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Gray tensor product and Kontsevich's Swiss-Cheese conjecture

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posted on 29.03.2022, 01:34 by Hyeon Tai Jung
We study connections between two seemingly very distant constructions: Gray-product of higher categories and famous Kontsevich Swiss-Cheese conjecture. Gray-product of 2-categories is known for almost 50 years and it is an extremely important construction in 2-category theory. It was proved by Crans and later by Bourke and Gurski that a naive analogue of Gray-product in higher dimensions does not exist. Nevertheless, there is a conjecture that there exists a weaker version of this product in all dimensions such that it descends to a closed structure on homotopy level. Swiss-Cheese conjecture was proposed by Fields medalist M. Kontsevich in 1998 to handle a problem of the existence of higher order Hochschild complexes. It is geometrical in nature and is very important in deformation quantisation theory. In the thesis we outline a surprising relationship between these two important conjectures, which was not observed before. Namely, the existence of a homotopically closed Gray-product of V-enriched categories implies the Swiss-Cheese conjecture in V. We provide a full proof of this statement for V = Set, Ab and Cat using the idea of categorification.


Table of Contents

1. Introduction -- 2. Background -- 3. Result in (Cat,×, I) -- 4. Symmetric closed monoidal structure on 2Cat with Gray tensor product -- 5. Result in (2Cat,⊗G,I) -- 6. Kontsevich's Swiss-Cheese conjecture.


Theoretical thesis. Bibliography: pages 49-50

Awarding Institution

Macquarie University

Degree Type

Thesis MRes


MRes, Macquarie University, Faculty of Science and Engineering, Department of Mathematics

Department, Centre or School

Department of Mathematics

Year of Award


Principal Supervisor

Michael Batanin


Copyright Hyeon Tai Jung 2018. Copyright disclaimer: http://mq.edu.au/library/copyright




1 online resource (x, 50 pages)

Former Identifiers

mq:70527 http://hdl.handle.net/1959.14/1265146