Monoidal Category Research Articles

Unitary braided-enriched monoidal categories

Braided-enriched monoidal categories were introduced in the work of Morrison–Penneys, where they were characterized using braided central functors. The recent work of Kong–Yuan–Zhang–Zheng and Dell extended this characterization to an equivalence of 2-categories. Since their introduction, braided-enriched fusion categories have been used to describe certain phenomena in topologically ordered systems in theoretical condensed matter physics. While these systems are unitary, there was previously no general notion of unitarity for enriched categories in the literature. We supply the notion of unitarity for enriched categories and braided-enriched monoidal categories and extend the above 2-equivalence to the unitary setting.

Influence of Entrepreneur Social Network on New Product Development Performance of Enterprises Under Intelligent Big Data

In the complex market environment, it is difficult for enterprises to innovate only by their own internal product resources. In order to improve the performance level of new product development, this paper puts forward the influence of entrepreneur social network on the performance of new product development under smart big data. The process of new product development is modeled, and the influence of entrepreneur social network on enterprise new product development performance is analyzed from two aspects: vertical network quality and horizontal network quality. According to the analysis results, build an intelligent big data platform, and use this platform to realize the mining and analysis of enterprise-related data; On this basis, EM algorithm is used to mine the performance data of enterprise new product development, DEA/AHP model is used to evaluate the performance level of enterprise new product development, and the evaluation results are combined with the influence model of entrepreneur social network on enterprise new product development performance to realize the influence analysis of entrepreneur social network on enterprise new product development performance. The experimental results show that the method is highly recognized by experts, which shows that the analysis results are reliable, and the correlation between entrepreneur social network and enterprise new product development performance can be obtained, which provides reference for enterprise new product development.

Development Directions of Fire-Fighting Equipment Using Aircraft Engines Abroad

In the last century, outstanding fire-fighting technology developments have occurred in Hungary. Mobile, quickly deployable, efficient, environmentally friendly tools and technologies with optimal fire extinguishing material use have come to the fore. This included equipment using exhaust gas from internal combustion engines and combustion products from aircraft engines for fire-fighting. The developers’ goal is to reduce environmental and secondary damage during fire-fighting, and at the same time, they sought to increase the efficiency of fire-fighting processes by using measurably less fire-fighting material. In our article, we present the foreign developments of fire-extinguishing equipment that uses aircraft engines with a unique operating principle and is suitable for applying multiple types of fire-extinguishing materials, thus also for implementing complex fire-fighting. We review the design and construction of the equipment, their operating principle, examine their practical application possibilities, and present directions for future development.

Coherence via focusing for symmetric skew monoidal and symmetric skew closed categories

Abstract The symmetric skew monoidal categories of Bourke and Lack are a weakening of Mac Lane’s symmetric monoidal categories where (i) the three structural laws of left and right unitality and associativity are not required to be invertible, they are merely natural transformations with a specific orientation; (ii) the structural law of symmetry is a natural isomorphism involving three objects rather than two. In a similar fashion, symmetric skew closed categories are a weakening of de Schipper’s symmetric closed categories with non-invertible structural laws of left and right unitality. In this paper, we study the structural proof theory of symmetric skew monoidal and symmetric skew closed categories, progressing the project initiated by Uustalu et al. on deductive systems for categories with skew structure. We discuss three equivalent presentations of the free symmetric skew monoidal (resp. closed) category on a set of generating objects: a Hilbert-style categorical calculus; a cut-free sequent calculus; a focused subsystem of derivations, corresponding to a sound and complete goal-directed proof search strategy for the cut-free sequent calculus. Focusing defines an effective normalization procedure for maps in the free symmetric skew monoidal (resp. closed) category, as such solving the coherence problem for symmetric skew monoidal (resp. closed) categories.

Optimization of Proportions of Alkali-Activated Slag-Fly Ash-Based Cemented Tailings Backfill and Its Strength Characteristics and Microstructure under Combined Action of Dry-Wet and Freeze-Thaw Cycles.

