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Differential graded manifolds of finite positive amplitude

Kai Behrend, Hsuan-Yi Liao, Ping Xu

Abstract

We prove that dg manifolds of finite positive amplitude, i.e. bundles of positively graded curved $L_\infty[1]$-algebras, form a category of fibrant objects. As a main step in the proof, we obtain a factorization theorem using path spaces. First we construct an infinite-dimensional factorization of a diagonal morphism using actual path spaces motivated by the AKSZ construction. Then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient in our method is the homotopy transfer theorem for curved $L_\infty[1]$-algebras. As an application, we study the derived intersections of manifolds.

Differential graded manifolds of finite positive amplitude

Abstract

We prove that dg manifolds of finite positive amplitude, i.e. bundles of positively graded curved -algebras, form a category of fibrant objects. As a main step in the proof, we obtain a factorization theorem using path spaces. First we construct an infinite-dimensional factorization of a diagonal morphism using actual path spaces motivated by the AKSZ construction. Then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient in our method is the homotopy transfer theorem for curved -algebras. As an application, we study the derived intersections of manifolds.
Paper Structure (27 sections, 40 theorems, 107 equations, 2 figures)

This paper contains 27 sections, 40 theorems, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Categories of fibrant objects
  4. Curved $L_\infty[1]$-algebras
  5. The transfer theorem for curved $L_\infty[1]$-algebras
  6. The category of $L_\infty$-bundles
  7. $L_\infty$-bundles
  8. Algebra model
  9. Étale morphisms and weak equivalences
  10. Fibrations
  11. The shifted tangent $L_\infty$-bundles
  12. The derived path space
  13. Short geodesic paths
  14. The derived path space of a manifold
  15. The derived path space of an $L_\infty$-bundle
  16. ...and 12 more sections

Key Result

Lemma 2.4

Given any object $X$ in a category of fibrant objects ${\mathscr C}$, the decomposition eq:PathSpDecomp is unique in the sense that if $X \to P_X' \to X \times X$ is another path space decomposition, then there exists a third path space object $P_X"$ together with trivial fibrations $P_X \leftarrow

Figures (2)

  • Figure 1: Decorated trees in the formulas of inclusion morphism $\phi$
  • Figure 2: Decorated trees in the formulas of transferred operations

Theorems & Definitions (87)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 2.9
  • Proposition 2.10: Transfer Theorem
  • ...and 77 more