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Abstract

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.