The (right) homotopy between two A_∞-morphisms

Abstract

Let f, g: A→ B be two A_∞-morphisms between two A_∞-algebras (linear over a commutative ring 𝕂). In this paper, we compare different notions of (right) homotopy between f and g. We note that the A_∞-algebra B⊗ R, where R:=𝕂[t,dt], is a model for the interval [0,1]× B in the sense of Fukaya-Oh-Ohta-Ono, and we give a short alternative proof, in the unfiltered case, of the existence of comparison morphisms between such models. We use it to prove that two A_∞-morphisms are homotopic à la Seidel if and only if, considered as morphisms of 2-colored DG operads, they are elementary homotopic à la Markl-Shnider-Stasheff through a homotopy which is constant on the two monochromatic parts. We also provide an example of two A_∞-morphisms which are elementary homotopic but not weakly equivalent (and so homotopic à la Seidel). Moreover, over a field, we observe that the models for the interval are good path objects in the category of A_∞-algebras with (non strict) A_∞-morphisms, and we explain why weakly equivalent A_∞-morphisms need not be homotopic.

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BibTeX RIS

BibTeX

@misc{ornaghi2026right,
  title = {{The (right) homotopy between two A$_\infty$-morphisms}},
  author = {Mattia Ornaghi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13803},
  eprint = {2610.13803},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13803},
}

RIS

TY  - UNPB
TI  - The (right) homotopy between two A_∞-morphisms
AU  - Mattia Ornaghi
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13803
UR  - https://arxiv.org/abs/2610.13803
AB  - Let f, g: A→ B be two A_∞-morphisms between two A_∞-algebras (linear over a commutative ring 𝕂). In this paper, we compare different notions of (right) homotopy between f and g. We note that the A_∞-algebra B⊗ R, where R:=𝕂[t,dt], is a model for the interval [0,1]× B in the sense of Fukaya-Oh-Ohta-Ono, and we give a short alternative proof, in the unfiltered case, of the existence of comparison morphisms between such models. We use it to prove that two A_∞-morphisms are homotopic à la Seidel if and only if, considered as morphisms of 2-colored DG operads, they are elementary homotopic à la Markl-Shnider-Stasheff through a homotopy which is constant on the two monochromatic parts. We also provide an example of two A_∞-morphisms which are elementary homotopic but not weakly equivalent (and so homotopic à la Seidel). Moreover, over a field, we observe that the models for the interval are good path objects in the category of A_∞-algebras with (non strict) A_∞-morphisms, and we explain why weakly equivalent A_∞-morphisms need not be homotopic.
ER  -

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