# Homotopy algebras in homotopy theory and higher category theory

## Final Report Summary - HOMALGHIGH (Homotopy algebras in homotopy theory and higher category theory)

This project studied 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; higher category theory is used in computer science and physics.
Homotopy theory studies topological spaces: these are sets with a suitable notion of nearness of points. In homotopy theory two spaces are considered the same if one can be transformed into the other using a particular transformation, called a homotopy equivalence, which is a kind of deformation which allows shrinking. 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.

Category theory is an intrinsically inter-disciplinary subject, as it develops a common language for different disciplines beyond mathematics, such as computer science, logic, geometry and physics. The progress made on the understanding of higher categorical structures makes a contribution to bridging between different areas of science. This is very important for our society, which is increasingly dependent on communication and inter-connectivity.

In this project we studied certain structures which resembles simple algebraic ones but which are in fact much more complex because the defining data are specified up to homotopy: that is, without using equalities, but a with weaker notion. We then study ways in which they can be made equivalent to simpler structures. This process is called rigidification.

We studied such rigidification for a model of weak n-categories, due to Tamsamani. This produces a new important type of higher categorical structures, called weakly globular n-fold categories. These are then used in applications.

In this project, weakly globular n-fold categories were first been explored in low dimension, where they are called weakly globular double categories. These have been shown to be equivalent to a classical structure, the bicategories. The latter are ubiquitous in category theory and in homotopy theory, and this new model opened the road to applications. One is the construction of a weakly globular double category of fractions of a category. This process, called localization, is of paramount importance in homotopy theory and in geometry, and our results offer a novel approach to it.

We also explored the notion of weak globularity in homotopy theory, for the modeling of the building blocks of spaces, called n-types. The higher categorical structures must then satisfy invertibility conditions, and are called weakly globular n-fold groupoids. This affords an algebraic and combinatorial model of spaces with many desirable properties.

Weakly globular n-fold categories were then explored in full generality, leading to a new model of weak higher categories, which we proved to be suitably equivalent to one of the existing models, due to Tamsamani. Our approach is based on a new paradigm for weakening higher categorical structures, given by the notion of weak globularity. This new approach lead to the introduction of many novel ideas and techniques in higher category theory, which in turn are be useful in pursuing applications.

The work of this project lead to significant publications and was disseminated through invited talks at high level conferences in Europe, North America and Australia. This contributed to the visibility of EU science, therefore enhancing European scientific excellence and competitiveness.

This project involved several collaborations within and outside the EU with world leading research centres in category theory and its applications. This contributed to the strengthening of the scientific links between the EU and the rest of the world.
The project also contributed to European excellence with a considerable transfer of knowledge to the Host institution. The Fellow re-integrated very successfully into the EU, and has an excellent career development at the University of Leicester, where she now holds a permanent position.