🔢 mathematics

From $L_\infty$ algebroids to $L_\infty$ spaces: Part I

This paper develops the notion of $L_\infty$ spaces over dg manifolds and establishes an equivalence between the category of transitive $L_\infty$ algebroids and that of $L_\infty$ spaces, while also constructing a faithful functor between them, with both mappings detecting weak equivalences.

Original authors: Alberto S. Cattaneo, Shuhan Jiang

Published 2026-06-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alberto S. Cattaneo, Shuhan Jiang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to describe a complex, bumpy landscape. In mathematics and physics, we often use "smooth" maps to describe these landscapes. But sometimes, the landscape has sharp corners, holes, or infinite layers of detail that a simple smooth map can't capture. This is where the concepts in this paper come in.

The authors, Alberto S. Cattaneo and Shuhan Jiang, are building a new "translator" between two different ways of describing these complex, bumpy landscapes. They want to show that two seemingly different mathematical languages are actually saying the same thing, just with different vocabulary.

Here is the breakdown of their work using simple analogies:

1. The Two Languages: "Algebroids" and "Spaces"

The paper connects two mathematical worlds:

$L_\infty$ -algebroids: Think of these as instruction manuals or blueprints. They describe how different parts of a system interact, move, and change. They are very detailed, listing rules for how to combine things (like adding numbers or mixing paints) but allowing for "fuzzy" or "approximate" rules that get more precise the closer you look.
$L_\infty$ -spaces: Think of these as the actual physical models or 3D prints built from those blueprints. Instead of just listing rules, they represent the system as a "space" where you can walk around and observe properties.

The paper's main goal is to prove that every valid blueprint ( $L_\infty$ -algebroid) corresponds perfectly to a physical model ( $L_\infty$ -space), and vice versa.

2. The Problem: "Smooth" vs. "Derived"

In the past, mathematicians could only build these models for "smooth" landscapes (like a perfect sphere). But in modern physics (specifically Quantum Field Theory), the landscapes are often "derived" or "singular"—they have wrinkles, folds, and hidden dimensions that aren't smooth.

The authors realized that the old way of building models ( $L_\infty$ -spaces) broke down when the landscape wasn't smooth. They needed a new way to handle these "bumpy" terrains.

3. The Middleman: "Quasi-dg Manifolds"

To bridge the gap, the authors invented a middleman concept called a Quasi-dg manifold.

The Analogy: Imagine you have a rough sketch of a building (the "quasi" part) that looks like a normal building once you ignore the scaffolding and construction errors (the "dg" part).
They show that you can translate a Blueprint directly into this Rough Sketch, and you can also translate a Physical Model directly into that same Rough Sketch.
Because both the Blueprint and the Model can be turned into the same Rough Sketch, they must be equivalent to each other.

4. The "Jet Space" Machine

One of the paper's most creative tools is the Jet Space Functor.

The Analogy: Imagine you have a photograph of a face. It's a flat, 2D image. Now, imagine a machine that takes that photo and adds layers of detail: the texture of the skin, the pores, the microscopic hairs, and the way light bends around every tiny bump. This machine creates an "infinite resolution" version of the face.
In math, this machine is called the Jet Space. It takes a "rough" mathematical object and expands it to include all its infinite layers of detail (called "infinite jets").
The authors prove that this machine is a faithful translator. If you take a blueprint, run it through the Jet Space machine, and then translate it back, you get a perfect physical model. Crucially, if two blueprints are "almost the same" (a concept called a "weak equivalence"), their resulting models will also be "almost the same."

5. The Main Results

The paper claims three big things:

Equivalence: There is a perfect, one-to-one match between the category of "Transitive $L_\infty$ -algebroids" (the blueprints) and " $L_\infty$ -spaces" (the models). If you understand one, you understand the other.
Detection: This match is sensitive. If two blueprints are slightly different in a meaningful way, the resulting models will also be slightly different in that same way. The translation doesn't lose information.
Fibrant Replacement: They built a specific tool (the "Fibrant Replacement Functor") that takes any messy, incomplete blueprint and turns it into a "perfect" version that fits the rules of the physical model world. This is like taking a rough draft of a story and polishing it until it's ready for publication without changing the plot.

Why Does This Matter? (According to the Paper)

The authors mention that this work is motivated by Quantum Field Theory (the study of how particles and forces work).

In these theories, scientists often look at "families of solutions" (ways the universe could behave). Sometimes these families are smooth, but often they are "derived" (bumpy and complex).
The authors' new framework allows physicists to describe these complex families of solutions more accurately.
Specifically, they mention this helps with globalizing problems: taking a local description of a quantum system and making it work for the whole system, even when there are "singularities" (breakdowns or infinite points) involved.

Summary

Think of this paper as the construction of a new universal adapter.

Before, you had a "Blueprint" language and a "Model" language, but they didn't talk to each other well when the terrain was rough.
The authors built a Quasi-dg adapter that translates both languages into a common format.
They then built a Jet Space machine that polishes the rough drafts into perfect models.
The result is a guarantee that for every complex mathematical blueprint in this field, there is a corresponding, equally complex physical model, and you can move between them without losing any of the subtle, "bumpy" details that make the physics work.

