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Innovation narratives serve as linguistic tools that audience use to make sense of innovation activities, events and actions, which play a crucial role in firm growth. However, prior studies have primarily focused on independent effects of innovation narrative strategy and firm growth, with relatively limited research exploring their interdependent effects, often yielding conflicting results. This study addresses this gap by employing fuzzy-set qualitative comparative analysis (fsQCA) to adopt a configurational approach, analysing the complex interdependencies between innovation narrative strategies (innovation distinctiveness from category prototypes and exemplars), external conditions (institutional quality, environmental uncertainty), and firm attributes (equity concentration, R&D investment intensity) in relation to high firm growth. Data from 242 Chinese high-tech firms reveal four distinct configurational solutions. These solutions clarify the conditions under which firms should adopt specific innovation narrative strategies to achieve high growth. This study fills the gap left by prior research, which failed to account for the configurational nature of innovation narrative strategies. Additionally, it provides valuable managerial insights on how to strategically disclose innovation-related information to maximise firm growth.

Labelled transition systems are a simple yet powerful formalism for describing the operational behaviour of computing systems. They can be extended to model concurrency faithfully by permitting transitions between states to be labelled by a collection of actions, denoting a concurrent step.

Petri nets (or Place/Transition nets) give rise to such step transition systems in a natural way—the marking diagram of a Petri net is the canonical transition system associated with it. In this paper, we characterize the class of PN-transition systems, which are precisely those step transition systems generated by Petri nets.

We express the correspondence between PN-transition systems and Petri nets in terms of an adjunction between a category of PN-transition systems and a category of Petri nets in which the associated morphisms are behaviour-preserving in a strong and natural sense.

Recent work on neutrino oscillations suggests that the three generations of fermions in the standard model are related by representations of the finite group A(4), the group of symmetries of the tetrahedron. Motivated by this, we explore models which extend the EPRL model for quantum gravity by coupling it to a bosonic quantum field of representations of A(4). This coupling is possible because the representation category of A(4) is a module category over the representation categories used to construct the EPRL model. The vertex operators which interchange vacua in the resulting quantum field theory reproduce the bosons and fermions of the standard model, up to issues of symmetry breaking which we do not resolve. We are led to the hypothesis that physical particles in nature represent vacuum changing operators on a sea of invisible excitations which are only observable in the A(4) representation labels which govern the horizontal symmetry revealed in neutrino oscillations. The quantum field theory of the A(4) representations is just the dual model on the extended lattice of the Lie group E₆, as explained by the quantum McKay correspondence of Frenkel, Jing and Wang.

The coupled model can be thought of as string field theory, but propagating on a discretized quantum spacetime rather than a classical manifold.

A Q-system is a unitary version of a separable Frobenius algebra object in a C*-tensor category or a C*-2-category. We prove that, for C*-2-categories ${𝒞}$ and ${𝒟}$, the C*-2-category $Fun({𝒞},{𝒟})$ of ∗-$2$-functors, ∗-$2$-transformations and ∗-$2$-modifications is Q-system complete, whenever ${𝒟}$ is Q-system complete. We use this result to provide a characterization of Q-system complete categories in terms of ∗-$2$-functors and to prove that the $2$-category of actions of a unitary fusion category ${𝒞}$ on C*-algebras is Q-system complete.

The unification of quantum mechanics and general relativity remains the primary goal of theoretical physics, with string theory appearing as the only plausible unifying scheme. In the present work, in a search of the conceptual foundations of string theory, we analyze the relational logic developed by C. S. Peirce in the late 19th century. The Peircean logic has the mathematical structure of a category with the relation R_ij among two individual terms S_i and S_j, serving as an arrow (or morphism). We introduce a realization of the corresponding categorical algebra of compositions, which naturally gives rise to the fundamental quantum laws, thus indicating category theory as the foundation of quantum mechanics. The same relational algebra generates a number of group structures, among them W_∞. The group W_∞ is embodied and realized by the matrix models, themselves closely linked with string theory. It is suggested that relational logic and in general category theory may provide a new paradigm, within which to develop modern physical theories.

The first part is an introductory description of a small cross-section of the literature on algebraic methods in nonperturbative quantum gravity with a specific focus on viewing algebra as a laboratory in which to deepen understanding of the nature of geometry. This helps to set the context for the second part, in which we describe a new algebraic characterization of the Dirac operator in noncommutative geometry and then use it in a calculation on the form of the fermion mass matrix. Assimilating and building on the various ideas described in the first part, the final part consists of an outline of a speculative perspective on (noncommutative) quantum spectral gravity. This is the second of a pair of papers so far on this project.

