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：受可计算分析中应用的启发，我们研究了容器范畴上特定自函子的不动点。具体而言，我们关注于在余定义纤维化的纤维反向上的纤维化自函子，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、终结余代数以及另一类不动点$ζ$。随后，我们探讨了一系列受Weihrauch复杂度和自由多项式幺半群通常构造启发的派生算子示例。我们引入$ζ$-表达式作为$μ$-双完备范畴的语法，扩展了$ζ$-绑定器和并行积，从而在容器中具有自然的指称语义。通过在类型2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，范围从$\{0, 1\}$上的闭选择到无限奇偶博弈的确定性，这通过一个"可回答部分"算子实现。