Periodic Report Summary - LAXALGHOMOTOP (Lax algebras in homotopy theory)
This project proposes to develop the theory of lax algebras in the context of homotopy. The former exploits a monadic approach to topological structures that allows for a simultaneous treatment of both neighbourhood and convergence structures: the filter monad F on the category set of sets and set-maps yields a presentation of a topological space either as a set X equipped with a map from X to F X that associates to an element of X its neighbourhood system, or as a set X equipped with a relation in F X × X that specifies which filters converge to which points. This formalised perspective opens up unexplored relations between topological theories and categorical homotopy or higher-dimensional categorical concepts. The monadic and higher-dimensional categorical aspects of lax algebras will be explored in relation with homotopical themes with as guiding objective: the development of a homotopy theory for lax algebras.
In the context of this project, the grant beneficiary has obtained promising results, and is currently working on further developments with an array of colleagues. In particular, with R. Lucyshyn-Wright he has written a new chapter in the monograph Monoidal Topology, and is completing existing chapters in collaboration with the other authors of the volume. He is regularly taking part in the topology seminar at the host institute and has attended and participated in a variety of scienti?c meetings; he has given six oral presentations in Swizerland, Belgium, and France at seminars or conferences. In parallel with his research activities, the grant beneficiary has taught a one-semester math course at the master level, has developed and is currently teaching two two-semester math courses for first year students in architecture at the host institute. He has also supervised four semester-long student research projects at the host institute.
Shortly before the start of the project, the grant recipient's work order-adjoint monads and injective objects (Journal Pure and Applied Algebra, 214, 2010) demonstrated the power of the monadic approach to topology by describing the injective objects of lax algebras. As a by-product, he obtained classical and novel results: for instance, continuous lattices form the initially determined injective objects of the category top of topological spaces and continuous maps, frames play the same role in the category of finitary closure spaces, as do sup-semilattices in the category of generalised metric spaces. Since injectivity is an important theme in homotopy theory, this work also laid out the groundwork for the current project.
In 'On the monadic nature of categories of ordered sets' (accepted), the grant recipient takes the previous work one step further and demonstrates how monadic structures are implicit in categories of lax algebras. In particular, the category of lax algebras for which the associated Eilenberg-Moore category describes the initially determined injective objects is the category that carries the most significant monadic information, while the category of pre-ordered sets and monotone maps underlies all categories of lax algebras, and consequently all associated Eilenberg-Moore categories. The main theorem allows a systematic treatment of monadicity results, uniting particular instances scattered throughout the literature with new ones and suggests potential developments with directed homotopy theory.
The expected results of the project concern exponentiability in categories of lax algebras and to the development of an enriched theory of lax algebras in the context of homotopy-theoretic themes. Potential developments for directed homotopy theory are aimed at, as well as results for descent theory. The impact of the project naturally depends on the results of forthcoming investigations. In the wake of the early successes of the project, the theory of monads is expected to ascertain its role a central tool in the development of unifying models in topological settings.
In the context of this project, the grant beneficiary has obtained promising results, and is currently working on further developments with an array of colleagues. In particular, with R. Lucyshyn-Wright he has written a new chapter in the monograph Monoidal Topology, and is completing existing chapters in collaboration with the other authors of the volume. He is regularly taking part in the topology seminar at the host institute and has attended and participated in a variety of scienti?c meetings; he has given six oral presentations in Swizerland, Belgium, and France at seminars or conferences. In parallel with his research activities, the grant beneficiary has taught a one-semester math course at the master level, has developed and is currently teaching two two-semester math courses for first year students in architecture at the host institute. He has also supervised four semester-long student research projects at the host institute.
Shortly before the start of the project, the grant recipient's work order-adjoint monads and injective objects (Journal Pure and Applied Algebra, 214, 2010) demonstrated the power of the monadic approach to topology by describing the injective objects of lax algebras. As a by-product, he obtained classical and novel results: for instance, continuous lattices form the initially determined injective objects of the category top of topological spaces and continuous maps, frames play the same role in the category of finitary closure spaces, as do sup-semilattices in the category of generalised metric spaces. Since injectivity is an important theme in homotopy theory, this work also laid out the groundwork for the current project.
In 'On the monadic nature of categories of ordered sets' (accepted), the grant recipient takes the previous work one step further and demonstrates how monadic structures are implicit in categories of lax algebras. In particular, the category of lax algebras for which the associated Eilenberg-Moore category describes the initially determined injective objects is the category that carries the most significant monadic information, while the category of pre-ordered sets and monotone maps underlies all categories of lax algebras, and consequently all associated Eilenberg-Moore categories. The main theorem allows a systematic treatment of monadicity results, uniting particular instances scattered throughout the literature with new ones and suggests potential developments with directed homotopy theory.
The expected results of the project concern exponentiability in categories of lax algebras and to the development of an enriched theory of lax algebras in the context of homotopy-theoretic themes. Potential developments for directed homotopy theory are aimed at, as well as results for descent theory. The impact of the project naturally depends on the results of forthcoming investigations. In the wake of the early successes of the project, the theory of monads is expected to ascertain its role a central tool in the development of unifying models in topological settings.
