Quillen cohomology of enriched operads
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arXiv
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| Format: | Preprint |
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2020
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| author | Truong, Hoang |
| author_facet | Truong, Hoang |
| contents | A modern insight due to Quillen, which is further developed by Lurie, asserts that many cohomology theories of interest are particular cases of a single construction, which allows one to define cohomology groups in an abstract setting using only intrinsic properties of the category (or $\infty$-category) at hand. This universal cohomology theory is known as Quillen cohomology. In any setting, Quillen cohomology of a given object is classified by its cotangent complex. The main purpose of this paper is to study Quillen cohomology of operads enriched over a general base category. Our main result provides an explicit formula for computing Quillen cohomology of enriched operads, based on a procedure of taking certain infinitesimal models of their cotangent complexes. Furthermore, we propose a natural construction of the twisted arrow $\infty$-categories of simplicial operads. We then assert that the cotangent complex of a simplicial operad can be represented as a spectrum valued functor on its twisted arrow $\infty$-category.
When working in stable base categories such as chain complexes and spectra, Francis and Lurie proved the existence of a fiber sequence relating the cotangent complex and Hochschild complex of an $E_n$-algebra, from which a conjecture of Kontsevich is verified. We establish an analogous fiber sequence for the operad $E_n$ itself, in the topological setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2005_01198 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Quillen cohomology of enriched operads Truong, Hoang Algebraic Topology 55P42, 18M60, 18N60, 18M75 A modern insight due to Quillen, which is further developed by Lurie, asserts that many cohomology theories of interest are particular cases of a single construction, which allows one to define cohomology groups in an abstract setting using only intrinsic properties of the category (or $\infty$-category) at hand. This universal cohomology theory is known as Quillen cohomology. In any setting, Quillen cohomology of a given object is classified by its cotangent complex. The main purpose of this paper is to study Quillen cohomology of operads enriched over a general base category. Our main result provides an explicit formula for computing Quillen cohomology of enriched operads, based on a procedure of taking certain infinitesimal models of their cotangent complexes. Furthermore, we propose a natural construction of the twisted arrow $\infty$-categories of simplicial operads. We then assert that the cotangent complex of a simplicial operad can be represented as a spectrum valued functor on its twisted arrow $\infty$-category. When working in stable base categories such as chain complexes and spectra, Francis and Lurie proved the existence of a fiber sequence relating the cotangent complex and Hochschild complex of an $E_n$-algebra, from which a conjecture of Kontsevich is verified. We establish an analogous fiber sequence for the operad $E_n$ itself, in the topological setting. |
| title | Quillen cohomology of enriched operads |
| topic | Algebraic Topology 55P42, 18M60, 18N60, 18M75 |
| url | https://arxiv.org/abs/2005.01198 |