On the Kelly monoidal structure of $Λ$-sequences and unital operads
Aowen Fan, Foling Zou
TL;DR
The paper develops a comprehensive framework for understanding operads via monoids in $\Lambda$-sequences under the Kelly product, including unital and based variants. It proves the existence of a universal normal oplax monoidal structure on $\Lambda$-sequences (extending the Kelly product) and shows the resulting equivalence between categories of operads and monoids in $\Lambda$-sequences, generalizing prior work to arbitrary complete and cocomplete symmetric monoidal categories $\mathscr{V}$. A key technical advance is the monoidal localization theorem, which ensures that tensoring commutes with (κ-small) colimits after localization, enabling a robust transfer of monoidal structure through $L: \mathscr{V} \to L\mathscr{V}$. Consequently, unital (resp. based) operads in $\mathscr{V}$ are identified with monoids in appropriate $\Lambda$-sequence categories, providing a unified categorical machinery for operad theory that remains valid in non-closed or non-Cartesian settings. The results connect operadic algebra with operator categories, monads, and Day convolution, offering a versatile toolkit for studying algebras over operads and their modules in broad categorical contexts.
Abstract
Let $Λ$ be the category of based finite sets $\mathbf{n}$ and based injections. We study properties of monoids and modules in $Λ$-sequences under the Kelly monoidal structure. In particular, we show that the forgetful functor from right modules in $Λ$-sequences to right modules in symmetric sequences is an isomorphism. We show that any compatible lower data extends to a normal oplax monoidal structure and use this to establish a universal normal oplax monoidal structure on $Λ$-sequences extending the Kelly product, identifying unital operads to monoids in unital $Λ$-sequences for a general symmetric monoidal category $\mathscr{V}$. We also establish a closed monoidal localization theorem.