A minimal model for prestacks and morphisms of operadic algebras via Koszul duality for box operads · arXivDesk
2312.04185Dec 7, 202355 pages, lots of pictures; Added second application to morphisms of operadic algebras (10 pages). Corrected typos and technical adjustments of section on Koszul dual box (co)operads. All comments are welcome!
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 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
Nearby in the stack
-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)^
and
Lax^
to obtain our two main applications. As our second main result we establish a minimal model
Morph(P)∞
for the box operad
Morph(P)
as the cobar construction on a box subcooperad
Morph(P)^_p≤ 1
of
Morph(P)^
, hereby answering an open question by Markl in arXiv:math/0103052 (Problem 9). As expected, it encodes
∞
-morphisms of
P∞
-algebras. As our third main result we establish a minimal model
Lax∞
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.