Cellular approximations for nonsymmetric homotopy operads
Abstract
In this note, for every topological colored nonsymmetric operad with CW spaces in each arity we construct a homotopy isomorphism with a CW homotopy operad (i.e. all higher homotopies are cellular with respect to the product structure and CW structure on cubes) not changing its objects. By using the two-sided bar construction, we rectify this to a homotopy isomorphism with a strict CW operad. This applies more generally to topological algebras over colored operads in sets. In the case of diagrams of finite CW complexes over finite posets, the resulting diagram consists of finite CW spaces enhancing the Quillen adjunction between simplicial sets and topological spaces. As a byproduct, we produce an embedding of topological homotopy algebras over operads into the presheaf category over the branch resolution for the category of P-trees generalizing the known case of topological operads.
Related
BibTeX
@misc{hoefel2026cellular,
title = {{Cellular approximations for nonsymmetric homotopy operads}},
author = {Eduardo Hoefel and Grigory Solomadin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09945},
eprint = {2610.09945},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09945},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cellular approximations for nonsymmetric homotopy operads
AU - Eduardo Hoefel
AU - Grigory Solomadin
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09945
UR - https://arxiv.org/abs/2610.09945
AB - In this note, for every topological colored nonsymmetric operad with CW spaces in each arity we construct a homotopy isomorphism with a CW homotopy operad (i.e. all higher homotopies are cellular with respect to the product structure and CW structure on cubes) not changing its objects. By using the two-sided bar construction, we rectify this to a homotopy isomorphism with a strict CW operad. This applies more generally to topological algebras over colored operads in sets. In the case of diagrams of finite CW complexes over finite posets, the resulting diagram consists of finite CW spaces enhancing the Quillen adjunction between simplicial sets and topological spaces. As a byproduct, we produce an embedding of topological homotopy algebras over operads into the presheaf category over the branch resolution for the category of P-trees generalizing the known case of topological operads.
ER -