  We define a category $\mathsf{List}$ whose objects are sets and morphisms are
mappings which assign to an element in the domain an ordered sequence (list) of
elements in the codomain. We introduce and study a category of simplicial
objects $\mathsf{slist}$ whose objects are functors $\Delta^{op} \to
\mathsf{List}$, which we call simplicial lists, and morphisms are natural
transformations which have functions as components. We demonstrate that
$\mathsf{sList}$ supports the combinatorics of (non-symmetric) operads by
constructing a fully-faithful nerve functor $N^l : \mathsf{Operad} \to
\mathsf{sList}$ from the category of operads. This leads to a reasonable model
for the theory of non-symmetric $\infty$-operads.
  We also demonstrate that $\mathsf{sList}$ has the structure of a presheaf
category. In particular, we study a subcategory $\mathsf{sList}_{\text{op}}$ of
operadic simplicial lists, in which the nerve functor takes values. The latter
category is also a presheaf category over a base whose objects may be
interpreted as levelled trees. We construct a coherent nerve functor which
outputs an $\infty$-operad for each operad enriched in Kan complexes. We also
define homology groups of simplicial lists and study first properties.
