{
  "id": "1709.02973",
  "version": "v1",
  "published": "2017-09-09T16:24:28.000Z",
  "updated": "2017-09-09T16:24:28.000Z",
  "title": "(Co)monads in Free Probability Theory",
  "authors": [
    "Roland M. Friedrich"
  ],
  "comment": "32 pages at 11pt",
  "categories": [
    "math.PR",
    "math.CT"
  ],
  "abstract": "We discuss free probability theory and free harmonic analysis from a categorical perspective. In order to do so, we extend first the set of analytic convolutions and operations and then show that the comonadic structure governing free probability is isomorphic to several well-known categories of algebras, such as, e.g., Witt vectors, differential algebras, etc. Within this framework moment-cumulant formulae are shown to correspond to natural transformations and not to be exclusive to probability theories. Finally, we start to discuss free probability and in particular free harmonic analysis from the point of view of algebraic theories and operads.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-09T16:24:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "free probability theory",
      "free harmonic analysis",
      "comonadic structure governing free probability",
      "framework moment-cumulant formulae",
      "analytic convolutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}