Fernández Ternero, DesamparadosGarcía Calcines, José ManuelMacías Virgós, EnriqueVilches Alarcón, José Antonio2025-01-282025-01-282021-01-12Fernández Ternero, D., García Calcines, J.M., Macías Virgós, E. y Vilches Alarcón, J.A. (2021). Simplicial fibrations. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 115 (2), 54. https://doi.org/10.1007/s13398-020-00966-5.1578-73031579-1505https://hdl.handle.net/11441/167711We 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a systematic study of fibrations in the setting of abstract simplicial complexes, where the concept of “homotopy” has been replaced by that of “contiguity”.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 . 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 is a finite-fibration, where 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.application/pdf26 p.engAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Simplicial complexesContiguous simplicial mapsFibrationsLS-categoryTopological complexityinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccesshttps://doi.org/10.1007/s13398-020-00966-5