Simplicial properadic homotopy
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916763994685440 |
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| author | Hoffbeck, Eric Leray, Johan Vallette, Bruno |
| author_facet | Hoffbeck, Eric Leray, Johan Vallette, Bruno |
| contents | In this paper, we settle the homotopy properties of the infinity-morphisms of homotopy (bial)-gebras over properads, i.e. algebraic structures made up of operations with several inputs and outputs. We start by providing the literature with characterizations for the various types of infinity-morphisms, the most seminal one being the equivalence between infinity-quasi-isomorphisms and zig-zags of quasi-isomorphisms which plays a key role in the study the formality property. We establish a simplicial enrichment for the categories of gebras over some cofibrant properads together with their infinity-morphisms, whose homotopy category provides us with the localisation with respect to infinity-quasi-isomorphisms. These results extend to the properadic level known properties for operads, but the lack of the rectification procedure in this setting forces us to use different methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_22004 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Simplicial properadic homotopy Hoffbeck, Eric Leray, Johan Vallette, Bruno Algebraic Topology Category Theory Quantum Algebra 18M85, 14D15, 16T10, 17B55, 18M70, 18N40, 18N50 In this paper, we settle the homotopy properties of the infinity-morphisms of homotopy (bial)-gebras over properads, i.e. algebraic structures made up of operations with several inputs and outputs. We start by providing the literature with characterizations for the various types of infinity-morphisms, the most seminal one being the equivalence between infinity-quasi-isomorphisms and zig-zags of quasi-isomorphisms which plays a key role in the study the formality property. We establish a simplicial enrichment for the categories of gebras over some cofibrant properads together with their infinity-morphisms, whose homotopy category provides us with the localisation with respect to infinity-quasi-isomorphisms. These results extend to the properadic level known properties for operads, but the lack of the rectification procedure in this setting forces us to use different methods. |
| title | Simplicial properadic homotopy |
| topic | Algebraic Topology Category Theory Quantum Algebra 18M85, 14D15, 16T10, 17B55, 18M70, 18N40, 18N50 |
| url | https://arxiv.org/abs/2505.22004 |