Descent of Deligne groupoids

To any non-negatively graded dg Lie algebra $g$ over a field $k$ of characteristic zero we assign a functor $\Sigma_g: art/k \to Kan$ from the category of commutative local artinian $k$-algebras with the residue field $k$ to the category of Kan simplicial sets. There is a natural homotopy equivalence between $\Sigma_g$ and the Deligne groupoid corresponding to $g$. The main result of the paper claims that the functor $\Sigma$ commutes up to homotopy with the "total space" functors which assign a dg Lie algebra to a cosimplicial dg Lie algebra and a simplicial set to a cosimplicial simplicial set. This proves a conjecture of Schechtman which implies that if a deformation problem is described ``locally'' by a sheaf of dg Lie algebras $g$ on a topological space $X$ then the global deformation problem is described by the homotopy Lie algebra $R\Gamma(X,g)$.Comment: Minor corrections made AMSLaTeX v 1.2 (Compatibility mode

Deformations of sheaves of algebras

A construction of the tangent dg Lie algebra of a sheaf of operad algebras on a site is presented. The requirements on the site are very mild; the requirements on the algebra are more substantial. A few applications including the description of deformatins of a scheme and equivariant deformations are considered. The construction is based upon a model structure on the category of presheaves which should be of an independent interest.Comment: 57 pages, references adde

Deligne categories and the limit of categories Rep(GL(m∣n))Rep(GL(m|n))

For each integer $t$ a tensor category $V_t$ is constructed, such that exact tensor functors $V_t \longrightarrow C$ classify dualizable $t$-dimensional objects in $C$ not annihilated by any Schur functor. This means that $V_t$ is the "abelian envelope" of the Deligne category $Rep(GL_t)$. Any tensor functor $Rep(GL_t)\longrightarrow C$ is proved to factor either through $V_t$ or through one of the classical categories $Rep(GL(m|n))$ with $m-n=t$. The universal property of $V_t$ implies that it is equivalent to the categories $Rep_{Rep(GL_{t_1})\otimes Rep(GL_{t_2})}(GL(X),\epsilon)$, ($t=t_1+t_2$, $t_1$ not integer) suggested by Deligne as candidates for the role of abelian envelope.Comment: v3: lemma added to section 9, v4: some typos fixed, v5: fixed support acknowledgement, v:6 minor fix in definition of abelian envelop

On the equivalence of the Lurie's ∞\infty-operads and dendroidal ∞\infty-operads

In this paper we prove the equivalence of two symmetric monoidal $\infty$-categories of $\infty$-operads, the one defined in Lurie's book on Higher Algebra and the one based on dendroidal spaces. V.2 Some corrections made and exposition slightly altered.Comment: 30 page