoting or inhibiting interactions between molecules in a biochemical system. We introduce a category-theoretic formalism for regulatory networks, using signed graphs to model the networks and signed functors to describe occurrences of one network in another, especially occurrences of network motifs. With this foundation, we establish functorial mappings between regulatory networks and other mathematical models in biochemistry. We construct a functor from reaction networks, modeled as Petri nets with signed links, to regulatory networks, enabling us to precisely define when a reaction network could be a physical mechanism underlying a regulatory network. Turning to quantitative models, we associate a regulatory network with a Lotka-Volterra system of differential equations, defining a functor from the category of signed graphs to a category of parameterized dynamical systems. We extend this result from closed to open systems, demonstrating that Lotka-Volterra dynamics respects not only inclusions and collapsings of regulatory networks, but also the process of building up complex regulatory networks by gluing together simpler pieces. Formally, we use the theory of structured cospans to produce a lax double functor from the double category of open signed graphs to that of open parameterized dynamical systems. Throughout the paper, we ground the categorical formalism in examples inspired by systems biology.Comment: 33 pages. Added several examples, plus minor revision

Altered visual and haptic verticality perception in posterior cortical atrophy and Alzheimer's disease

Abstract: There is increasing theoretical and empirical support for the brain combining multisensory information to determine the direction of gravity and hence uprightness. A fundamental part of the process is the spatial transformation of sensory signals between reference frames: eye-centred, head-centred, body-centred, etc. The question ‘Am I the right way up?’ posed by a patient with posterior cortical atrophy (PCA) suggests disturbances in upright perception, subsequently investigated in PCA and typical Alzheimer's disease (tAD) based on what looks or feels upright. Participants repeatedly aligned to vertical a rod presented either visually (visual-vertical) or haptically (haptic-vertical). Visual-vertical involved orienting a projected rod presented without or with a visual orientation cue (circle, tilted square (±18°)). Haptic-vertical involved orientating a grasped rod with eyes closed using a combination of side (left, right) and hand (unimanual, bimanual) configurations. Intraindividual uncertainty and bias defined verticality perception. Uncertainty was consistently greater in both patient groups than in control groups, and greater in PCA than tAD. Bias in the frontal plane was strongly directionally affected by visual cue tilt (visual-vertical) and grip side (haptic-vertical). A model was developed that assumed verticality information from multiple sources is combined in a statistically optimal way to produce observed uncertainties and biases. Model results suggest the mechanism that spatially transforms graviceptive information between body parts is disturbed in both patient groups. Despite visual dysfunction being typically considered the primary feature of PCA, disturbances were greater in PCA than tAD particularly for haptic-vertical, and are considered in light of posterior parietal vulnerability. (Figure presented.). Key points: The perception of upright requires accurate and precise estimates of orientation based on multiple noisy sensory signals. The question ‘Am I the right way up?’ posed by a patient with posterior cortical atrophy (PCA; purported ‘visual variant Alzheimer's’) suggests disturbances in the perception of upright. What looks or feels upright in PCA and typical Alzheimer's disease (tAD) was investigated by asking participants to repeatedly align to vertical a rod presented visually (visual-vertical) or haptically (haptic-vertical). PCA and tAD groups exhibited not only greater perceptual uncertainty than controls, but also exaggerated bias induced by tilted visual orientation cues (visual-vertical) and grip side (haptic-vertical). When modelled, these abnormalities, which were particularly evident in PCA haptic-vertical performance, were compatible with disruption of a mechanism that spatially transforms verticality information between body parts. The findings suggest an important role of posterior parietal cortex in verticality perception, and have implications for understanding spatial disorientation in dementia. © 2021 The Authors. The Journal of Physiology published by John Wiley & Sons Ltd on behalf of The Physiological Society

ReadClear: An Assistive Reading Tool for People Living with Posterior Cortical Atrophy

BACKGROUND: Progressive reading impairment is an early and debilitating symptom of posterior cortical atrophy (PCA) arising from the progressive deterioration of visual processing skills. OBJECTIVE: The goal of this study was to test the effectiveness of a purpose-built reading app (ReadClear) co-produced with people living with PCA and designed to reduce the reading difficulties experienced by this population (e.g., getting lost in the page and missing words when reading). METHODS: Twenty subjects with PCA were included in a cross-over design home-based study aimed at determining whether ReadClear could 1) enhance the subjective reading experience (reading pleasantness) and 2) improve reading accuracy (reducing the number of reading errors) compared with a sham condition (a standard e-reader). RESULTS: Reading using ReadClear provided a better subjective reading experience than sham (p = 0.018, d = 0.5) and significantly reduced the percentage of reading errors (p <  0.0001, r = 0.82), particularly errors due to omissions (p = 0.01, r = 0.50), repeated words (p = 0.002, r = 0.69), and regressions in the text (p = 0.003, r = 0.69). We found that different kinds of reading errors were related to specific neuropsychological profiles. CONCLUSION: ReadClear can assist reading in people living with PCA by reducing the number of reading errors and improving the subjective reading experience of users