Over the past several years, software architecture representation and analysis has become an active area of research. However, most approaches to software architecture representation and analysis have been informal. We postulate that through formality, the term "architecture" can be precisely defined and important properties of systems, such as semantic compatibility between connected entities, can be investigated with precision. In this paper, we use category theory and algebraic specifications to develop a formal definition of architecture and show how architecture theory can be used in the construction of software specifications.

This paper proposes a formal representation of modeling languages based on category theory. These languages are generally described by "metamodels", i.e. structures composed by classes and relations, and related by "transformations". Thus, this paper studies how the key categorical concepts such as functors and relations between functors (called natural transformations) can be used for equational reasoning about modeling artifacts (models, metamodels, transformations). As a result, this paper proposes a formal point of view of models usable to specify/prove equivalence between models or transformations (with an application to refactoring).

Software reuse may only be put into practice if we make available matching mechanisms between user needs and reusable components. The matching mechanisms should also be powerful enough to allow transformations of the user components that, while maintaining correctness, increase the possibilities of use of repository components by the user. In this paper we propose to use the mathematical framework of category theory to develop matching mechanisms for single-sort algebraic specifications that would allow the automatic retrieval and data adaptation of correct programs. We identify two types of specification matching, isomorphic and composite.

We define and explore rack objects internal to categories with products. In demonstration, we classify the group-racks, and use homotopy to prove both existence and exclusion theorems for path-connected topological racks.

We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either “annihilator algebras” — algebras whose socle is a principal ideal — or field extensions. The relationship between two-dimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras.

These results together provide a classification of the indecomposable two-dimensional topological quantum field theories.

Theoretical and computational frameworks of complexity science are dominated by binary structures. This binary bias, seen in the ubiquity of pair-wise networks and formal binary operations in mathematical models, limits our capacity to faithfully capture irreducible polyadic interactions in higher-order systems. A paradigmatic example of a higher-order interaction is the Borromean link of three interlocking rings. In this paper, we propose a mathematical framework via hypergraphs and hypermatrix algebras that allows to formalize such forms of higher-order bonding and connectivity in a parsimonious way. Our framework builds on and extends current techniques in higher-order networks — still mostly rooted in binary structures such as adjacency matrices — and incorporates recent developments in higher-arity structures to articulate the compositional behavior of adjacency hypermatrices. Irreducible higher-order interactions turn out to be a widespread occurrence across natural sciences and socio-cultural knowledge representation. We demonstrate this by reviewing recent results in computer science, physics, chemistry, biology, ecology, social science, and cultural analysis through the conceptual lens of irreducible higher-order interactions. We further speculate that the general phenomenon of emergence in complex systems may be characterized by spatio-temporal discrepancies of interaction arity.

Decision making is a vital function in the age of machine learning and artificial intelligence; however, its physical realization and theoretical fundamentals are not yet well understood. In our former study, we demonstrated that single photons can be used to make decisions in uncertain, dynamically changing environments. The two-armed bandit problem was successfully solved using the dual probabilistic and particle attributes of single photons. In this study, we present a category theoretic modeling and analysis of single-photon-based decision making, including a quantitative analysis that agrees well with the experimental results. The category theoretic model unveils complex interdependencies of the entities of the subject matter in the most simplified manner, including a dynamically changing environment. In particular, the octahedral structure and the braid structure in triangulated categories provide better understandings and quantitative metrics of the underlying mechanisms for the single-photon decision maker. This study provides insight and a foundation for analyzing more complex and uncertain problems for machine learning and artificial intelligence.

This is the second part of a three-part overview, in which we derive the category-theoretic backbone of quantum theory from a process ontology, treating quantum theory as a theory of systems, processes and their interactions. In this part, we focus on classical–quantum interaction. Classical and quantum systems are treated as distinct types, of which the respective behavioral properties are specified in terms of processes and their compositions. In particular, classicality is witnessed by ‘spiders’ which fuse together whenever they connect. We define mixedness and show that pure processes are extremal in the space of all processes, and we define entanglement and show that quantum theory indeed exhibits entanglement. We discuss the classification of tripartite qubit entanglement and show that both the GHZ-state and the W-state come from spider-like families of processes, which differ only in how they behave when they are connected by two or more wires. We define measurements and provide fully comprehensive descriptions of several quantum protocols involving classical data flow. Finally, we give a notion of ‘genuine quantumness’, from which special processes called ‘phase spiders’ arise, and get a first glimpse of quantum nonlocality.

