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Comparing dg category models for path spaces via $A_\infty$-functors

Manuel Rivera, Yi Wang

TL;DR

The work develops a robust $A_infty$-categorical framework connecting path-space models to cobar constructions, providing a many-object dual of Chen's iterated integral map and linking it to Adams' cobar construction. It introduces three cobar variants, builds natural $A_infty$-transformations between them, and proves that the Chen-dual map $\mathcal{I}$ acts as a left $A_infty$-inverse to Adams' map in the simply connected case, with extensions to non-simply connected spaces via the fundamental groupoid. An elementary proof of Adams' extension through universal covers is given, and the results are extended to arbitrary spaces using a groupoid-based model $\mathsf{C^P}(X)$, yielding quasi-equivalences and natural $A_infty$-transformations $\widetilde{\mathcal{I}}$ and $\widetilde{\mathcal{H}}$. Overall, the paper provides explicit algebraic models for path and loop spaces that unify cobar-type constructions with path-space functors, valid in broad generality and grounded in $A_infty$-categorical techniques.

Abstract

We construct a many-object dual version of Chen's iterated integral map. For any topological space X, the construction takes the form of an A-infinity functor between two dg categories whose objects are the points of X: the domain has as morphisms the singular (cubical) chains on the space of (Moore) paths in X and the codomain has morphisms arising by totalizing a cosimplicial chain complex determined by the dg coalgebra of singular (simplicial) chains in X. When X is simply connected, we show this construction defines a homotopy inverse to a classical map of Adams, which sends ordered sequences of singular simplices in X linked by shared vertices to cubes of paths in X. When X is not necessarily simply connected, following an idea of Irie, we incorporate the fundamental groupoid of X into the construction and deduce analogous results. Along the way, we provide an elementary and new proof of the fact that the (direct-sum) cobar construction of the chains in X, suitably interpreted, models the dg category of paths in X, an extension of Adams's cobar theorem established by Rivera-Zeinalian using different methods.

Comparing dg category models for path spaces via $A_\infty$-functors

TL;DR

The work develops a robust -categorical framework connecting path-space models to cobar constructions, providing a many-object dual of Chen's iterated integral map and linking it to Adams' cobar construction. It introduces three cobar variants, builds natural -transformations between them, and proves that the Chen-dual map acts as a left -inverse to Adams' map in the simply connected case, with extensions to non-simply connected spaces via the fundamental groupoid. An elementary proof of Adams' extension through universal covers is given, and the results are extended to arbitrary spaces using a groupoid-based model , yielding quasi-equivalences and natural -transformations and . Overall, the paper provides explicit algebraic models for path and loop spaces that unify cobar-type constructions with path-space functors, valid in broad generality and grounded in -categorical techniques.

Abstract

We construct a many-object dual version of Chen's iterated integral map. For any topological space X, the construction takes the form of an A-infinity functor between two dg categories whose objects are the points of X: the domain has as morphisms the singular (cubical) chains on the space of (Moore) paths in X and the codomain has morphisms arising by totalizing a cosimplicial chain complex determined by the dg coalgebra of singular (simplicial) chains in X. When X is simply connected, we show this construction defines a homotopy inverse to a classical map of Adams, which sends ordered sequences of singular simplices in X linked by shared vertices to cubes of paths in X. When X is not necessarily simply connected, following an idea of Irie, we incorporate the fundamental groupoid of X into the construction and deduce analogous results. Along the way, we provide an elementary and new proof of the fact that the (direct-sum) cobar construction of the chains in X, suitably interpreted, models the dg category of paths in X, an extension of Adams's cobar theorem established by Rivera-Zeinalian using different methods.

Paper Structure

This paper contains 15 sections, 15 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Main Constructions
  3. Different versions of the cobar construction
  4. The "many object" version \ref{['item: version of cobar - many']}
  5. The totalized cosimplicial version \ref{['item: version of cobar - many prod']}
  6. The natural transformations A, G and the natural map $It_X$.
  7. The "many object" version of Adams' map.
  8. A formal dual of Chen's iterated integral map
  9. The natural transformation $\mathcal{G}$
  10. Construction of $\mathcal{I}$ (Theorem \ref{['theorem: A infinity map I']}) and $\mathcal{H}$ (Theorem \ref{['theorem: A infinity homotopy']})
  11. Proof of Theorem \ref{['theorem: A infinity map I']}
  12. Proof of Theorem \ref{['theorem: A infinity homotopy']}
  13. Elaboration on Remark \ref{['remark: Chen Adams inverse simply connected']}
  14. Adams' map and the universal cover
  15. Extension of the results to arbitrary spaces

Key Result

Theorem 1.2

The natural maps in equation: I_1 can be extended to a natural $A_\infty$-transformation in the sense that for any topological space $X$, is an $A_\infty$-functor with $\mathcal{I}_{X,1} = It_X$, and the family $\{\mathcal{I}_X\}_{X \in \mathsf{Top}}$ is natural in $X$.

Theorems & Definitions (35)

  • Remark 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9: RZ_cubicalrivera2022adams
  • Theorem 1.10
  • ...and 25 more