Higher operads, higher categories

Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. It draws its inspiration from areas as diverse as topology, quantum algebra, mathematical physics, logic, and theoretical computer science. This is the first book on the subject and lays its foundations. Many examples are given throughout. There is also an introductory chapter motivating the subject for topologists.Comment: Book, 410 page

On the magnitude of a finite dimensional algebra

There is a general notion of the magnitude of an enriched category, defined subject to hypotheses. In topological and geometric contexts, magnitude is already known to be closely related to classical invariants such as Euler characteristic and dimension. Here we establish its significance in an algebraic context. Specifically, in the representation theory of an associative algebra $A$, a central role is played by the indecomposable projective $A$-modules, which form a category enriched in vector spaces. We show that the magnitude of that category is a known homological invariant of the algebra: writing $\chi_A$ for the Euler form of $A$ and $S$ for the direct sum of the simple $A$-modules, it is $\chi_A(S,S)$

Towards a Java Subtyping Operad

The subtyping relation in Java exhibits self-similarity. The self-similarity in Java subtyping is interesting and intricate due to the existence of wildcard types and, accordingly, the existence of three subtyping rules for generic types: covariant subtyping, contravariant subtyping and invariant subtyping. Supporting bounded type variables also adds to the complexity of the subtyping relation in Java and in other generic nominally-typed OO languages such as C# and Scala. In this paper we explore defining an operad to model the construction of the subtyping relation in Java and in similar generic nominally-typed OO programming languages. Operads, from category theory, are frequently used to model self-similar phenomena. The Java subtyping operad, we hope, will shed more light on understanding the type systems of generic nominally-typed OO languages.Comment: 13 page

Appraising longitudinal trends in the strategic risks cited by risk managers in the international water utility sector, 2005-2015

We report dynamic changes in the priorities for strategic risks faced by international water utilities over a 10 year period, as cited by managers responsible for managing them. A content analysis of interviews with three cohorts of risk managers in the water sector was undertaken. Interviews probed the focus risk managers' were giving to strategic risks within utilities, as well as specific questions on risk analysis tools (2005); risk management cultures (2011) and the integration of risk management with corporate decision-making (2015). The coding frequency of strategic (business, enterprise, corporate) risk terms from 18 structured interviews (2005) and 28 semi-structured interviews (12 in 2011; 16 in 2015) was used to appraise changes in the perceived importance of strategic risks within the sector. The aggregated coding frequency across the study period, and changes in the frequency of strategic risks cited at three interview periods identified infrastructure assets as the most significant risk over the period and suggests an emergence of extrinsic risk over time. Extended interviews with three utility risk managers (2016) from the UK, Canada and the US were then used to contextualise the findings. This research supports the ongoing focus on infrastructure resilience and the increasing prevalence of extrinsic risk within the water sector, as reported by the insurance sector and by water research organisations. The extended interviews provided insight into how strategic risks are now driving the implementation agenda within utilities, and into how utilities can secure tangible business value from proactive risk governance. Strategic external risks affecting the sector are on the rise, involve more players and are less controllable from within a utility's own organisational boundaries. Proportionate risk management processes and structures provide oversight and assurance, whilst allowing a focus on the tangible business value that comes from managing strategic risks well

Classical Structures Based on Unitaries

Starting from the observation that distinct notions of copying have arisen in different categorical fields (logic and computation, contrasted with quantum mechanics) this paper addresses the question of when, or whether, they may coincide. Provided all definitions are strict in the categorical sense, we show that this can never be the case. However, allowing for the defining axioms to be taken up to canonical isomorphism, a close connection between the classical structures of categorical quantum mechanics, and the categorical property of self-similarity familiar from logical and computational models becomes apparent. The required canonical isomorphisms are non-trivial, and mix both typed (multi-object) and untyped (single-object) tensors and structural isomorphisms; we give coherence results that justify this approach. We then give a class of examples where distinct self-similar structures at an object determine distinct matrix representations of arrows, in the same way as classical structures determine matrix representations in Hilbert space. We also give analogues of familiar notions from linear algebra in this setting such as changes of basis, and diagonalisation.Comment: 24 pages,7 diagram

Higher Structures in M-Theory

The key open problem of string theory remains its non-perturbative completion to M-theory. A decisive hint to its inner workings comes from numerous appearances of higher structures in the limits of M-theory that are already understood, such as higher degree flux fields and their dualities, or the higher algebraic structures governing closed string field theory. These are all controlled by the higher homotopy theory of derived categories, generalised cohomology theories, and $L_\infty$-algebras. This is the introductory chapter to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in M-Theory. We first review higher structures as well as their motivation in string theory and beyond. Then we list the contributions in this volume, putting them into context.Comment: 22 pages, Introductory Article to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018, references update