To enhance the application of alkali-activated materials in mine filling, cemented tailings backfill was prepared using slag, fly ash, sodium silicate, and NaOH as primary constituents. The effects of the raw material type and dosage on the backfill were examined through a single-factor experiment. Additionally, response surface methodology (RSM) was utilized to optimize the mixing ratios of the backfill, with a focus on fluidity and compressive strength as key objectives. The evolution of backfill quality and compressive strength under the combined effects of dry-wet and freeze-thaw (DW-FT) cycles was analyzed. The hydration products, microstructure, and pore characteristics of the specimens were analyzed using X-ray diffraction (XRD), scanning electron microscopy with energy dispersive spectroscopy (SEM-EDS), and nitrogen adsorption tests (NATs) across varying cycles. The results demonstrate that the optimal backfill composition includes 47.8% fly ash, 6.10% alkali equivalent, and a 1.44 sodium silicate modulus. The macroscopic behavior of the backfill under DW-FT coupling followed this progression: pore initiation → pore expansion → crack formation → crack propagation → structural damage. After a minor initial increase, the backfill strength steadily decreased. Microscopic analysis revealed that the decline in internal cementation products and the deterioration of pore structure were the primary causes of this strength reduction. Thus, the DW-FT coupling can cause significant erosion of the backfill. The technical solutions presented in this paper offer a reference for solid waste utilization and provide valuable insights into the durability of backfill under DW-FT coupling.

Rank-preserving multidimensional mechanisms: An equivalence between identical-object and heterogeneous-object models

We show that the mechanism-design problem for a monopolist selling multiple, heterogeneous objects to a buyer with ex ante symmetric and additive values is equivalent to the mechanism-design problem for a monopolist selling identical objects to a buyer with decreasing marginal values. We derive three new results for the identical-objects model: (i) a new condition for revenue monotonicity of stochastic mechanisms, (ii) a sufficient condition on priors, such that prices in optimal deterministic mechanism are not increasing, and (iii) a simplification of incentive constraints for deterministic mechanisms. We use the equivalence to establish corresponding results in the heterogeneous-objects model.

On a general notion of a polynomial identity and codimensions

Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category C as well as their codimensions in the case when C is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter–Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.

Home energy management system for residential fuel cell-based combined heat and power system

The goal of this paper is to propose an optimization scheme for enhancing power dispatch and load scheduling for residential fuel cell-based combined heat and power systems (FC-CHPS) using mixed integer linear programming (MILP), considering the lifetime degradation of both the fuel cell system (FCS) and battery energy storage systems (BESS). The scheme is applied in the home energy management system that oversees the electric and thermal power of residential FC-CHPS. First, the nonlinearity in the relationship between the natural gas consumption and output electric power, and between the residual thermal power and output electric power of the fuel cell system (FCS) in the FC-CHPS is approximated using piecewise linearization. Then, the piecewise linearization is integrated with MILP to derive optimal power dispatch and load scheduling solutions, while accounting for the nonlinearity of the FCS. Next, cost functions representing the lifetime degradation of FCS and BESS are formulated. These cost functions result in nonlinear optimization constraints. Therefore, innovative methods are proposed to linearize these constraints, allowing the continued use of MILP as the optimization method and factoring in the lifetime degradation of FCS and BESS. Compared to the mixed-integer non-linear programming (MINLP) method that does not linearize the non-linearity in the FCS or the lifetime degradation of the FCS and BESS, the proposed MILP scheme achieves the identical objective function value. Furthermore, the computation time for the proposed MILP scheme is 99.94 % lower than that required by the MINLP method. The proposed scheme provides accurate results in short computation times.

