The equifibered approach to $\infty$-properads
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910066721947648 |
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| author | Barkan, Shaul Steinebrunner, Jan |
| author_facet | Barkan, Shaul Steinebrunner, Jan |
| contents | We define a notion of $\infty$-properads that generalises $\infty$-operads by allowing operations with multiple outputs. Specializing to the case where each operation has a single output provides a simple new perspective on $\infty$-operads, but at the same time the extra generality allows for examples such as bordism categories. We also give an interpretation of our $\infty$-properads as Segal presheaves on a category of graphs by comparing them to the Segal $\infty$-properads of Hackney-Robertson-Yau. Combining these two approaches yields a flexible tool for doing higher algebra with operations that have multiple inputs and outputs. Crucially, this allows for a definition of algebras over an $\infty$-properad such that, for example, topological field theories are algebras over the bordism $\infty$-properad. The key ingredient to this paper is the notion of an equifibered map between $E_\infty$-monoids, which is a well-behaved generalisation of free maps. We also use this to prove facts about free $E_\infty$-monoids, for example that free $E_\infty$-monoids are closed under pullbacks along arbitrary maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_02576 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The equifibered approach to $\infty$-properads Barkan, Shaul Steinebrunner, Jan Algebraic Topology Category Theory 18M85, 55P99, 18N70 We define a notion of $\infty$-properads that generalises $\infty$-operads by allowing operations with multiple outputs. Specializing to the case where each operation has a single output provides a simple new perspective on $\infty$-operads, but at the same time the extra generality allows for examples such as bordism categories. We also give an interpretation of our $\infty$-properads as Segal presheaves on a category of graphs by comparing them to the Segal $\infty$-properads of Hackney-Robertson-Yau. Combining these two approaches yields a flexible tool for doing higher algebra with operations that have multiple inputs and outputs. Crucially, this allows for a definition of algebras over an $\infty$-properad such that, for example, topological field theories are algebras over the bordism $\infty$-properad. The key ingredient to this paper is the notion of an equifibered map between $E_\infty$-monoids, which is a well-behaved generalisation of free maps. We also use this to prove facts about free $E_\infty$-monoids, for example that free $E_\infty$-monoids are closed under pullbacks along arbitrary maps. |
| title | The equifibered approach to $\infty$-properads |
| topic | Algebraic Topology Category Theory 18M85, 55P99, 18N70 |
| url | https://arxiv.org/abs/2211.02576 |