{
  "id": "2009.02428",
  "version": "v1",
  "published": "2020-09-05T01:08:54.000Z",
  "updated": "2020-09-05T01:08:54.000Z",
  "title": "An equivalence between enriched $\\infty$-categories and $\\infty$-categories with weak action",
  "authors": [
    "Hadrian Heine"
  ],
  "categories": [
    "math.AT",
    "math.CT"
  ],
  "abstract": "We show that an $\\infty$-category $\\mathcal{M}$ left tensored over a monoidal $\\infty$-category $\\mathcal{V}$ is completely determined by its graph $$\\mathcal{M}^\\simeq \\times \\mathcal{M}^\\simeq \\to \\mathcal{P}(\\mathcal{V}), \\ (\\mathrm{X},\\mathrm{Y}) \\mapsto \\mathcal{M}((-)\\otimes\\mathrm{X},\\mathrm{Y}),$$ parametrized by the maximal subspace $\\mathcal{M}^\\simeq$ in $\\mathcal{M}$, equipped with the structure of an enrichment in the sense of Gepner-Haugseng in the Day-convolution monoidal structure on the $\\infty$-category $\\mathcal{P}(\\mathcal{V})$ of presheaves on $\\mathcal{V}.$ Precisely, we prove that sending an $\\infty$-category left tensored over $\\mathcal{V}$ to its graph defines an equivalence between $\\infty$-categories left tensored over $\\mathcal{V}$ and a subcategory of all $\\infty$-categories enriched in presheaves on $\\mathcal{V}.$ More generally we consider a generalization of $\\infty$-categories left tensored over $\\mathcal{V}$, which Lurie calls $\\infty$-categories pseudo-enriched in $\\mathcal{V}$, and extend the former equivalence to an equivalence $\\chi$ between $\\infty$-categories pseudo-enriched in $\\mathcal{V}$ and all $\\infty$-categories enriched in presheaves on $\\mathcal{V}.$ The equivalence $\\chi$ identifies $\\mathcal{V}$-enriched $\\infty$-categories in the sense of Lurie with $\\mathcal{V}$-enriched $\\infty$-categories in the sense of Gepner-Haugseng. Moreover if $\\mathcal{V}$ is symmetric monoidal, we prove that sending an $\\infty$-category left tensored over $\\mathcal{V}$ to its graph is lax symmetric monoidal with respect to the relative tensorproduct on $\\infty$-categories left tensored over $\\mathcal{V}$ and the canonical tensorproduct on $\\mathcal{P}(\\mathcal{V})$-enriched $\\infty$-categories.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-05T01:08:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "equivalence",
      "weak action",
      "categories left",
      "category left",
      "day-convolution monoidal structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}