一类奇异子类的富单元和富变种的图解表示

IF 0.6 4区 数学 Q3 MATHEMATICS
Rory B. B. Lucyshyn-Wright, Jason Parker
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引用次数: 7

摘要

丰富单子的表示理论是由Kelly、Power和Lack在Lawvere的经典著作之后发展起来的,并在Bourke–Garner及其作者最近的著作中被推广应用于arities的子类别。我们认为,虽然在理论上很优雅,结构上也很基本,但在实践中直接构建这种丰富单元的表示可能很不方便,因为它们与通过运算和方程构建许多类别的丰富代数结构时使用的定义过程不直接匹配。保留上述演示方法作为关键技术基础,我们建立了一种灵活的形式主义,用于直接描述由(\mathscr{V})-范畴(\mathscr{C})的对象所承载的丰富代数结构,用参数化的(\mathscr{J})运算和适当的arities子类的图解方程来描述(\mathscr{J}\hookrightarrow\mathscr{C})。在此基础上,我们引入了图解表示和变异范畴的概念,并证明了变异范畴与单元范畴是对偶等价的。我们建立了几个与数学和理论计算机科学相关的图解表示和变体的例子,并定义了图解表示的和和张量积。我们证明了\(\mathscr{J}\)-相对单元和\(\math scr{J}\)-预理论都产生了直接描述其代数的图解表示。使用图解表示法作为证明方法,我们将Bourke和Garner的前理论单元附加推广到局部表示环境之外。最后,我们将代数类和方程组之间的Birkhoff Galois连接推广到上述设置。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Diagrammatic Presentations of Enriched Monads and Varieties for a Subcategory of Arities

The theory of presentations of enriched monads was developed by Kelly, Power, and Lack, following classic work of Lawvere, and has been generalized to apply to subcategories of arities in recent work of Bourke–Garner and the authors. We argue that, while theoretically elegant and structurally fundamental, such presentations of enriched monads can be inconvenient to construct directly in practice, as they do not directly match the definitional procedures used in constructing many categories of enriched algebraic structures via operations and equations. Retaining the above approach to presentations as a key technical underpinning, we establish a flexible formalism for directly describing enriched algebraic structure borne by an object of a \(\mathscr {V}\)-category \(\mathscr {C}\) in terms of parametrized \(\mathscr {J}\)-ary operations and diagrammatic equations for a suitable subcategory of arities \(\mathscr {J}\hookrightarrow \mathscr {C}\). On this basis we introduce the notions of diagrammatic \(\mathscr {J}\)-presentation and \(\mathscr {J}\)-ary variety, and we show that the category of \(\mathscr {J}\)-ary varieties is dually equivalent to the category of \(\mathscr {J}\)-ary \(\mathscr {V}\)-monads. We establish several examples of diagrammatic \(\mathscr {J}\)-presentations and \(\mathscr {J}\)-ary varieties relevant in both mathematics and theoretical computer science, and we define the sum and tensor product of diagrammatic \(\mathscr {J}\)-presentations. We show that both \(\mathscr {J}\)-relative monads and \(\mathscr {J}\)-pretheories give rise to diagrammatic \(\mathscr {J}\)-presentations that directly describe their algebras. Using diagrammatic \(\mathscr {J}\)-presentations as a method of proof, we generalize the pretheories-monads adjunction of Bourke and Garner beyond the locally presentable setting. Lastly, we generalize Birkhoff’s Galois connection between classes of algebras and sets of equations to the above setting.

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来源期刊
CiteScore
1.30
自引率
16.70%
发文量
29
审稿时长
>12 weeks
期刊介绍: Applied Categorical Structures focuses on applications of results, techniques and ideas from category theory to mathematics, physics and computer science. These include the study of topological and algebraic categories, representation theory, algebraic geometry, homological and homotopical algebra, derived and triangulated categories, categorification of (geometric) invariants, categorical investigations in mathematical physics, higher category theory and applications, categorical investigations in functional analysis, in continuous order theory and in theoretical computer science. In addition, the journal also follows the development of emerging fields in which the application of categorical methods proves to be relevant. Applied Categorical Structures publishes both carefully refereed research papers and survey papers. It promotes communication and increases the dissemination of new results and ideas among mathematicians and computer scientists who use categorical methods in their research.
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