A combinatorial model for the path fibration

Manuel Rivera; Samson Saneblidze

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arXiv · 1706.00983

A combinatorial model for the path fibration

Manuel Rivera·Samson Saneblidze

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Abstract

We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set $X$ we associate a necklical set $\widehat{\mathbf{\Omega}}X$ such that its geometric realization $|\widehat{\mathbf{\Omega}}X|$, a space built out of gluing cubical cells, is homotopy equivalent to the based loop space on $|X|$ and the differential graded module of chains $C_*(\widehat{\mathbf{\Omega}}X)$ is a differential graded associative algebra generalizing Adams' cobar construction.

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Manuel Rivera, Samson Saneblidze. 2017-06-03. A combinatorial model for the path fibration. https://arxiv.org/abs/1706.00983

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