高值系数
Coends of higher arity
摘要：
我们专门介绍了一个最近引入的概念，广义二度函子$T : (\mathcal{C}^\text{op})^p \times \mathcal{C}^q \to \mathcal{D}$的情况下，域（分别为，余域）是常数，获得端的概念（分别，coends），在此称为$(p,q)$-末端（分别，$(p,q)$-coends）。虽然高arity余尾是"完全对称"（普通）余尾的特殊情况，但它们在研究许多新的范畴现象中起着重要的技术作用，这些新的范畴现象可以被广泛地归类为范畴论的两个新变体。其中第一个，加权范畴论，包括研究加权变种的经典概念和建设发现在普通范畴论，除了一个限制。这导致了许多不同的和丰富的概念，如加权Kan扩张，加权删除和加权结束。第二种是对角范畴论，它沿着一个不同的（尽管是相关的）方向前进，在这个方向上，人们用关于双自然变换的普遍性代替关于自然变换的普遍性，模仿从极限到目的的过程。在这样做的时候，我们又遇到了许多新的有趣的概念，其中我们同样发现了对角Kan扩张、对角扩张和对角端点。
英文摘要：
We specialise a recently introduced notion of generalised dinaturality for functors $T : (\mathcal{C}^\text{op})^p \times \mathcal{C}^q \to \mathcal{D}$ to the case where the domain (resp., codomain) is constant, obtaining notions of ends (resp., coends) of higher arity, dubbed herein $(p,q)$-ends (resp., $(p,q)$-coends). While higher arity co/ends are particular instances of "totally symmetrised" (ordinary) co/ends, they serve an important technical role in the study of a number of new categorical phenomena, which may be broadly classified as two new variants of category theory. The first of these, weighted category theory, consists of the study of weighted variants of the classical notions and construction found in ordinary category theory, besides that of a limit. This leads to a host of varied and rich notions, such as weighted Kan extensions, weighted adjunctions, and weighted ends. The second, diagonal category theory, proceeds in a different (albeit related) direction, in which one replaces universality with respect to natural transformations with universality with respect to dinatural transformations, mimicking the passage from limits to ends. In doing so, one again encounters a number of new interesting notions, among which one similarly finds diagonal Kan extensions, diagonal adjunctions, and diagonal ends.
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