This is a comment on the Kuranishi method of constructing analytic deformation spaces. It is based on a simple observation that the Kuranishi map can always be inverted in the category of $L_{\infty}$-algebras. The $L_{\infty}$-structure obtained by this inversion is used to define an ''unobstructed'' deformation functor which is always representable by a smooth pointed moduli space.
The singular nature of the original Kuranishi deformation space emerges in this setting merely as a result of the truncation of this ``naive'' $L_{\infty}$-algebra controlling the deformations to a usual differential Lie algebra.
A dg-natural transformation between dg-functors is called an objectwise homotopy equivalence if its induced morphism on each object admits a homotopy inverse. In general an objectwise homotopy equivalence does not have a dg-inverse but has an $A_{\infty}$ quasi-inverse. In this note we give a recurrent formula of the $A_{\infty}$ quasi-inverse. This result is useful in studying the compositions of dg-lifts of derived functors of schemes.
We study linear Batalin-Vilkovisky (BV) quantization, which is a derived and shifted version of the Weyl quantization of symplectic vector spaces. Using a variety of homotopical machinery, we implement this construction as a symmetric monoidal functor of $\infty$-categories. We also show that this construction has a number of pleasant properties: It has a natural extension to derived algebraic geometry, it can be fed into the higher Morita category of $E_n$-algebras to produce a "higher BV quantization" functor, and when restricted to formal moduli problems, it behaves like a determinant. Alon…
We discuss the following problem: how can an arbitrary Fourier-Mukai transform $ϕ: \mathrm{D}^{\mathrm{b}}( \mathbb{P}^a ) \rightarrow \mathrm{D}^{\mathrm{b}}( \mathbb{P}^b )$ between the bounded derived categories of two projective spaces of dimensions $a$ and $b$ be expressed in explicit terms as an exact functor between the homotopy categories $\mathrm{K}^{\mathrm{b}}( \mathbb{B}^a ) \rightarrow \mathrm{K}^{\mathrm{b}}( \mathbb{B}^b )$ generated by the full strong exceptional sequences of the line bundles $\mathbb{B}^a = \{\mathcal{O}(-a), \dots, \mathcal{O}\}$ and $\mathbb{B}^b = \{\mathca…
Let $G$ be a compact connected Lie group. We show that the category $\mathbf{Loc}_{\infty}(BG)$ of $\infty$-local systems on the classifying space of $G$, can be described infinitesimally as the category $\mathbf{InfLoc}_{\infty}(\mathfrak{g})$ of basic $\mathfrak{g}$-$L_\infty$ spaces. Moreover, we show that, given a principal bundle $π\colon P \rightarrow X$ with structure group $G$ and any connection $θ$ on $P$, there is a DG functor $$\mathcal{CW}_θ \colon \mathbf{InfLoc}_{\infty}(\mathfrak{g}) \longrightarrow \mathbf{Loc}_{\infty}(X), $$ which corresponds to the pullback functor by the cl…
Bivariant theory is a unified framework for cohomology and Borel-Moore homology theories. In this paper, we extract an $\infty$-enhanced bivariant homology theory from Gaitsgory-Rozenblyum's six functor formalism.
In \cite{Goto}, Ryushi Goto has constructed the deformation space for a manifold equipped with a collection of closed differential forms and showed that in some important cases (Calabi-Yau, $G_2$- and $Spin(7)$-structures) this deformation space is smooth. This result unifies the classical Bogomolov-Tian-Todorov and Joyce theorems about unobstructedness of deformations. Using the work of Fiorenza and Manetti, we show that this deformation space could be obtained as the deformation space associated to a certain $L_{\infty}$-algebra. We also show that for Calabi-Yau, $G_2$- and $Spin(7)$-structu…
In this paper, we relate Lie algebroids to Costello's version of derived geometry. For instance, we show that each Lie algebroid $L$-and the natural generalization to dg Lie algebroids-provides an (essentially unique) $L_\infty$ space. More precisely, we construct a faithful functor from the category of Lie algebroids to the category of $L_\infty$ spaces. Then we show that for each Lie algebroid $L$, there is a fully faithful functor from the category of representations up to homotopy of $L$ to the category of vector bundles over the associated $L_\infty$ space. Indeed, this functor sends the …
Let $A$ be a finite-dimensional smooth unital cyclic $A_\infty$-algebra. Assume furthermore that $A$ satisfies the Hodge-to-de-Rham degeneration property. In this short note, we prove the non-commutative analogue of the Bogomolov-Tian-Todorov theorem: the deformation functor associated with the differential graded Lie algebra of Hochschild cochains of $A$ is smooth. Furthermore, the deformation functor associated with the DGLA of cyclic Hochschild cochains of $A$ is also smooth.
