Homotopy Type Theory in Lean

We discuss the homotopy type theory library in the Lean proof assistant. The library is especially geared toward synthetic homotopy theory. Of particular interest is the use of just a few primitive notions of higher inductive types, namely quotients and truncations, and the use of cubical methods.Comment: 17 pages, accepted for ITP 201

Modal dependent type theory and dependent right adjoints

In recent years we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory, and spatial and cohesive type theory. In this paper we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent version of) the K-axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any finite limit category with an adjunction of endofunctors gives rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal lambda-calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes

Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory

In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters

Glucose Gradients Influence Zonal Matrix Deposition in 3D Cartilage Constructs

Reproducing the native collagen structure and glycosaminoglycan (GAG) distribution in tissue-engineered cartilage constructs is still a challenge. Articular cartilage has a specific nutrient supply and mechanical environment due to its location and function in the body. Efforts to simulate this native environment have been reported through the use of bioreactor systems. However, few of these devices take into account the existence of gradients over cartilage as a consequence of the nutrient supply by diffusion. We hypothesized that culturing chondrocytes in an environment, in which gradients of nutrients can be mimicked, would induce zonal differentiation. Indeed, we show that glucose gradients facilitating a concentration distribution as low as physiological glucose levels enhanced a zonal chondrogenic capacity similar to the one found in native cartilage. Furthermore, we found that the glucose consumption rates of cultured chondrocytes were higher under physiological glucose concentrations and that GAG production rates were highest in 5 mM glucose. From these findings, we concluded that this condition is better suited for matrix deposition compared to 20 mM glucose standard used in a chondrocyte culture system. Reconsidering the culture conditions in cartilage tissue engineering strategies can lead to cartilaginous constructs that have better mechanical and structural properties, thus holding the potential of further enhancing integration with the host tissue

Virtual care pathways for people living with HIV:A mixed‐methods systematic review

Background: The COVID-19 pandemic prompted an unprecedented surge in virtual services, necessitating a rapid shift to digital healthcare approaches. This review focuses on evaluating the evidence of virtual care (VC) in delivering HIV care, considering the complex nature of HIV and the need for tailored-approaches, especially for marginalized populations.Methods: A mixed-methods systematic review was performed with searches on five databases, covering studies from January 1946 to May 2022. Inclusion criteria involved two-way virtual consultations between healthcare workers and people living with HIV (PLHIV), with detailed descriptions and outcomes. Qualitative and quantitative studies were included, and the risk of bias was assessed using the Newcastle–Ottawa score and Stenfors' framework.Results: Among 4143 identified records, 26 studies met the criteria, with various models of care described. The majority of studies were observational, and videoconferencing was the primary mode of virtual consultation employed. Quantitative analysis revealed PLHIV generally accept VC, with high attendance rates (87%). Mean acceptability and satisfaction rates were 80% and 85%, respectively, while 87% achieved HIV viral suppression. The setting and models of VC implementation varied, with some introduced in response to COVID-19 while others were as part of trials.Conclusions: VC for PLHIV is deemed an acceptable and effective approach and is associated with good virological outcomes. Data on other health outcomes is lacking. The review underscores the importance of diverse models of care, patient choice and comprehensive training initiatives for both staff and patients. Establishing a ‘gold standard’ for VC models is crucial for ensuring appropriate and effective reviews of PLHIV in virtual settings.</div

Developing a policy game intervention to enhance collaboration in public health policymaking in three European countries

Background: One of the key elements to enhance the uptake of evidence in public health policies is stimulating cross-sector collaboration. An intervention stimulating collaboration is a policy game. The aim of this study was to describe the design and methods of the development process of the policy game 'In2Action' within a real-life setting of public health policymaking networks in the Netherlands, Denmark and Romania.Methods: The development of the policy game intervention consisted of three phases, pre intervention, designing the game intervention and tailoring the intervention.Results: In2Action was developed as a role-play game of one day, with main focus to develop in collaboration a cross-sector implementation plan based on the approved strategic local public health policy.Conclusions: This study introduced an innovative intervention for public health policymaking. It described the design and development of the generic frame of the In2Action game focusing on enhancing collaboration in local public health policymaking networks. By keeping the game generic, it became suitable for each of the three country cases with only minor changes. The generic frame of the game is expected to be generalizable for other European countries to stimulate interaction and collaboration in the policy process

Bohrification of operator algebras and quantum logic

Following Birkhoff and von Neumann, quantum logic has traditionally been based on the lattice of closed linear subspaces of some Hilbert space, or, more generally, on the lattice of projections in a von Neumann algebra A. Unfortunately, the logical interpretation of these lattices is impaired by their nondistributivity and by various other problems. We show that a possible resolution of these difficulties, suggested by the ideas of Bohr, emerges if instead of single projections one considers elementary propositions to be families of projections indexed by a partially ordered set C(A) of appropriate commutative subalgebras of A. In fact, to achieve both maximal generality and ease of use within topos theory, we assume that A is a so-called Rickart C*-algebra and that C(A) consists of all unital commutative Rickart C*-subalgebras of A. Such families of projections form a Heyting algebra in a natural way, so that the associated propositional logic is intuitionistic: distributivity is recovered at the expense of the law of the excluded middle. Subsequently, generalizing an earlier computation for n-by-n matrices, we prove that the Heyting algebra thus associated to A arises as a basis for the internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the "Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of functors from C(A) to the category of sets. We explain the relationship of this construction to partial Boolean algebras and Bruns-Lakser completions. Finally, we establish a connection between probability measure on the lattice of projections on a Hilbert space H and probability valuations on the internal Gelfand spectrum of A for A = B(H).Comment: 31 page