Pushouts of Dwyer maps are $(\infty,1)$-categorical
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916331999199232 |
|---|---|
| author | Hackney, Philip Ozornova, Viktoriya Riehl, Emily Rovelli, Martina |
| author_facet | Hackney, Philip Ozornova, Viktoriya Riehl, Emily Rovelli, Martina |
| contents | The inclusion of 1-categories into $(\infty,1)$-categories fails to preserve colimits in general, and pushouts in particular. In this note, we observe that if one functor in a span of categories belongs to a certain previously-identified class of functors, then the 1-categorical pushout is preserved under this inclusion. Dwyer maps, a kind of neighborhood deformation retract of categories, were used by Thomason in the construction of his model structure on 1-categories. Thomason previously observed that the nerves of such pushouts have the correct weak homotopy type. We refine this result and show that the weak homotopical equivalence is a weak categorical equivalence. We also identify a more general class of functors along which 1-categorical pushouts are $(\infty,1)$-categorical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_02353 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Pushouts of Dwyer maps are $(\infty,1)$-categorical Hackney, Philip Ozornova, Viktoriya Riehl, Emily Rovelli, Martina Algebraic Topology Category Theory 18N60, 55U35 The inclusion of 1-categories into $(\infty,1)$-categories fails to preserve colimits in general, and pushouts in particular. In this note, we observe that if one functor in a span of categories belongs to a certain previously-identified class of functors, then the 1-categorical pushout is preserved under this inclusion. Dwyer maps, a kind of neighborhood deformation retract of categories, were used by Thomason in the construction of his model structure on 1-categories. Thomason previously observed that the nerves of such pushouts have the correct weak homotopy type. We refine this result and show that the weak homotopical equivalence is a weak categorical equivalence. We also identify a more general class of functors along which 1-categorical pushouts are $(\infty,1)$-categorical. |
| title | Pushouts of Dwyer maps are $(\infty,1)$-categorical |
| topic | Algebraic Topology Category Theory 18N60, 55U35 |
| url | https://arxiv.org/abs/2205.02353 |