HOMALGHIGH

Homotopy algebras in homotopy theory and higher category theory

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 Obiettivo del progetto (Objective)

'We will study in this project exciting new interactions and applications between two fundamental modern research areas of mathematics, homotopy theory and higher category theory. These areas of mathematics are used in applied sciences. For instance, homotopy theory is used in robotics and in computer science. Higher category theory studies the way in which complex structures arising for instance in physics, computer science, biology, can be described by a common language, the one of ‘weak n-categories’. In this project, we apply ideas and techniques from homotopy theory to higher category theory. This will provide new and groundbreaking insights into the latter and will return homotopical applications. We will study certain structures which resembles simple algebraic ones but which are in fact much more complex because the defining data are specified ‘up to homotopy’. These structures are called homotopy algebras. We will then study ways in which a homotoy algebra can be made suitably equivalent to a simpler structure, a strict algebra. This process is called rigidification. We will then apply this theory to weak n-categories. We will view one of the models of weak n-categories, due to Tamsamani, as homotopy algebras, and study its rigidification. This will produce a new important type of higher categorical structure, called weakly globular n-fold categories. These will then be used in applications. We will obtain a new way to describe the building blocks of topological spaces, called n-types, and we will understand their connection with iterated loop spaces. We will also pursue other homotopical applications which will lead to the computation of important invariants used to describe topological spaces.'

Introduzione (Teaser)

From the surprising connection between geometry, algebra and logic, a new approach to the development of mathematics on the basis of homotopy theory emerged. Discovered quite recently, EU-funded researchers are now investigating various aspects of it.

Descrizione progetto (Article)

In mathematical logic, correct proofs must be derived from basic axioms and other already proven statements. The homotopy theory allows mathematical proofs to be translated into a computer programming language and have computers check the most complex proofs. Similarly to the equivalence arising in the interpretation of mathematical equations, the theory of topological spaces adds the equivalence axiom.

In homotopy theory, two topological spaces are considered the same if one can be deformed into the other. With the support of the EU, mathematicians now aim to capitalise on this idea of homotopy between topological spaces through work on the 'Homotopy algebras in homotopy theory and higher category theory' (HOMALGHIGH) project.

Modern mathematics organises its objects of study into categories and searches for a common language to describe them. There are categories of topological spaces and other complex structures arising, for instance, in physics. The HOMALGHIGH scientists have focused on structures that resemble simple algebraic ones, but are, in fact, more complex.

Through the so-called rigidification process, the HOMALGHIGH mathematicians searched for ways to make them homotopy-equivalent to simpler structures. New ways are also sought to describe their building blocks and compute invariants of a specific class of topological spaces, called iterated loop spaces. As topological spaces, these are equivalent to continuous maps from a circle to another topological space, the space of loops.

During the first reporting period, the first crucial steps in understanding of these homotopological applications have been made and are described in two papers uploaded to the http://arxiv.org/ (arXiv repository). However, further steps are needed to address the many challenges faced, ranging from practical implementation to computer science applications.