Commutative 2-algebra and applications.

点击翻译按钮获取中文摘要

Commutative algebra is a well-established area of pure mathematics, with several applications across the whole of mathematics. In particular, commutative algebra provides a foundation for the development of algebraic geometry. A useful way to understand and study commutative algebra is via category theory. Indeed, category theory provides a general approach to characterise categories of algebraic objects, establish their key properties, and describe in abstract terms the idea of commutativity and distributivity. This is achieved via monad theory, established in the '60s and '70s in the now classical works of Eilenberg, Lawvere, Beck, and others. This project seeks to develop further a relatively new subject, which may be referred to as commutative 2-algebra. This can be understood as a counterpart of commutative algebra in which instead of considering algebraic structures carried by sets, one considers algebraic structures carried by categories. As such, the subject is closely related in spirit to the idea of `categorification' that has been very prominent in representation theory in recent years. Just as commutative algebra could be studied under the lens of category theory via monads, 2-commutative algebra can be be studied using 2-dimensional category theory via 2-monads and their generalisations, which have been studied extensively by the Australian category theory school.The overall goal of the project is to develop further commutative 2-algebra by establishing counterparts of some fundamental results of classical monad theory for relative pseudo-monads, a generalisation of 2-monads that the project supervisor and some collaborators have introduced recently. Specific goals include the proof of counterparts of Beck's fundamental result on distributive laws and of Kock's characterisation of commutative monads. As an application, the project will push further the theory of operads and analytic functors, as developed by the project supervisor and his collaborators. One specific goal here is to enhance the bicategory of symmetric operads and analytic functors introduced by the supervisor and Andre' Joyal to a pseudo double category, building on work of Dwyer and Hess.The novelty of the project derives from the idea of developing the theory on the basis of the notion of a relative pseudomonad rather than that of a 2-monad, as traditionally done. This is useful because it allows us to capture important examples that are beyond the standard theory. These examples are of fundamental for our applications and are of interest also for current research in theoretical computer science.