Not every pseudoalgebra is equivalent to a strict one
Not every pseudoalgebra is equivalent to a strict one
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并非所有伪代数都等同于严格代数
DOI：
10.1016/j.aim.2011.01.010
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发表时间：
2010
影响因子：
1.7
通讯作者：
Michael Shulman
中科院分区：
文献类型：
--
作者：
Michael Shulman
在一个局部有限可表示的2范畴上，我们描述了一个有限2-单子，对于这个单子，并不是所有的伪代数都等价于严格的伪代数。这表明在2-单子上，有秩并不是每个伪代数可严格的充分条件。我们的反例来自于高等范畴论：严格代数是严格3-范畴，而伪代数是介于灰色范畴和三范畴之间的半严格3-范畴。因此，并非每个灰色范畴都等价于一个严格的三范畴，从而连接了二范畴和更高范畴的连贯理论。特别地，任何非平凡编织的一元范畴都给出了一个不等同于严格代数的伪代数的例子。
We describe a finitary 2-monad on a locally finitely presentable 2-category for which not every pseudoalgebra is equivalent to a strict one. This shows that having rank is not a sufficient condition on a 2-monad for every pseudoalgebra to be strictifiable. Our counterexample comes from higher category theory: the strict algebras are strict 3-categories, and the pseudoalgebras are a type of semi-strict 3-category lying in between Gray-categories and tricategories. Thus, the result follows from the fact that not every Gray-category is equivalent to a strict 3-category, connecting 2-categorical and higher-categorical coherence theory. In particular, any nontrivially braided monoidal category gives an example of a pseudoalgebra that is not equivalent to a strict one.