Microlocal sheaf theory of \cite{KS90} makes an essential use of an extension lemma for sheaves due to Kashiwara, and this lemma is based on a criterion of the same author giving conditions in order that a functor defined in $\mathbb{R}$ with values in the category $Sets$ of sets be constant. In a first part of this paper, using classical tools, we show how to generalize the extension lemma to the case of the unbounded derived category. In a second part, we extend Kashiwara's result on constant functors by replacing the category $Sets$ with the $\infty$-category of spaces and apply it to gener…
We introduce the basic elements of the theory of parametrized $\infty$-categories and functors between them. These notions are defined as suitable fibrations of $\infty$-categories and functors between them. We give as many examples as we are able at this stage. Simple operations, such as the formation of opposites and the formation of functor $\infty$-categories, become slightly more involved in the parametrized setting, but we explain precisely how to perform these constructions. All of these constructions can be performed explicitly, without resorting to such acts of desperation as straight…
In this paper, we construct a refined, relative version of the étale realization functor of motivic spaces, first studied by Isaksen and Schmidt. Their functor goes from the $\infty$-category of motivic spaces over a base scheme $S$ to the $\infty$-category of $p$-profinite spaces, where $p$ is a prime which is invertible in all residue fields of $S$. In the first part of this paper, we refine the target of this functor to an $\infty$-category where $p$-profinite spaces is a further completion. Roughly speaking, this $\infty$-category is generated under cofiltered limits by those spaces whose …
Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($\infty$-)categories of spans (or correspondences). In this paper we study the more complicated setup where we have two pushforwards (an "additive" and a "multiplicative" one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(\infty,2)$-categories of bispans, characterized by a universal property: they corepresent functors out of $\infty$-cat…
We study constructive $A_\infty$-models of the DG-category of matrix factorisations of a potential over a commutative $\mathbb{Q}$-algebra $k$, consisting of a Hom-finite $A_\infty$-category equipped with an $A_\infty$-idempotent functor.
We extend Ayoub's formalism of motivic nearby cycle functor to the $\infty$-categorical level, and prove some desired cohomological properties by relating the motivic nearby cycle functor to the notion of local acyclicity in motivic homotopy.
We develop the theory of semi-orthogonal decompositions and spherical functors in the framework of stable $\infty$-categories. Building on this, we study the relative Waldhausen S-construction $S_\bullet(F)$ of a spherical functor $F$ and equip it with a natural paracyclic structure (``rotational symmetry''). This fulfills a part of the general program to provide a rigorous account of perverse schobers which are (thus far conjectural) categorifications of perverse sheaves. Namely, in terms of our previous identification of perverse sheaves on Riemann surfaces with Milnor sheaves, the relative …
We give an algebro-geometric interpretation of $C_2$-equivariant stable homotopy theory by means of the $b$-topology introduced by Claus Scheiderer in his study of $2$-torsion phenomena in étale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a $\infty$-topos $\mathscr{X}$ with $G$-action a presentable stable $\infty$-category $\mathrm{Sp}^G(\mathscr{X})$, which recovers the $\infty$-category $\mathrm{Sp}^G$ of genuine $G$-spectra when $\mathscr{X}$ is the terminal $G$-$\infty$-topos. Given a scheme $X$ w…
