内部交叉模的三值换向器阻塞
The ternary commutator obstruction for internal crossed modules
摘要：
在余余完备同调范畴中，余smash积产生（可能更高阶的）子smash子。我们利用二元和三元的co-smash积和相关的分解子分别刻画了内部交叉模和内部范畴。如果范畴具有Smith is Huq性质，则三元项是冗余的，这意味着给定对象上的两个等价关系恰好在它们的规范化交换时交换。事实上，我们表明，这种关系的史密斯换向器和Huq换向器的规范化之间的差异是衡量一个三元换向器，使史密斯是Huq属性本身可以由后两个换向器之间的关系。这允许表明循环的范畴不具有Smith is Huq性质，这也意味着三元循环通常不能分解为嵌套的二元循环。因此，在Smith是Huq的情况下，我们得到了内部范畴、Beck模和双中心扩张的新描述，以及Smith换位子的分解公式。三元换向器现在也出现在霍普夫公式的第三个同源性与系数的abelianisation函子。
英文摘要：
In finitely cocomplete homological categories, co-smash products give rise to (possibly higher-order) commutators of subobjects. We use binary and ternary co-smash products and the associated commutators to give characterisations of internal crossed modules and internal categories, respectively. The ternary terms are redundant if the category has the Smith is Huq property, which means that two equivalence relations on a given object commute precisely when their normalisations do. In fact, we show that the difference between the Smith commutator of such relations and the Huq commutator of their normalisations is measured by a ternary commutator, so that the Smith is Huq property itself can be characterised by the relation between the latter two commutators. This allows to show that the category of loops does not have the Smith is Huq property, which also implies that ternary commutators are generally not decomposable into nested binary ones. Thus, in contexts where Smith is Huq need not hold, we obtain a new description of internal categories, Beck modules and double central extensions, as well as a decomposition formula for the Smith commutator. The ternary commutator now also appears in the Hopf formula for the third homology with coefficients in the abelianisation functor.
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