三次Ω范畴中的薄元与交换壳

Moore hyperrectangles on a space form a strict cubical omega-category

A question of Jack Morava is answered by generalising the notion of Moore paths to that of Moore hyperrectangles, so obtaining a strict cubical omega-category. This also has the structure of connections in the sense of Brown and Higgins, but cancellation of connections does not hold.

多重范畴：球形和立方体方法的等价性

Multiple categories: the equivalence of a globular and a cubical approach

我们证明了两类严格重范畴，即著名的球形Ω范畴和带联络的立方Ω范畴的等价性。

Ω范畴立方神经中的薄填充物

Thin fillers in the cubical nerves of omega-categories

证明了严格Ω范畴的立方神经是具有立方面运算的集合序列，且满足一定的薄填充条件的薄元的区别子类。它还表明，这种类型的序列是一个严格的欧米茄范畴唯一同构的立方神经;立方神经函子因此是范畴的等价。所涉及的集合序列实际上是适合于严格ω范畴的三次T-复形的类似物。在这些序列的定义中不需要简并，但实际上可以构造为薄填料。薄填充条件的证明使用链复形和链同伦。

On the Cubical Homology Groups of Free Partially Commutative Monoids

We study a Leech homology of a locally bounded free partially commutative monoid $M(E,I)$. Given a contravariant natural system of abelian groups $F$ on $M(E,I)$ we build a precubical set $T(E,I)$ with a homological system of abelian groups $\bar{F}$ and prove that the Leech homology groups $H_n(M(E,I),F)$ are isomorphic to the cubical homology groups $H_n(T(E,I),\bar{F})$, $n\geq 0$. As a consequence we have confirmed a conjecture that if the free partially commutative monoid does not contain $>n$ mutually commuting generators, then its the homological dimension $\leq n$. We have built the co…

Homology groups of cubical sets with connections

Toward defining commutative cubes in all dimensions, Brown and Spencer introduced the notion of "connection" as a new kind of degeneracy. In this paper, for a cubical set with connections, we show that the connections generate an acyclic subcomplex of the chain complex of the cubical set. In particular, our results show that the homology groups of a cubical set with connections are independent of whether we normalize by the connections or we do not, that is, connections do not contribute to any nontrivial cycle in the homology groups of the cubical set.

切触范畴中立方形式的Cartan演算

Cartan calculus of cubical forms in tangent categories

本文在笛卡尔切触范畴中构造微分立方形式的Cartan演算，证明其构成CDGA，并在无2-挠条件下得到介于Kähler与Lie-Rinehart形式之间的演算，恢复de Rham复形。