论文标题
富集的单阶的表现和代数构造的贫苦子类别
Presentations and algebraic colimits of enriched monads for a subcategory of arities
论文作者
论文摘要
我们开发了一个一般框架，用于研究富集的单元的签名，演示文稿和代数构造，即使在富集$ \ Mathcal {v} $的基础上也不是本地呈现的。当$ \ nathcal {v} $满足局部界限的较弱需求时，所产生的框架足以适用于$φ$ - 可访问的缺乏和rosický和$ \ nathcal {j} $ - $ \ mathcal {j} $ - 不属于本地框架的捕获量的范围，同时还没有范围的捕获物，而不是全部框架，而又有框架的范围，而又有框架的范围。 Wolff的$ \ Mathcal {V} $ - 类别的演示。给定任何小的Arities的子类别$ J：\ MATHCAL {J} \ HOCKRIGHTARROW \ MATHCAL \ MATHCAL {C} $在丰富的类别$ \ Mathcal {C C} $中，满足某些假设，我们证明了免费$ \ Mathcal {J} $ - ARY MANDIC的$ MANDIC，我们证明了pree $ \ nathcal {j} $ \ MATHCAL {J} $ - 签名和$ \ Mathcal {J} $ - Ary Monads的代数colimits的存在。我们研究了$ \ Mathcal {J} $ - Ary Monads的演示概念，并表明每个此类演示文稿都呈现$ \ Mathcal {J} $ - Ary Monad。我们的某些结果概括了凯利（Kelly）的早期结果，权力和缺乏本地富有富含元素的单调的结果，以及凯利（Kelly）的类似结果，而缺乏笛卡尔封闭类别的强烈限制单子。我们的主要结果是在本地富有富集的类别中的各种贫民窟的子类别。
We develop a general framework for studying signatures, presentations, and algebraic colimits of enriched monads for a subcategory of arities, even when the base of enrichment $\mathcal{V}$ is not locally presentable. When $\mathcal{V}$ satisfies the weaker requirement of local boundedness, the resulting framework is sufficiently general to apply to the $Φ$-accessible monads of Lack and Rosický and the $\mathcal{J}$-ary monads of the first author, while even without local boundedness our framework captures in full generality the presentations of strongly finitary monads of Lack and Kelly as well as Wolff's presentations of $\mathcal{V}$-categories by generators and relations. Given any small subcategory of arities $j : \mathcal{J} \hookrightarrow \mathcal{C}$ in an enriched category $\mathcal{C}$, satisfying certain assumptions, we prove results on the existence of free $\mathcal{J}$-ary monads, the monadicity of $\mathcal{J}$-ary monads over $\mathcal{J}$-signatures, and the existence of algebraic colimits of $\mathcal{J}$-ary monads. We study a notion of presentation for $\mathcal{J}$-ary monads and show that every such presentation presents a $\mathcal{J}$-ary monad. Certain of our results generalize earlier results of Kelly, Power, and Lack for finitary enriched monads in the locally finitely presentable setting, as well as analogous results of Kelly and Lack for strongly finitary monads on cartesian closed categories. Our main results hold for a wide class of subcategories of arities in locally bounded enriched categories.
