We undertake a systematic study of fibrations in the setting of abstract simplicial complexes, where the concept of “homotopy” has been replaced by that of “contiguity”. Then, a fibration will be a simplicial map satisfying the “contiguity lifting property”. This definition turns out to be equivalent to that introduced by Minian, established in terms of a cylinder construction \(K \times I_m\). This allows us to prove several properties of simplicial fibrations which are analogous to the classical ones in the topological setting, for instance: all the fibers of a fibration with connected base have the same strong homotopy type and any fibration with a strongly collapsible base is fibrewise trivial. We also introduce the concept of “simplicial finite-fibration”, that is, a simplicial map which has the contiguity lifting property only for finite complexes. Then, we prove that the path fibration \(\mathrm {P}K \rightarrow K\times K\) is a finite-fibration, where \(\mathrm {P}K\) is the simplicial complex of Moore paths introduced by Grandis. This result allows us to prove that any simplicial map factors through a finite-fibration, up to a P-homotopy equivalence. Moreover, we prove a simplicial version of a Varadarajan result for fibrations, relating the LS-category of the total space, the base and the generic fiber. Finally, we introduce a definition of “Švarc genus” of a simplicial map and we are able to compare the Švarc genus of path fibrations with the notions of simplicial LS-category and simplicial topological complexity introduced by the authors in several previous papers.

我们对抽象简单复合体中的纤维化进行了系统的研究，其中“同态”的概念已被“连续性”的概念取代。然后，纤维化将是满足“邻接提升特性”的简单图。事实证明，此定义与Minian引入的定义相同，该定义是根据圆柱结构\（K \ times I_m \）建立的。这使我们能够证明简单纤维的几种属性，它们类似于拓扑环境中的经典纤维，例如：具有连接碱基的纤维的所有纤维都具有相同的强同伦类型，并且具有强可折叠碱基的纤维都是纤维同向的。不重要的。我们还介绍了“单纯有限纤维化”的概念，即仅对有限复合体具有连续性提升性质的单纯映射。然后，我们证明路径振动\（\ mathrm {P} K \ rightarrow K \ times K \）是有限振动，其中\（\ mathrm {P} K \）是Grandis引入的摩尔路径的简单复杂形式。该结果使我们能够证明任何简单映射因数都通过有限纤维化而达到P同伦等效。此外，我们证明了Varadarajan结果的简化形式，涉及总空间，基础和通用纤维的LS分类。最后，我们介绍了简单映射的“Švarc属”的定义，并且能够将路径纤维化的Švarc属与简单LS分类和作者在先前几篇论文中引入的简单拓扑复杂性的概念进行比较。