Mini-Workshop: Bridging Number Theory and Nichols Algebras via Deformations

Nichols algebras are graded Hopf algebra objects in braided tensor categories. They appeared first in a paper by Nichols in 1978 in the search for new examples of Hopf algebras. Rediscovered later several times, they also provide a conceptual explanation of the construction of quantum groups. The aim of the workshop is to review recent developments in the field, initiate collaborations, and discuss new approaches to open problems.

Effect of surface orientation on the corrosion behavior of Ni-based single-crystal alloy in a simulated marine atmosphere at 750 °C

DD5 single-crystal alloy samples with different surface orientations were obtained by machining the alloy ingot perpendicular (PP) and parallel (P), respectively, to its solidification direction (SD). The corrosion behaviors of the PPSD and PSD alloy samples were investigated in the atmosphere of moist air + NaCl aerosol at 750 °C. Results showed that during initial corrosion, γ′ phase developed a thin NiO layer with Al internal corrosion product. However, a NiO nodule with an internal Cr2O3 layer was formed on γ phase. As corrosion progressed, the corrosion product gradually grew to a NiO/(Al, Cr)2O3 bilayer, accompanied by internal corrosion consisting of Al2O3. It was also found that the PSD sample with a lower fraction and smaller size of γ’ phases exhibited better corrosion resistance than the PPSD sample. The effect of alloy surface orientation on the corrosion mechanism was discussed.

On fusing matrices associated with conformal boundary conditions

In the context of rational conformal field theories (RCFT) we look at the fusing matrices that arise when a topological defect is attached to a conformal boundary condition. We call such junctions open topological defects. One type of fusing matrices arises when two open defects fuse while another arises when an open defect passes through a boundary operator. We use the topological field theory approach to RCFTs based on Frobenius algebra objects in modular tensor categories to describe the general structure associated with such matrices and how to compute them from a given Frobenius algebra object and its representation theory. We illustrate the computational process on the rational free boson theories. Applications to boundary renormalisation group flows are briefly discussed.

The category $\Theta_{2}$, derived modifications, and deformation theory of monoidal categories

A complex C^{\bullet}(C,D)(F,G)(\eta,\theta) , generalising the Davydov–Yetter complex of a monoidal category (Davydov (1998) and Yetter (1998)), is constructed. Here, C,D are \Bbbk -linear (corresp., dg) bicategories, F,G\colon C\to D are \Bbbk -linear (corresp., dg) strong functors, and \eta,\theta\colon F\Rightarrow G are strong natural transformations. Morally, it is a complex of “derived modifications” \eta\Rrightarrow\theta ; likewise for the case of dg categories, one has the complex of “derived natural transformations” F\Rightarrow G , given by the Hochschild cochain complex of C with coefficients in C -bimodule D(F-,G=) .The complex C^{\bullet}(C,D)(F,G)(\eta,\theta) naturally arises from a 2-cocellular dg vector space A(C,D)(F,G)(\eta,\theta)\colon\Theta_{2}\to C^{\bullet}(\Bbbk) , as its \Theta_{2} -totalisation (here, \Theta_{2} is the category dual to the category of Joyal 2-disks (Joyal (1997))).It is shown that for a \Bbbk -linear monoidal category C , the third cohomology vector space H^{3}(C^{\bullet}(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id})) is isomorphic to the vector space of the outer (modulo twists) infinitesimal deformations of the \Bbbk -linear monoidal category which we call the full deformations. It means that the following data is to be deformed: (a) the underlying dg category structure, (b) the monoidal product on morphisms (the monoidal product on objects is a set-theoretical datum and is maintained under the deformation), and (c) the associator. The data (a), (b), (c) is subject to the (infinitesimal versions of) numerous monoidal compatibilities, which we interpret as the closeness of the corresponding degree 3 element. Similarly, H^{2}(C^{\bullet}(C,D)(F,F)(\mathrm{id},\mathrm{id})) is isomorphic to the vector space of the outer infinitesimal deformations of the strong monoidal functor F .A relative totalisation Rp_{*}A(C,D)(F,F)(\mathrm{id},\mathrm{id}) along the projection p\colon\Theta_{2}\to\Delta is defined, and it is shown to be a cosimplicial monoid, which fulfils the Batanin–Davydov 1-commutativity condition (Batanin and Davydov (2023)). Then it follows from loc. cit. that C^{\bullet}(C,D)(F,F)(\mathrm{id},\mathrm{id}) is a C_{\bullet}(E_{2};\Bbbk) -algebra. Conjecturally, C^{\bullet}(C,C)(\mathrm{Id},\mathrm{Id})(\mathrm{id},\mathrm{id}) is a C_{\bullet}(E_{3};\Bbbk) -algebra; however, the proof requires more sophisticated methods.

