论文标题
较高的Arity的含量
Coends of higher arity
论文作者
论文摘要
我们专门为函子提供了最近引入的广义性二尿性概念$ t：（\ Mathcal {c}^\ text {op}）^p \ times \ times \ times \ mathcal {c}^q \ to \ nathcal {d} $ formand（severmain is of deveber，co.scomain is efterions of end end of。 $（p，q）$ - 结束（分别，$（p，q）$ - coends）。虽然较高的Arity Co/End是“完全对称”（普通的）CO/ENEN的特殊实例，但它们在研究许多新的分类现象的研究中起着重要的技术作用，这些现象可以广泛地归类为类别理论的两个新变体。 其中的第一个加权类别理论包括对普通类别理论中经典概念和构造的加权变体的研究。这导致了许多多样化和丰富的概念，例如加权KAN扩展，加权套和加权末端。 第二个对角线类别理论是朝着不同（尽管相关）方向进行的，其中人们在自然转化方面取代了与自然转化相对于DIN Aturater Transformations的普遍性，模仿了从极限到末端的通用。这样一来，一个人再次遇到了许多新的有趣的概念，其中类似地发现对角线kan扩展，对角线辅助和对角线末端。
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.
