Un sch\'ema simplicial de Grothendieck-Pridham

TL;DR

This paper constructs an explicit simplicial scheme presentation of an Artin n-stack of Maurer-Cartan elements, verifying Grothendieck-Pridham smoothness conditions to establish its geometricity.

Contribution

It provides an explicit construction of the full simplicial scheme for the Artin n-stack of Maurer-Cartan elements and proves the Grothendieck-Pridham smoothness conditions.

Findings

Explicit construction of the simplicial scheme

Verification of Grothendieck-Pridham smoothness conditions

Establishment of the stack's geometricity

Abstract

Pridham has shown that any Artin $n$-stack $M$ has a presentation as a simplicial scheme $X$ satisfying certain smoothness properties originally introduced by Grothendieck. In the previous paper we introduced an Artin $n$-stack $M$ of Maurer-Cartan elements of a dg-category and constructed a chart, and have already proven the first conditions of smoothness of $X_1\rightarrow X_0$ explicitly. In this paper, we will construct explicitly the full simplicial scheme and show the Grothendieck-Pridham smoothness conditions, implying that $M$ is a geometric $n$-stack.