Kan扩张是部分余极限
Kan extensions are partial colimits
AI总结：
本文借助单子bar构造中的部分赋值将左Kan扩张是部分余极限的直觉严格化，定义图表单子与小预层单子间的“像”态射推广共尾函子概念，证明逐点左Kan扩张对应余极限的部分赋值。
摘要：
左Kan扩张的一种解读方式是将其视作一种“部分余极限”，即通过余极限替换图表的部分内容。我们借助位于单子的所谓bar构造中的“部分赋值”，将这一直觉严格化。用于构造余极限的相关（伪）单子是图表单子和小预层单子，二者都定义在局部小范畴构成的（巨型）范畴CAT上。全文特别注意处理大小问题，这类问题在自由余完备化的语境中是出了名的棘手。\n我们详细阐明了这些伪单子的结构映射，包含所有二维细节。随后，基于对单子的“标量限制”构造如何推广到伪单子上的伪代数情形的详细通用证明，我们在二者之间定义了一个单子态射，称之为“像”。该态射尤其让我们能够推广“共尾函子”的概念，即那些以绝对方式保持余极限不变的函子。这一推广将绝对余极限的概念作为特例包含在内。\n本文的主要结果阐明了一个图表的逐点左Kan扩张如何精确对应于其余极限的一次部分赋值。这一范畴论结果与概率单子中的情形类似：在概率单子中，随机变量的条件期望对应于其质心的部分赋值。
英文摘要：
One way of interpreting a left Kan extension is as taking a kind of "partial colimit", whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the "partial evaluations" sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the "restriction-of-scalars" construction of monads extends to the case of pseudoalgebras over pseudomonads, we define a morphism of monads between them, which we call "image". This morphism allows us in particular to generalize the idea of "confinal functors" i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this paper spells out how a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass.
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