A minimal model for prestacks via Koszul duality for box operads

Prestacks are algebro-geometric objects whose defining relations are far from quadratic. Indeed, they are cubic and quartic, and moreover inhomogeneous. We show that box operads, a rectangular type of operads introduced in arXiv:2305.20036, constitute the correct framework to encode them and resolve their relations up to homotopy. Our first main result is a Koszul duality theory for box operads, extending the duality for (nonsymmetric) operads. In this new theory, the classical restriction of being quadratic is replaced by the notion of being $\textit{thin-quadratic}$, a condition referring to a particular class of ``thin'' operations. Our main case of interest is the box operad $Lax$ encoding lax prestacks as algebras. We show that $Lax$ is not Koszul by explicitly computing its Koszul complex. We then go on to remedy the situation by suitably restricting the Koszul dual box cooperad Lax^{\antishriek} to obtain our second main result: we establish a minimal (in particular cofibrant) model $Lax_{\infty}$ for the box operad $Lax$ encoding lax prestacks. This sheds new light on Markl's question on the existence of a cofibrant model for the operad encoding presheaves of algebras from arXiv:math/0103052. Indeed, we answer the parallel question with presheaves viewed as prestacks, and prestacks considered as algebras over a box operad, in the positive.Comment: 45 pages, lots of pictures; All comments are welcome

Box operads and higher Gerstenhaber brackets

We introduce a symmetric operad $\square p$ ("box-op") which describes a certain calculus of rectangular labeled ``boxes''. Algebras over $\square p$, which we call box operads, have appeared under the name of fc multicategories in work by Leinster \cite{LeinsterFcmulticategories1999}. In our main result, we endow a suitable (graded, zero differential) totalisation $\square p_{\mathrm{td}}$ with a morphism $L_{\infty} \rightarrow \square p_{\mathrm{td}}$. We show that $\square p$ acts on an $\mathbb{N}^3$-graded enlargement of the $\mathbb{N}^2$-graded Gerstenhaber-Schack object $\mathbf{C}_{GS}(\mathbb{A})$ of a quiver $\mathbb{A}$ on a small category from \cite{DinhVanLowen2018}. This action restricts to an $L_{\infty}$-structure on $\mathbf{C}_{GS}(\mathbb{A})$ (with zero differential). For an element $\alpha = (m,f,c) \in \mathbf{C}_{GS}^2(\mathbb{A})$, the Maurer-Cartan equation holds precisely when $(\mathbb{A}, m, f, c)$ is a lax prestack with multiplications $m$, restrictions $f$, and twists $c$. As a consequence, the $\alpha$-twisted $L_{\infty}$-structure on $\mathbf{C}_{GS}(\mathbb{A})$ controls the deformation theory of $(\mathbb{A}, \alpha)$ as a lax prestack.Comment: 22 pages + 8 appendix. Minor changes, references update

The influence of risk perceptions on close contact frequency during the SARS-CoV-2 pandemic.

Human behaviour is known to be crucial in the propagation of infectious diseases through respiratory or close-contact routes like the current SARS-CoV-2 virus. Intervention measures implemented to curb the spread of the virus mainly aim at limiting the number of close contacts, until vaccine roll-out is complete. Our main objective was to assess the relationships between SARS-CoV-2 perceptions and social contact behaviour in Belgium. Understanding these relationships is crucial to maximize interventions' effectiveness, e.g. by tailoring public health communication campaigns. In this study, we surveyed a representative sample of adults in Belgium in two longitudinal surveys (survey 1 in April 2020 to August 2020, and survey 2 in November 2020 to April 2021). Generalized linear mixed effects models were used to analyse the two surveys. Participants with low and neutral perceptions on perceived severity made a significantly higher number of social contacts as compared to participants with high levels of perceived severity after controlling for other variables. Our results highlight the key role of perceived severity on social contact behaviour during a pandemic. Nevertheless, additional research is required to investigate the impact of public health communication on severity of COVID-19 in terms of changes in social contact behaviour

SOCRATES-CoMix: a platform for timely and open-source contact mixing data during and in between COVID-19 surges and interventions in over 20 European countries.

BACKGROUND: SARS-CoV-2 dynamics are driven by human behaviour. Social contact data are of utmost importance in the context of transmission models of close-contact infections. METHODS: Using online representative panels of adults reporting on their own behaviour as well as parents reporting on the behaviour of one of their children, we collect contact mixing (CoMix) behaviour in various phases of the COVID-19 pandemic in over 20 European countries. We provide these timely, repeated observations using an online platform: SOCRATES-CoMix. In addition to providing cleaned datasets to researchers, the platform allows users to extract contact matrices that can be stratified by age, type of day, intensity of the contact and gender. These observations provide insights on the relative impact of recommended or imposed social distance measures on contacts and can inform mathematical models on epidemic spread. CONCLUSION: These data provide essential information for policymakers to balance non-pharmaceutical interventions, economic activity, mental health and wellbeing, during vaccine rollout. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1186/s12916-021-02133-y