Limits of $(\infty, 1)$-categories with structure and their lax morphisms
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909618782863360 |
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| author | Ko, Joanna |
| author_facet | Ko, Joanna |
| contents | Riehl and Verity have established that for a quasi-category $A$ that admits limits, and a homotopy coherent monad on $A$ which does not preserve limits, the Eilenberg-Moore object still admits limits; this can be interpreted as a completeness result involving lax morphisms. We generalise their result to different models for $(\infty, 1)$-categories, with an abundant variety of structures. For instance, $(\infty, 1)$-categories with limits, Cartesian fibrations between $(\infty, 1)$-categories, and adjunctions between $(\infty, 1)$-categories. In addition, we show that these $(\infty, 1)$-categories with structure in fact possess an important class of limits of lax morphisms, including $\infty$-categorical versions of inserters and equifiers, when only one morphism in the diagram is required to be structure-preserving. Our approach provides a minimal requirement and a transparent explanation for several kinds of limits of $(\infty, 1)$-categories and their lax morphisms to exist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15598 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Limits of $(\infty, 1)$-categories with structure and their lax morphisms Ko, Joanna Category Theory Algebraic Topology 18N60 Riehl and Verity have established that for a quasi-category $A$ that admits limits, and a homotopy coherent monad on $A$ which does not preserve limits, the Eilenberg-Moore object still admits limits; this can be interpreted as a completeness result involving lax morphisms. We generalise their result to different models for $(\infty, 1)$-categories, with an abundant variety of structures. For instance, $(\infty, 1)$-categories with limits, Cartesian fibrations between $(\infty, 1)$-categories, and adjunctions between $(\infty, 1)$-categories. In addition, we show that these $(\infty, 1)$-categories with structure in fact possess an important class of limits of lax morphisms, including $\infty$-categorical versions of inserters and equifiers, when only one morphism in the diagram is required to be structure-preserving. Our approach provides a minimal requirement and a transparent explanation for several kinds of limits of $(\infty, 1)$-categories and their lax morphisms to exist. |
| title | Limits of $(\infty, 1)$-categories with structure and their lax morphisms |
| topic | Category Theory Algebraic Topology 18N60 |
| url | https://arxiv.org/abs/2505.15598 |