Commutative 2-algebra and applications.

交换代数是纯数学中一个建立良好的领域，在整个数学中有几个应用。特别是，交换代数为代数几何的发展奠定了基础。范畴论是理解和学习交换代数的一个有用的方法。事实上，范畴论提供了一种通用的方法来描述代数对象的范畴，建立它们的关键属性，并用抽象的术语描述交换性和分配性的概念。这是通过单一性理论实现的，该理论建立于60年代和70年代，现在是Eilenberg， Lawvere， Beck和其他人的经典作品。这个项目寻求进一步发展一个相对较新的主题，这可能被称为交换2代数。这可以被理解为交换代数的对立物，在交换代数中，人们不考虑由集合携带的代数结构，而是考虑由范畴携带的代数结构。因此，这个主题在精神上与近年来在表征理论中非常突出的“分类”概念密切相关。正如交换代数可以在范畴论的透镜下通过单子来研究一样，2-交换代数也可以通过二维范畴论的透镜通过2-单子及其推广来研究，这已经被澳大利亚范畴论学派广泛研究。该项目的总体目标是通过建立相对伪单子的经典单子理论的一些基本结果的对应物来进一步发展交换2-代数，这是项目主管和一些合作者最近介绍的2-单子的推广。具体目标包括证明贝克关于分配律的基本结果的对应物和科克对交换单子的表征。作为一项应用，该项目将进一步推动由项目主管及其合作者开发的操作数和解析函子理论。这里的一个具体目标是在Dwyer和Hess的工作基础上，将导师和Andre' Joyal引入的对称操作数和解析函子的双范畴增强为伪双范畴。这个项目的新颖之处在于，它是在相对伪单峰的基础上发展理论的，而不是像传统上那样是在2单峰的基础上发展理论的。这很有用，因为它使我们能够捕捉到超出标准理论的重要例子。这些例子是我们应用的基础，也是当前理论计算机科学研究的兴趣所在。

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.