We compare two possible ways of defining a category of 1-combs, the first intensionally as coend optics and the second extensionally as a quotient by the operational behaviour of 1-combs on lower-order maps. We show that there is a full and bijective on objects functor quotienting the intensional definition to the extensional one and give some sufficient conditions for this functor to be an isomorphism of categories. We also show how the constructions for 1-combs can be extended to produce polycategories of n-combs with similar results about when these polycategories are equivalent. The extensional definition is of particular interest in the study of quantum combs and we hope this work might produce further interest in the usage of optics for modelling these structures in quantum theory.

翻译：我们比较了定义1-梳范畴的两种可能方式：第一种是内涵性地作为协端光学，第二种是外延性地作为1-梳在低阶映射上的操作行为的商。我们证明存在一个从内涵定义到外延定义的满的、对象双射的函子，并给出了该函子成为范畴同构的充分条件。我们还展示了如何将1-梳的构造推广为n-梳的多范畴，并得到了这些多范畴何时等价的相似结论。外延定义在量子梳研究中具有特殊意义，我们希望这项工作能进一步激发在量子理论中使用光学建模这些结构的兴趣。