Motivated by applications in computable analysis, we study fixpoints of certain endofunctors over categories of containers. More specifically, we focus on fibred endofunctors over the fibrewise opposite of the codomain fibration that can be themselves be represented by families of polynomial endofunctors. In this setting, we show how to compute initial algebras, terminal coalgebras and another kind of fixpoint $ζ$. We then explore a number of examples of derived operators inspired by Weihrauch complexity and the usual construction of the free polynomial monad. We introduce $ζ$-expressions as the syntax of $μ$-bicomplete categories, extended with $ζ$-binders and parallel products, which thus have a natural denotation in containers. By interpreting certain $ζ$-expressions in a category of type 2 computable maps, we are able to capture a number of meaningful Weihrauch degrees, ranging from closed choice on $\{0, 1\}$ to determinacy of infinite parity games, via an "answerable part" operator.

翻译：受可计算分析应用的启发，我们研究了容器范畴上某些自函子的不动点问题。具体而言，我们聚焦于纤维化自函子（其定义在余域纤维化逐纤维对偶范畴上），且这些自函子本身可由多项式自函子族表示。在此设定下，我们展示了如何计算初始代数、终余代数以及另一类不动点$ζ$。随后，我们借鉴Weihrauch复杂度与自由多项式单子的标准构造，探索了大量派生算子示例。我们将$ζ$-表达式引入为$μ$-双完备范畴的语法，并扩展了$ζ$-绑定子与并行乘积，从而在容器中具有自然指称。通过在第二类可计算映射范畴中解释某些$ζ$-表达式，我们能够捕获一系列有意义的Weihrauch度，范围涵盖从$\{0, 1\}$上的闭选择到无限奇偶博弈的确定性，其间通过一个"可回答部分"算子实现。