# Applications of Higher Dimensional Algebra to Stable Homotopy Theory

Lead Research Organisation:
University of Sheffield

Department Name: Mathematics and Statistics

### Abstract

Category theory is the abstract study of relationships between mathematical objects. These relationships can be simple, such as one number being larger or smaller than another, or they can be more complicated and many-layered, such as spatial relationships between two points on a surface. Higher-dimensional category theory is then the study of similar kinds of relationships, but we now allow the existence of relationships between relationships, and so on. One example might describe all the ways to walk from one point to another. If the space between these two points is an empty field, say, then any two different paths are essentially the same, as we could devise a series of paths that gradually changed from the first given path to the second. On the other hand, if there is a pond between these two points, then there are at least two essentially different paths, one for each way around the pond. The higher dimensional category theory in this proposal is in the subfield of higher dimensional algebra which is the study of objects with many different kinds of algebraic operations on them, and the interactions between those; a simple example would just be the real numbers with the operations plus and times, and one example interaction between these is a(b+c)=ab+ac.

The other field of mathematics in this proposal is topology, which is the study of shapes. In topology, aspects like distance or smoothness do not matter, so a circle is the same as a square and a doughnut is the same as a coffee cup. One of the most profitable ways of understanding shapes is by assigning them algebraic invariants. For instance, there is an invariant called the fundamental group which counts certain kinds of holes in a shape; computing the fundamental group of both a circle and a square will give the same answer, as they both have a single hole in the middle. On the other hand, computing the fundamental group of a filled-in square will give a different answer, precisely because the hole has now been filled.

The research in this proposal is about using higher dimensional algebra to describe shapes. As an example, imagine two flexible tubes. We can combine these in a variety of ways: we can just set them next to each to get a pair of tubes, we could glue the end of one tube to the other and get a single very long tube, or we could even glue both ends of the first tube to the corresponding ends of the second tube to get a circular tube (a shape called a torus). Gluing surfaces together is one kind of algebraic operation, and setting surfaces next to each other is another, and these two operations behave in ways that are governed by laws appearing in higher dimesional algebra. If we additionally take into account symmetries of the shapes involved (for our example, tubes can be rotated or the ends can be swapped), then both the topological and algebraic descriptions become more complicated. These constructions have been studied using topology, but this research is aimed at utilising the tools of higher dimensional algebra in order to shed new light on old problems.

The other field of mathematics in this proposal is topology, which is the study of shapes. In topology, aspects like distance or smoothness do not matter, so a circle is the same as a square and a doughnut is the same as a coffee cup. One of the most profitable ways of understanding shapes is by assigning them algebraic invariants. For instance, there is an invariant called the fundamental group which counts certain kinds of holes in a shape; computing the fundamental group of both a circle and a square will give the same answer, as they both have a single hole in the middle. On the other hand, computing the fundamental group of a filled-in square will give a different answer, precisely because the hole has now been filled.

The research in this proposal is about using higher dimensional algebra to describe shapes. As an example, imagine two flexible tubes. We can combine these in a variety of ways: we can just set them next to each to get a pair of tubes, we could glue the end of one tube to the other and get a single very long tube, or we could even glue both ends of the first tube to the corresponding ends of the second tube to get a circular tube (a shape called a torus). Gluing surfaces together is one kind of algebraic operation, and setting surfaces next to each other is another, and these two operations behave in ways that are governed by laws appearing in higher dimesional algebra. If we additionally take into account symmetries of the shapes involved (for our example, tubes can be rotated or the ends can be swapped), then both the topological and algebraic descriptions become more complicated. These constructions have been studied using topology, but this research is aimed at utilising the tools of higher dimensional algebra in order to shed new light on old problems.

