Constructing Initial Algebras Using Inflationary Iteration

A. Pitts, S. Steenkamp
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引用次数: 2

Abstract

An old theorem of Adámek constructs initial algebras for sufficiently cocontinuous endofunctors via transfinite iteration over ordinals in classical set theory. We prove a new version that works in constructive logic, using “inflationary” iteration over a notion of size that abstracts from limit ordinals just their transitive, directed and well-founded properties. Borrowing from Taylor’s constructive treatment of ordinals, we show that sizes exist with upper bounds for any given signature of indexes. From this it follows that there is a rich class of endofunctors to which the new theorem applies, provided one admits a weak form of choice (WISC) due to Streicher, Moerdijk, van den Berg and Palmgren, and which is known to hold in the internal constructive logic of many kinds of topos.
用膨胀迭代构造初始代数
在经典集合论中,Adámek的一个老定理通过序上的超限迭代构造了充分共连续内函子的初始代数。我们证明了一个新的版本,它在构造逻辑中工作,使用“膨胀”迭代的大小概念,从极限序数中抽象出它们的传递性,有向性和有根据的性质。利用泰勒对序数的构造处理,我们证明了对于任何给定的指标签名,大小都存在上界。由此可以得出,如果承认Streicher, Moerdijk, van den Berg和Palmgren提出的弱选择形式(WISC),并且已知在许多拓扑的内部构造逻辑中成立,则存在一类丰富的内函子可以应用新定理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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