Simplicial lists in operad theory I
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911244646088704 |
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| author | Haderi, Redi Ünlü, Özgün |
| author_facet | Haderi, Redi Ünlü, Özgün |
| contents | 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 $Δ^{op} \to \mathsf{List}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_10072 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Simplicial lists in operad theory I Haderi, Redi Ünlü, Özgün Algebraic Topology Category Theory 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 $Δ^{op} \to \mathsf{List}$, 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. |
| title | Simplicial lists in operad theory I |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2405.10072 |