### Planned Impact

The primary benefactors of the impact of this proposal will be academic and the general public. The academic beneficiaries have already been discussed, but this proposal has a large side-benefit in the form of training for PhD students, some of which will likely continue in academia and some of which will certainly enter the workforce. This training will have a large academic component which underlies the development of crucial skills that can affect both the PhD students themselves and those around them in any work environment. As an intradisciplinary project, an important aspect of this research is communication with two very different mathematical communities. While these communities have had many interactions in the past, this proposal will introduces a wide array of techniques and results from category theory into algebraic topology via the study of equivariant cohomology. This effort to bring these two communities closer together will involve both academic outreach and an attention to making this research usable by a wide audience; one example of how this will be accomplished is by posting accounts of this and other research in a form suitable for outsiders to the discipline to read on websites such as the nCategory cafe (http://golem.ph.utexas.edu/category/). This drive to make my research accessible and usable to the widest possible audience will also be instilled in PhD students at the University of Sheffield, who will then take that desire to communicate outside of academia after they have completed their degree. Apart from this specific impact, there will also be the general impact of the training of skilled people (in this case, PhD students) for non-academic professions. This training will be of a general nature, including but not limited to writing skills, communication skills, and presentation skills, and will follow as a natural consequence of the discussion of this research with students both in Sheffield and in the larger community.

### Organisations

- University of Sheffield, United Kingdom (Collaboration, Fellow, Lead Research Organisation)
- University of Hamburg, Germany (Collaboration)
- University of Leeds, United Kingdom (Collaboration)
- Reed College (Collaboration)
- Masaryk University, Czech Republic (Collaboration)
- London Mathematical Society, United Kingdom (Collaboration)
- Kansas State University, United States (Collaboration)
- Ohio State University, United States (Collaboration)
- University of Leicester, United Kingdom (Collaboration)

## People |
## ORCID iD |

Michael Nicholas Gurski (Principal Investigator / Fellow) |

### Publications

Bourke J
(2016)

*The Gray Tensor Product Via Factorisation*in Applied Categorical Structures
Bourke John
(2015)

*A COCATEGORICAL OBSTRUCTION TO TENSOR PRODUCTS OF GRAY-CATEGORIES*in THEORY AND APPLICATIONS OF CATEGORIES
Bourke, J.

*The Gray tensor product via factorisation*in Applied Category Structures
Cheng, E.
(2014)

*Iterated icons*in Theory and Applications of Categories
Corner, A S

*OPERADS WITH GENERAL GROUPS OF EQUIVARIANCE, AND SOME 2-CATEGORICAL ASPECTS OF OPERADS IN Cat*in Journal of Pure and Applied Algebra
Gurski N
(2017)

*K -theory for 2-categories*in Advances in Mathematics
Gurski, MN

*Stable Postnikov data of Picard 2-categories*in Algebraic and Geometric Topology
Gurski, MN
(2016)

*Extending homotopy theories from strict maps to lax maps*in Homology, Homotopy and Applications
Gurski, MN

*K-theory for 2-categories*in Acta MathematicaDescription | I have developed tools and broad expertise to study foundational questions through applications of higher dimensional category theory. This is an ongoing project, and will broaden as it goes into research in topology, algebra, and mathematical physics. |

Exploitation Route | This is an open area of research, with a variety of questions that could be tackled by other researchers. |

Sectors | Other |

Description | London Mathematical Society Scheme 3 |

Amount | £4,000 (GBP) |

Organisation | London Mathematical Society |

Sector | Academic/University |

Country | United Kingdom |

Start | 11/2014 |

End | 10/2016 |

Description | Action operads |

Organisation | University of Sheffield |

Department | School of Mathematics and Statistics |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | This collaboration involves ongoing work with two of my post-graduate research students. It has motivated new applications of ideas in my research to solve problems they were studying as part of their PhD research. I have provided direction as well as key background knowledge. |

Collaborator Contribution | My partners (Alex Corner and Edward Prior) have postulated some key problems for us to attack, and helped apply methods I have helped develop to those problems. |

Impact | The preprint "Operads with general groups of equivariance, and some 2-categorical aspects of operads in Cat" is a direct outcome of this collaboration, as is "Operads, tensor products, and the categorical Borel construction." We expect another submitted paper in 2016. |

Start Year | 2014 |

Description | Daniel Schaeppi |

Organisation | University of Sheffield |

Department | School of Mathematics and Statistics |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | I mentored Daniel Schaeppi as an RA in Sheffield during 2014-15, and we began a joint research project. |

Collaborator Contribution | Daniel suggested the topic of research which also led to a series of seminar talks. |

