arXiv subjects

Lander Hermans

Publications and source records attributed to Lander Hermans.

4 recordsLinked to original sources

arXiv2024

A Deligne conjecture for prestacks

We prove an analog of the Deligne conjecture for prestacks. We show that given a prestack $\mathbb A$, its Gerstenhaber--Schack complex $\mathbf{C}_{\mathsf{GS}}(\mathbb A)$ is naturally an $E_2$-algebra. This structure generalises both the known $\mathsf{L}_\infty$-algebra structure on $\mathbf{C}_{\mathsf{GS}}(\mathbb A)$, as well as the Gerstenhaber algebra structure on its cohomology $\mathbf{H}_{\mathsf{GS}}(\mathbb A)$. The main ingredient is the proof of a conjecture of Hawkins \cite{hawkins}, stating that the dg operad $\mathsf{Quilt}$ has vanishing homology in positive degrees. As a corollary, $\mathsf{Quilt}$ is quasi-isomorphic to the operad $\mathsf{Brace}$ encoding brace algebras. In addition, we improve the $L_\infty$-structure on $\mathsf{Quilt}$ by showing that it originates from a $\mathsf{PreLie}_\infty$-structure lifting the $\mathsf{PreLie}$-structure on $\mathsf{Brace}$ in homology.

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arXiv2023

A minimal model for prestacks and morphisms of operadic algebras via Koszul duality for box operads

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.

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arXiv2023

Box operads and higher Gerstenhaber brackets

We introduce a symmetric operad $\square p$ ("box-op") which describes a certain calculus of rectangular labeled ``boxes''. Algebras over $\square p$, which we call box operads, have appeared under the name of fc multicategories in work by Leinster \cite{LeinsterFcmulticategories1999}. In our main result, we endow a suitable (graded, zero differential) totalisation $\square p_{\mathrm{td}}$ with a morphism $L_{\infty} \rightarrow \square p_{\mathrm{td}}$. We show that $\square p$ acts on an $\mathbb{N}^3$-graded enlargement of the $\mathbb{N}^2$-graded Gerstenhaber-Schack object $\mathbf{C}_{GS}(\mathbb{A})$ of a quiver $\mathbb{A}$ on a small category from \cite{DinhVanLowen2018}. This action restricts to an $L_{\infty}$-structure on $\mathbf{C}_{GS}(\mathbb{A})$ (with zero differential). For an element $\alpha = (m,f,c) \in \mathbf{C}_{GS}^2(\mathbb{A})$, the Maurer-Cartan equation holds precisely when $(\mathbb{A}, m, f, c)$ is a lax prestack with multiplications $m$, restrictions $f$, and twists $c$. As a consequence, the $\alpha$-twisted $L_{\infty}$-structure on $\mathbf{C}_{GS}(\mathbb{A})$ controls the deformation theory of $(\mathbb{A}, \alpha)$ as a lax prestack.

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arXiv2020

Operadic structure on the Gerstenhaber-Schack complex for prestacks

We introduce an operad which acts on the Gerstenhaber-Schack complex of a prestack as defined by Dinh Van and Lowen, and which in particular allows us to endow this complex with an underlying $L_{\infty}$-structure. We make use of the operad $\operatorname{Quilt}$ which was used by Hawkins in order to solve the presheaf case. Due to the additional difficulty posed by the presence of twists, we have to use $\operatorname{Quilt}$ in a fundamentally different way (even for presheaves) in order to allow for an extension to prestacks. The resulting $L_{\infty}$-algebra governs the deformation theory of the prestack.

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