Representations of large Mackey Lie algebras and universal tensor categories

Representations of large Mackey Lie algebras and universal tensor categories

Erratum to “Poisson geometry, monoidal Fukaya categories, and commutative Floer cohomology rings”

This erratum corrects the false submisson date displayed on the last page of [Enseign. Math. 70, 313–381 (2024)].

Key capabilities for closed-loop supply chain: Empirical evidence from manufacturing firms

The objective of this paper is to explore the capabilities to implement closed-loop supply chain (CLSC). In this regard, a theoretical model grounded in natural resource-based view is proposed, which depicts inter-relationships among the capabilities and CLSC. The model is tested using survey data from Indian manufacturing firms by partial least squares (PLS) approach. The findings show information technology and organizational learning as important lower-order capabilities, and internal integration, demand management and product design as significant higher-order capabilities for CLSC implementation. To the best of authors' knowledge, this is the first study to examine key capabilities for CLSC. The study contributes to CLSC literature by providing an integrative framework, classifying capabilities into lower-order capabilities and higher-order capabilities, and empirically examining the key capabilities for CLSC. The findings provide managers with insights about the hierarchical levels of capabilities for CLSC, which will help them to accordingly deploy the appropriate resources to build capabilities for CLSC.

Modular tensor categories arising from central extensions and related applications

A modular tensor category is a non-degenerate ribbon finite tensor category and a ribbon factorizable Hopf algebra is a Hopf algebra whose finite-dimensional representations form a modular tensor category. In this paper, we provide a method of constructing ribbon factorizable Hopf algebras using central extensions. We then apply this method to n-rank Taft algebras, which are considered finite-dimensional quantum groups associated with abelian Lie algebras (see Section 2 for the definition), and obtain a family of non-semisimple ribbon factorizable Hopf algebras Eq, thus producing non-semisimple modular tensor categories using their representation categories. And we provide a prime decomposition of Rep(Eq) (the representation category of Eq). By further studying the simplicity of Eq (whether it is a simple Hopf algebra or not), we conclude that(1)there exists a twist J of uq(sl2⊕3) such that uq(sl2⊕3)J is a simple Hopf algebra,(2)there is no relation between the simplicity of a Hopf algebra H and the primality of Rep(H),(3)there are many ribbon factorizable Hopf algebras that are distinct from some known ones, i.e., not isomorphic to any tensor products of trivial Hopf algebras (group algebras or their dual), Drinfeld doubles, and small quantum groups.

Study on the hydration characteristics, mechanical properties, and microstructure of thermally activated low-carbon recycled cement