We introduce the categories of quantale-valued approach uniform spaces and quantale-valued uniform gauge spaces, and prove that they are topological categories. We first show that the category of quantale-valued uniform gauge spaces is a full bireflective subcategory of the category of quantale-valued approach uniform spaces and; second, we prove that only under strong restrictions on the quantale these two categories are isomorphic. Besides presenting embeddings of the category of quantale-valued metric spaces into the categories of quantale-valued approach uniform spaces as well as quantale-valued uniform gauge spaces, we show that every quantale-valued approach system group and quantale-valued gauge group has a natural underlying quantale-valued approach uniform space, respectively, a quantale-valued uniform gauge space.

This work focuses on establishing the relationship between bipolar fuzzy automata, reverse bipolar fuzzy automata, and double bipolar fuzzy automata. It also introduces the concepts of bipolar fuzzy subsystems, reverse bipolar fuzzy subsystems, and double bipolar fuzzy subsystems, and explores various properties associated with these subsystems. Furthermore, the paper aims to introduce the categorical aspects of bipolar fuzzy automata and reverse bipolar fuzzy automata, along with their functorial relationship.

What does it mean for one mind to be a different version of another one, or a natural continuation of another one? Or put differently: when can two minds sensibly be considered versions of one another? This question occurs in relation to mind uploading, where one wants to be able to assess whether an approximate upload constitutes a genuine continuation of the uploaded mind or not. It also occurs in the context of the rapid mental growth that is likely to follow mind uploading, at least in some cases — here the question is, when is growth so rapid or discontinuous as to cause the new state of the mind to no longer be sensibly considerable as a continuation of the previous one? Provisional answers to these questions are sketched, using mathematical tools drawn from category theory and probability theory. It is argued that if a mind's growth is "approximately smooth", in a certain sense, then there will be "continuity of self" and the mind will have a rough comprehension of its growth and change process as it occurs. The treatment is somewhat abstract, and intended to point a direction for ongoing research rather than as a definitive practical solution. These ideas may have practical value in future, however, for those whose values favor neither strict self-preservation nor unrestricted growth, but rather growth that is constrained to be at least quasi-comprehensible to the minds doing the growing.

Design patterns are widely used to achieve large-scale reuse by capturing successful practice in software development, but their implementations still remain redundant and burdensome. We present a category theoretic approach to mechanize the refinement process from design pattern templates to executable programs. A pattern template and its “standard” refinement path, which are expressed with theory-based specifications and morphisms, can be saved in a library. When encountering another problem where the same pattern template needs to be applied, instead of performing a new refinement, we can work out its provably correct implementation using category theoretic operations.

The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valuation in quantum mechanics as exemplified, in particular, by Kochen-Specker's theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event algebras. We show explicitly that the latter category is equipped with an object of truth values, or classifying object, which constitutes the appropriate tool for assigning truth values to propositions describing the behavior of quantum systems. Effectively, this category-theoretic representation scheme circumvents consistently the semantic ambiguity with respect to truth valuation that is inherent in conventional quantum mechanics by inducing an objective contextual account of truth in the quantum domain of discourse. The philosophical implications of the resulting account are analyzed. We argue that it subscribes neither to a pragmatic instrumental nor to a relative notion of truth. Such an account essentially denies that there can be a universal context of reference or an Archimedean standpoint from which to evaluate logically the totality of facts of nature. In this light, the transcendence condition of the usual conception of correspondence truth is superseded by a reflective-like transcendental reasoning of the proposed account of truth that is suitable to the quantum domain of discourse.

The study of the foundations of Quantum Mechanics, especially after the advent of Quantum Computation and Information, has benefited from the application of category-theoretic tools and modal logics to the analysis of Quantum processes: we witness a wealth of theoretical frameworks casted in either of the two languages. This paper explores the interplay of the two formalisms in the peculiar context of Quantum Theory.

After a review of some influential abstract frameworks, we show how different modal logic frames can be extracted from the category of finite dimensional Hilbert spaces, connecting the Categorical Quantum Mechanics approach to some modal logics that have been proposed for Quantum Computing. We then apply a general version of the same technique to two other categorical frameworks, the ‘topos approach’ of Doering and Isham and the sheaf-theoretic work on contextuality by Abramsky and Brandenburger, suggesting how some key features can be expressed with modal languages.