The root functor
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910957267058688 |
|---|---|
| author | Pratali, Francesca |
| author_facet | Pratali, Francesca |
| contents | In this paper we show that any $\infty$-operad is equivalent to the localization of a discrete $Σ$-free operad, working in the formalism of dendroidal sets. The key point is defining the root functor of a dendroidal set $X$, a functor from the dendroidal nerve of a discrete operad $\mathbfΩ/X$ into $X$, which we show to be an operadic weak equivalence after localizing $\mathbfΩ/X$. This extends an analogous result for $\infty$-categories due to Joyal: when $X$ is a simplicial set, $\mathbfΩ/X$ is its category of elements, and the root functor is the last vertex map. As an application, we deduce that the $\infty$-category of algebras over an $\infty$-operad is equivalent to that of locally constant algebras over its discrete resolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_14288 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The root functor Pratali, Francesca Algebraic Topology Category Theory In this paper we show that any $\infty$-operad is equivalent to the localization of a discrete $Σ$-free operad, working in the formalism of dendroidal sets. The key point is defining the root functor of a dendroidal set $X$, a functor from the dendroidal nerve of a discrete operad $\mathbfΩ/X$ into $X$, which we show to be an operadic weak equivalence after localizing $\mathbfΩ/X$. This extends an analogous result for $\infty$-categories due to Joyal: when $X$ is a simplicial set, $\mathbfΩ/X$ is its category of elements, and the root functor is the last vertex map. As an application, we deduce that the $\infty$-category of algebras over an $\infty$-operad is equivalent to that of locally constant algebras over its discrete resolution. |
| title | The root functor |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2505.14288 |