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

An objective representation of the Gaussian integers

A rig is a riNg without Negatives. We analyse the free rig on a generator x subject to the equivalence x = 1 + x + x^2, showing that in it the non-constant polynomials form a ring. This ring can be identified with the Gaussian integers, which thus acquire objective meaning

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

Environmental regulation in transition: Policy officials’ views of regulatory instruments and their mapping to environmental risks

This study re-analysed 14 semi-structured interviews with policy officials from the UK Department for Environment, Food and Rural Affairs (Defra) to explore the use of a variety of regulatory instruments and different levels of risk across 14 policy domains and 18 separately named risks. Interviews took place within a policy environment of a better regulation agenda and of broader regulatory reform. Of 619 (n) coded references to 5 categories of regulatory instrument, ‘command and control’ regulation (n = 257) and support mechanisms (n = 118) dominated the discussions, with a preference for ‘command and control’ cited in 8 of the policy domains. A framing analysis revealed officials' views on instrument effectiveness, including for sub-categories of the 5 key instruments. Views were mixed, though notably positive for economic instruments including taxation, fiscal instruments and information provision. An overlap analysis explored officials' mapping of public environmental risks to instrument types suited to their management. While officials frequently cite risk concepts generally within discussions, the extent of overlap for risks of specific significance was low across all risks. Only ‘command and control’ was mapped to risks of moderate significance in likelihood and impact severity. These results show that policy makers still prefer ‘command and control’ approaches when a certainty of outcome is sought and that alternative means are sought for lower risk situations. The detailed reasons for selection, including the mapping of certain instruments to specific risk characteristics, is still developing

Homotopy Theoretic Models of Type Theory

We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it. On the other hand, those conditions are easy to check and provide a wide class of models some of which are listed in the paper.Comment: Corrected version of the published articl

A model structure for coloured operads in symmetric spectra

We describe a model structure for coloured operads with values in the category of symmetric spectra (with the positive model structure), in which fibrations and weak equivalences are defined at the level of the underlying collections. This allows us to treat R-module spectra (where R is a cofibrant ring spectrum) as algebras over a cofibrant spectrum-valued operad with R as its first term. Using this model structure, we give suficient conditions for homotopical localizations in the category of symmetric spectra to preserve module structures.Comment: 16 page

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