论文标题
小类的同型不变性
Homotopy invariants in small categories
论文作者
论文摘要
田中提出了一个小型类别$ \ mathcal {c} $的$ \ mathrm {ccat} \，\ mathcal {c} $的Lusternik Schnirelmann类别的概念。在其他属性中，他证明了Varadarajan定理的纤维化类似物，与总空间，底座和纤维的LS类别相关。 在本文中，我们回想起同型距离的概念$ \ mathrm {d}（f，g）$之间的两个函数$ f，g \ colon \ colon \ mathcal {c} \ to \ nathcal {d} $ \ Mathcal {C} = \ Mathrm {d}（\ Mathrm {id} _ {\ Mathcal {C}}，\ bulter）$作为特定情况。我们考虑另一种特殊情况，即两个投影之间的距离$ \ mathrm {d}（p_1，p_2）$此外，我们定义了小型类别的较高分类复杂性，我们表明它可以被描述为更高距离。 我们证明了这些不变的主要属性。作为最终结果，我们证明了Varadarajan的定理，用于在小型类别之间进行Grothendieck Bi纤维的同位距离。
Tanaka introduced a notion of Lusternik Schnirelmann category, denoted $\mathrm{ccat}\, \mathcal{C}$, of a small category $\mathcal{C}$. Among other properties, he proved an analog of Varadarajan's theorem for fibrations, relating the LS-categories of the total space, the base and the fiber. In this paper we recall the notion of homotopic distance $\mathrm{D}(F,G)$ between two functors $F,G\colon \mathcal{C} \to \mathcal{D}$, later introduced by us, which has $\mathrm{ccat} \mathcal{C}=\mathrm{D}(\mathrm{id}_{\mathcal{C}},\bullet)$ as a particular case. We consider another particular case, the distance $\mathrm{D}(p_1,p_2)$ between the two projections $p_1,p_2\colon \mathcal{C}\times \mathcal{C} \to \mathcal{C}$, which we call the categorical complexity of the small category $\mathcal{C}$. Moreover, we define the higher categorical complexity of a small category and we show that it can be characterized as a higher distance. We prove the main properties of those invariants. As a final result we prove a Varadarajan's theorem for the homotopic distance for Grothendieck bi-fibrations between small categories.
