The tube category of a modular tensor category is a variant of the tube algebra, first introduced by Ocneanu. As a category, it can be decomposed in two different, but related, senses. Firstly, via the Yoneda embedding, the Hom spaces decompose into summands factoring through irreducible functors, in a manner analogous to decomposing an algebra as a sum of matrix algebras. We describe these summands. Secondly, under the Yoneda embedding, each object decomposes into irreducibles, which correspond to primitive idempotents in the category itself. We identify these idempotents. We make extensive use of diagram calculus in the description and proof of these decompositions.

中文翻译：

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分解管类别

模张量范畴的管范畴是管代数的一种变体，最早由 Ocneanu 引入。作为一个类别，它可以分解为两种不同但相关的意义。首先，通过米田嵌入，Hom 空间通过不可约函子分解为和数，类似于将代数分解为矩阵代数的和。我们描述了这些命令。其次，在米田嵌入下，每个对象都分解为不可约数，它们对应于类别本身的原始幂等。我们识别这些幂等。我们在这些分解的描述和证明中广泛使用了图表演算。