The fibration sequences of the lattice path operad
Florian De Leger
TL;DR
The paper develops operads of complexity $m$ by embedding them as algebras for a suboperad of the $m$-th filtration of the lattice path operad and constructs a coherent theory of bimodules over these operads. It introduces a join-based composition framework (via $\hat{\mathcal{L}}_m$) and a corresponding notion of operads of complexity $m$, along with pointed bimodules and the Boardman–Vogt tensor product to control bimodule structure. A central technical achievement is a cofinality result for maps between polynomial monads, enabling the establishment of fibration sequences that extend known results (Turchin, Dwyer– Hess) to $m\le 3$, with the $m=3$ case presented as new. The work deepens understanding of Deligne-type phenomena in the lattice path filtration and provides tools for relating operads, bimodules, and mapping spaces within this higher-complexity setting, with potential implications for operadic homotopy theory and related deformation theories.
Abstract
We introduce a notion of an operad of complexity $m$, for $m \geq 1$. Operads of complexity $1$ are monoids in the category of $\mathbb{N}$-indexed collections, with monoidal product given by the Day convolution, and operads of complexity $2$ are non-symmetric operads. In general, we prove that the operad for operads of complexity $m$ is a suboperad of the $m$-th stage filtration of the lattice path operad introduced by Batanin and Berger. Finally, we exhibit fibration sequences involving this new notion, extending the results of Turchin and Dwyer-Hess.