Based on the thermal activation method, utilizing fine particles of waste concrete to prepare low-carbon recycled cement (RC) is currently one of the research hotspots. Based on existing literature research results, this paper identifies four calcination temperature conditions (120, 450, 750, 1000 ℃) and uses replacement rates of 10 %, 30 %, 50 %, 70 %, and 100 % of RC to replace ordinary Portland cement (OPC).The main physical properties, hydration characteristics, mechanical properties, and microstructure of RC are studied. The study elucidated the influence of RC thermal activation temperature, replacement rate, fineness, and age on its hydration characteristics, mechanical properties, and microstructure, establishing a macro-micro correlation for RC and revealing the mechanism of improvement in mechanical properties of thermal-activated RC. The research indicates that in the initial stage of hydration, RC exhibits a relatively fast hydration rate, with higher initial total heat release than OPC. Hydration products include internal hydration products (formation of C-(A)-S-H gel) and external hydration products (formation of plate-shaped AFm). Furthermore, RC exhibits a faster hydration rate at a hot activation temperature of 750°C, with CH and C-S-H gel transforming into the highly hydrated α´L-C2S, forming a denser microstructure at an early stage. Moreover, at a replacement rate of 30 %, the compressive strength of cement mortar reaches its optimum at 28 days, achieving 13.81 MPa, equivalent to 60.7 % of OPC cement mortar, with a 66.67 % reduction in carbon emissions. Microstructure analysis reveals that when the temperature reaches 1000°C, excessive combustion of RC leads to the formation of low-reactivity C2AS and β-C2S, resulting in a decrease in macroscopic mechanical properties. Additionally, increasing the fineness of RC significantly enhances its rehydration capability, especially as the age increases. This study is of great significance for the preparation of green cement-based composite cementitious materials.

Diagrammatics for Comodule Monads

We extend Willerton’s [24] graphical calculus for bimonads to comodule monads, a monadic interpretation of module categories over a monoidal category. As an application, we prove a version of Tannaka–Krein duality for these structures.

Hybrid electric aircraft design with optimal power management

The aviation industry has continuously improved its fuel efficiency over the last decades with innovations in technology, design and operation. Further improvements are becoming more difficult as current technology reaches complete maturity. Electrification of road vehicles is predicted to completely replace combustion engines in vehicles. As the efficiency and reliability of electric batteries and motors improve, so does the case for electric propulsion systems in aircraft. Jet fuel carries significantly more energy per weight than current state of the technology batteries, making complete replacement of fuel challenging. Hybrid electric propulsion, where both fuel and batteries are used to power propulsion systems could be feasible and improve fuel efficiency. Hybrid electric powertrains use electric power to reduce the power demands from the combustion engine. Electric batteries are discharged and charged during the operation based on power requirements.This paper presents an aircraft design method that includes optimized power management in the design loop. Power management determines battery discharging and charging throughout the design mission to reduce overall fuel consumption. Power management optimization provides the necessary feedback on battery and fuel performance to design battery pack size. This method is applied to regional aircraft, which typically fly the shortest routes of a commercial airline fleet. These short routes have proportionally longer climb and descent segments. These characteristics of regional aircraft routes lead to wider variations in power during operation, where hybrid electric powertrains are the most beneficial.The proposed hybrid electric aircraft design method finds that a serial hybrid electric regional aircraft could achieve significant fuel efficiency improvements to current regional aircraft. This result is in contrast of the current literature, which requires significant improvements to battery energy density, power distribution efficiency and electric motor efficiency to achieve similar fuel performance results. The hybrid electric regional aircraft does not outperform a traditional turboprop aircraft designed with identical objectives. The turboprop aircraft design improves even greater fuel efficiency improvements over current regional turboprop aircraft. This suggests two pathways are possible; the propulsion system of the next generation of regional aircraft could still be non-electric and achieve improved fuel burn improvements considering current technology or serial hybrid-electric powertrains configurations could be adopted in an expanded time horizon if optimistic technology development is done on battery power density to match the possible improvements of the proposed turboprop aircraft.

Twists of graded algebras in monoidal categories

Zhang twists are a common tool for deforming graded algebras over a field in a way that preserves important ring-theoretic properties. We generalize Zhang twists to the setting of closed monoidal categories equipped with their self-enriched structure. Along the way, we prove several key results about algebraic structures in closed monoidal categories missing from the literature. We use these to ultimately prove Morita-type results, showcasing when graded algebras with equivalent categories of graded modules can be related by Zhang twists.