A minimal model for prestacks and morphisms of operadic algebras via Koszul duality for box operads
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| Formato: | Preprint |
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2023
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| _version_ | 1866918147352690688 |
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| author | Hermans, Lander |
| author_facet | Hermans, Lander |
| contents | Prestacks are algebro-geometric objects whose defining relations are far from quadratic. Indeed, they are cubic and quartic, and moreover inhomogeneous. Similarly, a morphism of $P$-algebras for a (nonsymmetric) Koszul operad $P$ has inhomogeneous relations possibly of any arity. We show that box operads, a rectangular type of operads introduced in arXiv:2305.20036, constitute the correct framework to encode them and resolve their relations up to homotopy. Our first main result is a Koszul duality theory for box operads, extending the duality for (nonsymmetric) operads. In this new theory, the classical restriction of being quadratic is replaced by the notion of being \emph{thin-quadratic}, a condition referring to a particular class of ``thin'' operations. Our main cases of interest are the box operad $Morph(P)$ encoding morphisms of $P$-algebras and the box operad $Lax$ encoding lax prestacks. We show that both $Morph(P)$ and $Lax$ are not Koszul. We then go on to remedy the situation by suitably restricting the respective Koszul dual box cooperads $Morph(P)^{\antishriek}$ and $Lax^{\antishriek}$ to obtain our two main applications. As our second main result we establish a minimal model $Morph(P)_\infty$ for the box operad $Morph(P)$ as the cobar construction on a box subcooperad $Morph(P)^{\antishriek}_{p\leq 1}$ of $Morph(P)^{\antishriek}$, hereby answering an open question by Markl in arXiv:math/0103052 (Problem 9). As expected, it encodes $\infty$-morphisms of $P_\infty$-algebras. As our third main result we establish a minimal model $Lax_{\infty}$ for the box operad $Lax$ encoding lax prestacks. This sheds new light on Markl's question on the existence of an explicit cofibrant model for the operad encoding presheaves of algebras from arXiv:math/0103052 (Conjecture 31). Indeed, we answer the parallel question with presheaves viewed as prestacks in the positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_04185 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A minimal model for prestacks and morphisms of operadic algebras via Koszul duality for box operads Hermans, Lander Algebraic Topology Algebraic Geometry Category Theory 18M70 (Primary) 18M65, 14A20 (Secondary) Prestacks are algebro-geometric objects whose defining relations are far from quadratic. Indeed, they are cubic and quartic, and moreover inhomogeneous. Similarly, a morphism of $P$-algebras for a (nonsymmetric) Koszul operad $P$ has inhomogeneous relations possibly of any arity. We show that box operads, a rectangular type of operads introduced in arXiv:2305.20036, constitute the correct framework to encode them and resolve their relations up to homotopy. Our first main result is a Koszul duality theory for box operads, extending the duality for (nonsymmetric) operads. In this new theory, the classical restriction of being quadratic is replaced by the notion of being \emph{thin-quadratic}, a condition referring to a particular class of ``thin'' operations. Our main cases of interest are the box operad $Morph(P)$ encoding morphisms of $P$-algebras and the box operad $Lax$ encoding lax prestacks. We show that both $Morph(P)$ and $Lax$ are not Koszul. We then go on to remedy the situation by suitably restricting the respective Koszul dual box cooperads $Morph(P)^{\antishriek}$ and $Lax^{\antishriek}$ to obtain our two main applications. As our second main result we establish a minimal model $Morph(P)_\infty$ for the box operad $Morph(P)$ as the cobar construction on a box subcooperad $Morph(P)^{\antishriek}_{p\leq 1}$ of $Morph(P)^{\antishriek}$, hereby answering an open question by Markl in arXiv:math/0103052 (Problem 9). As expected, it encodes $\infty$-morphisms of $P_\infty$-algebras. As our third main result we establish a minimal model $Lax_{\infty}$ for the box operad $Lax$ encoding lax prestacks. This sheds new light on Markl's question on the existence of an explicit cofibrant model for the operad encoding presheaves of algebras from arXiv:math/0103052 (Conjecture 31). Indeed, we answer the parallel question with presheaves viewed as prestacks in the positive. |
| title | A minimal model for prestacks and morphisms of operadic algebras via Koszul duality for box operads |
| topic | Algebraic Topology Algebraic Geometry Category Theory 18M70 (Primary) 18M65, 14A20 (Secondary) |
| url | https://arxiv.org/abs/2312.04185 |