Impact | None yet. |

Start Year | 2014 |

Description | Gray tensor products |

Organisation | Masaryk University |

Country | Czech Republic |

Sector | Academic/University |

PI Contribution | This collaboration has explored some foundational questions in higher dimensional algebra. I have provided insight into these questions as well as helping to shape some of the central features of our outputs. |

Collaborator Contribution | My collaborator posed some of the initial questions and showed how some classic techniques could be updated to address them. |

Impact | J. Bourke and N. Gurski, The Gray tensor product via factorisation J. Bourke and N. Gurski, A cocategorical obstruction to tensor products of Gray-categories |

Start Year | 2013 |

Description | K-theory for 2-categories |

Organisation | Ohio State University |

Country | United States |

Sector | Academic/University |

PI Contribution | I continue to participate in a joint research project in which my particular research background plays a key role in our progress. |

Collaborator Contribution | My partners also contribute their own skills and knowledge to the project, and we were hosted at Ohio State for a week in October to pursue this joint project. I presented some of our findings at a local seminar in Ohio as an invited speaker. |

Impact | We submitted two papers in 2015 and expect to submit a further two papers in 2016. |

Start Year | 2014 |

Description | K-theory for 2-categories |

Organisation | Reed College |

Country | United States |

Sector | Academic/University |

PI Contribution | I continue to participate in a joint research project in which my particular research background plays a key role in our progress. |

Collaborator Contribution | My partners also contribute their own skills and knowledge to the project, and we were hosted at Ohio State for a week in October to pursue this joint project. I presented some of our findings at a local seminar in Ohio as an invited speaker. |

Impact | We submitted two papers in 2015 and expect to submit a further two papers in 2016. |

Start Year | 2014 |

Description | Modules over braided monoidal categories |

Organisation | Kansas State University |

Country | United States |

Sector | Academic/University |

PI Contribution | I have helped shape the research by providing expertise and linking discoveries with other areas of category theory. |

Collaborator Contribution | Manuel Barenz (Bamberg) provided the original direction of the research, while David Yetter (Kansas State) has provided extensive background knowledge. We have all contributed to content of the research equally. |

Impact | None yet. |

Start Year | 2015 |

Description | Modules over braided monoidal categories |

Organisation | University of Hamburg |

Country | Germany |

Sector | Academic/University |

PI Contribution | I have helped shape the research by providing expertise and linking discoveries with other areas of category theory. |

Collaborator Contribution | Manuel Barenz (Bamberg) provided the original direction of the research, while David Yetter (Kansas State) has provided extensive background knowledge. We have all contributed to content of the research equally. |

Impact | None yet. |

Start Year | 2015 |

Description | YaMCATS |

Organisation | London Mathematical Society |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | This collaboration set up an LMS Scheme 3 network between Leeds, Leicester and Sheffield called YaMCATS (Yorkshire and Midlands Category Theory Seminar) that is now in its second year of operation. I helped write the proposal, plan the meetings and organise the meetings that have been held in Sheffield. |

Collaborator Contribution | My partners have performed similar roles at their respective universities. |

Impact | We have held 6 meetings so far, with 2 more to come in the 2015-16 academic year. |

Start Year | 2014 |

Description | YaMCATS |

Organisation | University of Leeds |

Department | School of Mathematics Leeds |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | This collaboration set up an LMS Scheme 3 network between Leeds, Leicester and Sheffield called YaMCATS (Yorkshire and Midlands Category Theory Seminar) that is now in its second year of operation. I helped write the proposal, plan the meetings and organise the meetings that have been held in Sheffield. |

Collaborator Contribution | My partners have performed similar roles at their respective universities. |

Impact | We have held 6 meetings so far, with 2 more to come in the 2015-16 academic year. |

Start Year | 2014 |

Description | YaMCATS |

Organisation | University of Leicester |

Department | Department of Mathematics |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | This collaboration set up an LMS Scheme 3 network between Leeds, Leicester and Sheffield called YaMCATS (Yorkshire and Midlands Category Theory Seminar) that is now in its second year of operation. I helped write the proposal, plan the meetings and organise the meetings that have been held in Sheffield. |

Collaborator Contribution | My partners have performed similar roles at their respective universities. |

Impact | We have held 6 meetings so far, with 2 more to come in the 2015-16 academic year. |

Start Year | 2014 |