# Lax Algebras in Homotopy Theory

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# Lax algebras bring new tools for topology

By developing links between theories of lax algebras and topological problems, significant progress is being made in an area of mathematics that concerns how objects deform into other objects.

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Topology is a major area of mathematics concerned with properties that are preserved when objects are continuously deformed. Such deformations involve stretching but no tearing or gluing of the objects being deformed. In topology, two continuous functions are called homotopic, if one can be continuously deformed into the other. Such a deformation is called a homotopy between the two functions. Lax algebras allow a unified approach to the main fundamental structures of topology. The EU-funded project ‘Lax algebras in homotopy theory’ (Laxalghomotop) is contributing to the advancement of this area by exploiting links between lax algebras and categorical homotopy theory: the theory of locales, higher-dimensional category theory, model theory, convergence theory, descent theory and other advanced mathematical topics. In return, the abstraction allowed by lax algebras will provide a reference point for the development of generalisations of homotopy theories, ranging from metric to uniform interpretations. Prior to the project, the grant recipient's work on order-adjoint monads and injective objects demonstrated the power of the monadic approach to topology by describing the injective objects of lax algebras. Since injectivity is an important theme in homotopy theory, this work laid the groundwork for the current project. This approach has now been taken one step further by demonstrating how monadic structures are implicit in categories of lax algebras. The main theorem allows a systematic treatment of monadicity results, uniting particular instances mentioned throughout the literature with new results and suggests potential developments with directed homotopy theory. From the early successes of the project, it is expected that the theory of monads will become a central tool in the development of unifying models in topological settings. This will provide the European Union with a crucial mathematical asset.