INTERNAL NATURAL TRANSFORMATIONS AND FROBENIUS ALGEBRAS IN THE DRINFELD CENTER

For ℳ and $$ \mathcal{N} $$ N finite module categories over a finite tensor category $$ \mathcal{C} $$ C , the category $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) of right exact module functors is a finite module category over the Drinfeld center $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ). We study the internal Homs of this module category, which we call internal natural transformations. With the help of certain integration functors that map $$ \mathcal{C} $$ C -$$ \mathcal{C} $$ C -bimodule functors to objects of $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ), we express them as ends over internal Homs and define horizontal and vertical compositions. We show that if ℳ and $$ \mathcal{N} $$ N are exact $$ \mathcal{C} $$ C -modules and $$ \mathcal{C} $$ C is pivotal, then the $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C )-module $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) is exact. We compute its relative Serre functor and show that if ℳ and $$ \mathcal{N} $$ N are even pivotal module categories, then $$ \mathrm{\mathcal{R}}{ex}_{\mathcal{C}} $$ ℛ ex C (ℳ, $$ \mathcal{N} $$ N ) is pivotal as well. Its internal Ends are then a rich source for Frobenius algebras in $$ \mathcal{Z} $$ Z ($$ \mathcal{C} $$ C ).

Modified Traces and the Nakayama Functor

We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category ${\mathscr{M}}$ M , we introduce the notion of a Σ-twisted trace on the class $\text {Proj}({\mathscr{M}})$ Proj ( M ) of projective objects of ${\mathscr{M}}$ M . In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on $\text {Proj}({\mathscr{M}})$ Proj ( M ) and the set of natural transformations from Σ to the Nakayama functor of ${\mathscr{M}}$ M . Non-degeneracy and compatibility with the module structure (when ${\mathscr{M}}$ M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.

Information physics and quantum space technologies for natural hazard sensing, modelling and prediction

Disruptive socio-natural transformations and climatic change, where system invariants and symmetries break down, defy the traditional complexity paradigms such as machine learning and artificial intelligence. In order to overcome this, we introduced non-ergodic Information Physics, bringing physical meaning to inferential metrics, and a coevolving flexibility to the metrics of information transfer, resulting in new methods for causal discovery and attribution. With this in hand, we develop novel dynamic models and analysis algorithms natively built for quantum information technological platforms, expediting complex system computations and rigour. Moreover, we introduce novel quantum sensing technologies in our Meteoceanics satellite constellation, providing unprecedented spatiotemporal coverage, resolution and lead, whilst using exclusively sustainable materials and processes across the value chain. Our technologies bring out novel information physical fingerprints of extreme events, with recently proven records in capturing early warning signs for extreme hydro-meteorologic events and seismic events, and do so with unprecedented quantum-grade resolution, robustness, security, speed and fidelity in sensing, processing and communication. Our advances, from Earth to Space, further provide crucial predictive edge and added value to early warning systems of natural hazards and long-term predictions supporting climatic security and action.

Perception of soft materials relies on physics-based object representations: Behavioral and computational evidence

When encountering objects, we readily perceive not only low-level properties (e.g., color and orientation), but also seemingly higher-level ones--including aspects of physics (e.g., mass). Perhaps nowhere is this contrast more salient than in the perception of soft materials such as cloths: the dynamics of these objects (including how their three-dimensional forms vary) are determined by their physical properties such as stiffness, elasticity, and mass. Here we hypothesize that the perception of cloths and their physical properties must involve not only image statistics, but also abstract object representations that incorporate "intuitive physics". We provide behavioral and computational evidence for this hypothesis. We find that humans can visually match the stiffness of cloths with unfamiliar textures from the way they undergo natural transformations (e.g. flapping in the wind) across different scenarios. A computational model that casts cloth perception as mental physics simulation explains important aspects of this behavior.

Derived categories of functors and quiver sheaves

Let [Formula: see text] be small category and [Formula: see text] an arbitrary category. Consider the category [Formula: see text] whose objects are functors from [Formula: see text] to [Formula: see text] and whose morphisms are natural transformations. Let [Formula: see text] be another category, and again, consider the category [Formula: see text]. Now, given a functor [Formula: see text] we construct the induced functor [Formula: see text]. Assuming [Formula: see text] and [Formula: see text] to be abelian categories, it follows that the categories [Formula: see text] and [Formula: see text] are also abelian. We have two main goals: first, to find a relationship between the derived category [Formula: see text] and the category [Formula: see text]; second relate the functors [Formula: see text] and [Formula: see text]. We apply the general results obtained to the special case of quiver sheaves.

Functors, Transformations, and Modifications

This chapter discusses functors, transformations, and modifications that are bicategorical analogs of functors and natural transformations. The main concepts covered are lax functors, lax transformations, modifications, and icons. A section is devoted to representable pseudofunctors, representable transformations, and representable modifications, which will be used in the Bicategorical Yoneda Lemma.

Lines in the Mud: Tank Eco-restoration and Boundary Contestations in Chennai

This article analyses the politics of environmentalism revealed in struggles over the land–water boundary of an urbanising tank in Chennai. In contesting this boundary, property-less settlers on its banks called into question the tank’s ‘nature’ and functions in its urban milieu, and demanded a redrawing of boundaries to reflect the socio-natural transformations that had turned parts of it into land. Simultaneously, propertied residents, in concert with state eco-restoration schemes and court rulings, fought to restore the tank to its ‘original’ dimensions. In foregrounding the liminalities of the urbanising tank, this article suggests the limits of the contemporary property-determined eco-restoration discourse, premised on a return to pristine pasts and original boundaries. It proposes that the stand-off between water body restoration and the defence of working-class housing rights necessitates recognising the tank as an artefact assembled over time by socio-technical and natural processes.

On naturality of some construction of connections

Let \(F\) be a bundle functor on the category of all fibred manifolds and fibred maps. Let \(\Gamma\) be a general connection in a fibred manifold \(\mathrm{pr}:Y\to M\) and \(\nabla\) be a classical linear connection on \(M\). We prove that the well-known general connection \(\mathcal{F}(\Gamma,\nabla)\) in \(FY\to M\) is canonical with respect to fibred maps and with respect to natural transformations of bundle functors.