{
  "abstract": "Let $f$, $g\\colon A\\to B$ be two A$_{\\infty}$-morphisms between two A$_{\\infty}$-algebras (linear over a commutative ring $\\mathbb{K}$). In this paper, we compare different notions of (right) homotopy between $f$ and $g$. We note that the A$_{\\infty}$-algebra $B\\otimes R$, where $R:=\\mathbb{K}[t,dt]$, is a model for the interval $[0,1]\\times B$ 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$_{\\infty}$-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$_{\\infty}$-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$_{\\infty}$-algebras with (non strict) A$_{\\infty}$-morphisms, and we explain why weakly equivalent A$_{\\infty}$-morphisms need not be homotopic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13803",
  "author_total": 1,
  "authors": [
    {
      "name": "Mattia Ornaghi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7abh3wxyx82ha.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13803",
  "fulltext_url": null,
  "id": "7abh3wxyx82ha",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13803",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7221534893",
  "pdf_url": "https://arxiv.org/pdf/2610.13803",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7abh3wxyx82ha.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13803",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13803"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The (right) homotopy between two A$_{\\infty}$-morphisms",
  "title_text": "The (right) homotopy between two A_∞-morphisms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/7abh3wxyx82ha",
  "venue": "arXiv",
  "versions": []
}