## Our projects

## Current projects

## Monoidal categories and beyond: new contexts and new applications (2016-2019)

**Scheme**: ARC Discovery Projects, grant number DP160101519**Personnel**: Ross Street (CI), Stephen Lack (CI), Dominic Verity (CI), Richard Garner (CI)**Summary**: This project aims to develop a theory of generalised monoidal structures with applications to fields as diverse as combinatorics, representation theory, algebraic geometry, topology, theoretical physics and computer science.

Monoidal categories are a mathematical formalism for systems with operations of parallel and serial composition, such as the Hilbert space model of quantum mechanics; recent developments in topology and quantum algebra have led to more general notions of monoidal category, and further progress requires a comprehensive theory of such structures. By providing this, the project aims to give a foundation for future scientific endeavour and solidify Australia's position as a leading international force in abstract mathematics

## Structural homotopy theory: a category-theoretic study (2013-2016)

**Scheme**: ARC Discovery Projects, grant number DP130101969**Personnel**: Ross Street (CI), Stephen Lack (CI), Dominic Verity (CI), Richard Garner (CI)**Summary**: Our project will apply category theory to homotopy theory, using the vast body of work in higher and enriched category theory developed in Sydney over the last forty years.

A key tension in mathematics is that between algebra and geometry: whilst geometry gives spatial intuition, algebra provides computational tractability. Homotopy theory uses algebra to study that part of geometry known as topology. However, over the past 20 years, startling new discoveries have shown that it plays a much broader role, mediating between such disciplines as algebraic geometry, higher category theory and algebraic K-theory. These recent advances have almost all been cast in the language of category theory, without taking full advantage of its methodology.

## Algebraic Categories and Categorical Algebra (2012-2016)

**Scheme**: ARC Future Fellowship, grant number FT110100385**Personnel**: Stephen Lack (CI)**Summary**: This project will study two new branches of algebra – quantum algebra and postmodern algebra – using the latest techniques from category theory. It will lead to a deeper understanding in these related fields.

Algebra is the study of operations, such as addition and multiplication, and the relationships between these operations. It is fundamental to all areas of mathematics as well as quantitative and qualitative aspects of other fields. Despite 2000 years of study, many important questions remain open. Recently the types of operations considered and the sorts of relationships that hold between them have undergone a vast generalisation, motivated by the needs of related fields such as geometry, physics, and computing.

## Generalised Topological Spaces (2011-2016)

**Scheme**: ARC Australian Research Fellowship, grant number DP110102360**Personnel**: Richard Garner (CI)**Summary**: In this project, we aim to generalise the notion of topological space in order to obtain a framework within which we may describe and compare the weakened, perturbed and higher-dimensional topological structures that are playing an increasingly important role in modern mathematics.

A central concept in mathematics and physics is that of a continuous function: one for which small perturbations in input cause only small changes in output. To express the continuity of a function requires a notion of "nearness" between points, and this is encapsulated in the abstract notion of topological space. Such spaces are a powerful unifying tool in mathematics, allowing structures in algebra, topology, logic and geometry to be compared and classified.

## Recently funded projects

## Enriched higher category theory (2013-2015)

**Scheme**: ARC Discovery Projects, grant number DP130101172**Personnel**: Michael Batanin (CI), Martin Markl (PI), Clemens Berger (PI)**Summary**: Higher category theory is a very young branch of mathematics which has already become a vital tool in many areas of mathematics and theoretical physics such as algebra, geometry, topology, mathematical logic, quantum field theory and computer science. The impact of higher category theory for the future development of mathematics and physics will be immense. In its present shape, however, this theory is technically difficult. The challenge is to find an approach to this theory which would allow to make it transparent and accessible for the wider scientific community.

In this project we proposed such an approach and we studied its application to important open problems in geometry and topology.

## Applicable Categorical Structures (2010-2013)

**Scheme**: ARC Discovery Projects, grant number DP1094883**Personnel**: Ross Street (CI), Michael Johnson (CI), Stephen Lack (CI), Dominic Verity (CI)**Summary**: Sets with structure are to modern mathematics what numbers were to the ancients. Such structures can often be modelled in monoidal categories vastly different from the category of sets. In algebraic cases, there are free structures formed from symbols with no extra relations beyond those required to express the structure. Often, examples of such models in categories whose ingredients are geometric concepts (such as tangles on strings or cobordisms) have freeness properties that involve moving out of that base geometric category. This kind of freeness has allowed the construction of important invariants of geometric figures.

Our project studied both theoretical and applied aspects of this encompassing schema.

## Functorial operadic calculus (2010-2013)

**Scheme**: ARC Discovery Projects, grant number DP1095346**Personnel**: Michael Batanin (CI), Clemens Berger (PI)**Summary**: Substitution is one of the first mathematical operations everybody learns in primary school. It is so routine that we rarely notice it in our everyday life even though we perform this operation every time we count almost anything.

The aim of this project was to show that precisely because substitution is so basic and natural, it is one of the most fundamental and far reaching concepts in mathematics. A deep study of the algebra of such operations will lead to a solution of important open problems in the foundations of mathematics and mathematical physics.

## Cohomology enhanced: an application of enriched and higher categories (2007-2010)

**Scheme**: ARC Discovery Projects, grant number DP0771252**Personnel**: Ross Street (CI), Michael Johnson (CI), Stephen Lack (CI), Dominic Verity (CI)**Summary**: Cohomology has been one of the most powerful tools in the mathematics of the twentieth century, finding applications in all areas of modern mathematics. It is a technique for understanding and classifying complex mathematical structures in simpler terms.

This project involved a radical expansion in scope of the information extracted from these mathematical structures, using the most recent advances in enriched and higher dimensional category theory.

## Categorical structures in string theory (2006-2009)

**Scheme**: ARC Discovery Projects, grant number DP0663514**Personnel**: Ross Street (CI)**Summary**: General relativity and quantum field theory are used in physics to explain all the forces of nature. The most promising candidate for the unification of these two fundamental theories is string theory. String theory has exposed exciting mathematical challenges both on the geometric and algebraic side, and for linking those two sides.

Category theory has excelled in expressing such linkages in other fields and this project detailed how we planned to use our categorical expertise in string theory. Our results then fed back into physics.

## Foundation of higher dimensional homological algebra (2005-2008)

**Scheme**: ARC Discovery Projects, grant number DP0558372**Personnel**: Michael Batanin (CI), Mark Weber**Summary**: `Homotopical Mathematics' was a term introduced shortly before this project began to designate a rapidly developing methodology. It is based on the substitution of set theoretical notions by homotopy theoretical notions in a large part of mathematics relevant to geometry and physics. This approach previously produced spectacular applications in algebraic geometry, topology and mathematical physics. Homological algebra lies at the heart of this approach, yet its further development and application require clear and consistent foundations.

In our project we constructed such foundations, using methods of higher category theory. As an outcome, proof of important conjectures from both areas arose naturally.