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Keywords:
simplicial lists; operads; infinity operads; coherent nerve; homology of simplicial lists
simplicial lists; operads; infinity operads; coherent nerve; homology of simplicial lists
Summary:
We define a category 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 sList whose objects are functors $\Delta^{op} \rightarrow$ List, which we call simplicial lists, and morphisms are natural transformations which have functions as components. We demonstrate that sList supports the combinatorics of (non-symmetric) operads by constructing a fully-faithful nerve functor $N^l : Op \rightarrow$ sList from the category of operads. This leads to a reasonable model for the theory of non-symmetric $\infty$-operads. We also demonstrate that sList has the structure of a presheaf category. In particular, we study a subcategory sList$_{Op}$ 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 leveled 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.
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