The data for many useful bidirectional constructions in applied category theory (optics, learners, games, quantum combs) can be expressed in terms of diagrams containing "holes" or "incomplete parts", sometimes known as comb diagrams. We give a possible formalization of what these circuits with incomplete parts represent in terms of symmetric monoidal categories, using the dinaturality equivalence relations arising from a coend. Our main idea is to extend this formal description to allow for infinite circuits with holes indexed by the natural numbers. We show how infinite combs over an arbitrary symmetric monoidal category form again a symmetric monoidal category where notions of delay and feedback can be considered. The constructions presented here are still preliminary work.

中文翻译：

---

离散时间反馈的梳状图

应用范畴论（光学、学习器、游戏、量子梳）中许多有用的双向构造的数据可以用包含“洞”或“不完整部分”的图表来表达，有时也称为梳状图。我们给出了这些具有不完整部分的电路在对称幺半群范畴中所表示的可能形式化，使用由共端产生的双自然等价关系。我们的主要想法是扩展这种形式化的描述，以允许具有由自然数索引的孔的无限回路。我们展示了在任意对称幺半群类别上的无限组合如何再次形成对称幺半群类别，其中可以考虑延迟和反馈的概念。这里介绍的建筑仍然是初步工作。