Technical Summary: From $L_\infty$ -Algebroids to $L_\infty$ -Spaces: Part I

Problem Statement
The paper addresses a limitation in the existing theory of $L_\infty$ -spaces, originally introduced by Costello to describe quantization in the formal neighborhood of classical solutions in quantum field theory (QFT). Costello's formulation defines $L_\infty$ -spaces over smooth manifolds. However, in many QFT applications, the family of classical solutions is parameterized not by a smooth manifold but by a derived smooth manifold (or dg manifold). In the original formulation, the functor of points of an $L_\infty$ -space over a smooth manifold can only encode derived structure in nilpotent directions, failing to capture the full derived geometry required for these applications. The authors aim to adapt the notion of $L_\infty$ -spaces to the setting of dg manifolds and establish a rigorous correspondence with $L_\infty$ -algebroids over dg manifolds.

Methodology
The authors employ a unified approach using an intermediate category of quasi-dg manifolds.

Quasi-dg Manifolds: Defined as ringed spaces $(C, \mathcal{O}_C)$ where the structure sheaf is a sheaf of commutative differential graded algebras (cdgas). Crucially, modulo a proper dg ideal $\mathcal{I}_C$ , the space reduces to an ordinary dg manifold. This structure naturally admits the language of infinite jets and D-modules.
Chevalley–Eilenberg Functors: The authors construct functors between categories using Chevalley–Eilenberg algebras.
- They define a functor from $L_\infty$ -algebroids to quasi-dg manifolds by associating an $L_\infty$ -algebroid $L$ with its Chevalley–Eilenberg algebra $(\widehat{\Omega}_L, D_L)$ .
- They define a functor from $L_\infty$ -spaces (over dg manifolds) to fibrant quasi-dg manifolds, also via Chevalley–Eilenberg algebras of curved $L_\infty$ -algebras.
Jet Space Functor: To handle the "fibrant" condition (transitivity), the authors utilize the theory of infinite jets on dg manifolds. They construct a jet space functor $J^\infty$ that maps a quasi-dg manifold to a fibrant quasi-dg manifold. This functor is shown to be a weak equivalence (detecting weak equivalences) and serves as a fibrant replacement.

Key Contributions and Results

Categorical Equivalences:
The paper establishes three primary equivalences of categories:
- $L_\infty\text{Algd} \cong \text{QdgM}_{bp}$ : The category of $L_\infty$ -algebroids over dg manifolds is equivalent to the category of quasi-dg manifolds with base-preserving morphisms.
- $L_\infty\text{Algd}^{\text{fib}} \cong \text{QdgM}_{bp}^{\text{fib}}$ : The category of transitive $L_\infty$ -algebroids is equivalent to the category of fibrant quasi-dg manifolds.
- $L_\infty\text{Sp} \cong \text{QdgM}_{bp}^{\text{fib}}$ : The category of $L_\infty$ -spaces over dg manifolds is equivalent to the category of fibrant quasi-dg manifolds.
Theorem 1.1: Consequently, there is an equivalence between the category of transitive $L_\infty$ -algebroids and the category of $L_\infty$ -spaces over dg manifolds. This equivalence detects weak equivalences.
Fibrant Replacement Functor:
The authors construct a faithful functor, termed the fibrant replacement functor ( $\text{Fib}$ ), from the category of all $L_\infty$ -algebroids to the category of $L_\infty$ -spaces:
$\text{Fib}: L_\infty\text{Algd} \xrightarrow{\text{CE}_{\text{Algd}}} \text{QdgM}_{bp} \xrightarrow{J^\infty} \text{QdgM}_{bp}^{\text{fib}} \xrightarrow{\text{CE}_{\text{Sp}}^{-1}} L_\infty\text{Sp}$
Theorem 1.2: This functor $\text{Fib}$ detects weak equivalences. It restricts to the identity on $L_\infty$ -algebras and assigns to each dg manifold its Fedosov resolution.
Technical Developments:
- Whitehead-type Theorem: A proof that the kernel of a surjective morphism of graded vector bundles over a dg manifold is itself a graded vector bundle, utilizing a "Super Quillen–Suslin theorem."
- Curved $L_\infty$ -Algebras: The paper formalizes curved $L_\infty$ -algebras over cdgas with dg ideals, showing how the curvature term ( $l_0$ ) arises in the context of $L_\infty$ -spaces corresponding to transitive algebroids.
- Jet Spaces: A detailed construction of the infinite jet bundle for quasi-dg vector bundles and the associated jet space functor, proving it acts as a fibrant replacement in the category of quasi-dg manifolds.

Significance and Claims
The paper claims to resolve the limitation of Costello's original formulation by extending $L_\infty$ -spaces to the derived setting (dg manifolds). The primary significance lies in establishing a precise categorical bridge between $L_\infty$ -algebroids (Lie-theoretic structures) and $L_\infty$ -spaces (functorial/geometric structures) in the context of derived geometry.

The authors state that this framework is intended to provide an approach to the "globalization problem" for partition functions and observables in QFT within the Batalin–Vilkovisky formalism, specifically addressing cases involving derived and stacky singularities. They note that passing to $L_\infty$ -spaces renders certain structures (like characteristic classes and shifted symplectic structures) more transparent and computationally tractable.

The paper explicitly positions itself as Part I of a series. It does not claim to solve the globalization problem itself but rather to provide the necessary foundational machinery (the equivalence of categories and the fibrant replacement functor) for subsequent work. Future applications mentioned include the study of cohomological field theories and AKSZ theories with cotangent derived targets, which admit one-loop exact quantization.