On the Kelly monoidal structure of $\Lambda$-sequences and unital operads

Abstract

Let $\Lambda$ be the category of based finite sets $\mathbf{n}$ and based injections. We study properties of monoids and modules in $\Lambda$-sequences under the Kelly monoidal structure. In particular, we show that the forgetful functor from right modules in $\Lambda$-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 $\Lambda$-sequences extending the Kelly product, identifying unital operads to monoids in unital $\Lambda$-sequences for a general symmetric monoidal category $\mathcal{V}$. We also establish a closed monoidal localization theorem.