Deforming prestacks : a box operadic approach

This thesis is devoted to the study of prestacks and their deformation theory through box operads. Prestacks are algebro-geometric objects which generalize presheaves of algebras and appear in Non-Commutative Algebraic Geometry as non-commutative deformations of schemes. The innovation we present in this thesis is that prestacks and their deformation theory are governed by an abstract calculus of stacking 2-dimensional rectangular boxes. The new algebraic gadgets governing this combinatorial theory we call box operads. We start out the thesis by developing in detail three deformation perspectives (elementary, moduli and Lie) for associative algebras and show that the three associated deformation functors are isomorphic. Further, we move from deforming associative algebras to deforming prestacks, passing by commutative algebras, schemes and presheaves along the way. In the second part of the thesis, we provide a combinatorial description of the symmetric coloured operad “box-op” which encodes box operads, as stackings of rectangular boxes. In our first main result, we endow a suitable (graded, zero differential) totalisation of box-op with a morphism from the L-infinity operad. For a quiver V on a small category, we define an endomorphism box operad End(V) containing the Gerstenhaber-Schack object C_GS(V) as a L-infinity subalgebra of its totalisation Tot(End(V)). Further, we show that an element (m,f,c) of C_GS(V) satisfies the Maurer-Cartan equation precisely when A=(V,m,f,c) is a lax prestack. In the third part, we provide a box operad Lax encoding lax prestacks whose relations are far from quadratic. Indeed, their relations are cubic and quartic, and moreover inhomogeneous. We show that box operads constitute the correct framework to encode lax prestacks and resolve their relations up to homotopy. Our second 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. We show that Lax is not Koszul by explicitly computing its Koszul complex. We then go on to remedy the situation by suitably restricting the Koszul dual box cooperad to obtain our third main result: we establish a minimal (in particular cofibrant) model Lax-infinity for the box operad Lax.