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On analytic exponential functors on free groups

Minkyu Kim, Christine Vespa

Abstract

This paper concerns exponential contravariant functors on free groups. We obtain an equivalence of categories between analytic, exponential contravariant functors on free groups and conilpotent cocommutative Hopf algebras. This result explains how equivalences of categories obtained previously by Pirashvili and by Powell interact. Moreover, we obtain an equivalence between the categories of outer, exponential contravariant functors on free groups and bicommutative Hopf algebras. We also go further by introducing a subclass of analytic, contravariant functors on free groups, called primitive functors; and prove an equivalence between primitive, exponential contravariant functors and primitive cocommutative Hopf algebras.

On analytic exponential functors on free groups

Abstract

This paper concerns exponential contravariant functors on free groups. We obtain an equivalence of categories between analytic, exponential contravariant functors on free groups and conilpotent cocommutative Hopf algebras. This result explains how equivalences of categories obtained previously by Pirashvili and by Powell interact. Moreover, we obtain an equivalence between the categories of outer, exponential contravariant functors on free groups and bicommutative Hopf algebras. We also go further by introducing a subclass of analytic, contravariant functors on free groups, called primitive functors; and prove an equivalence between primitive, exponential contravariant functors and primitive cocommutative Hopf algebras.
Paper Structure (32 sections, 58 theorems, 102 equations)

This paper contains 32 sections, 58 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. The general case
  4. Polynomial filtration of $\mathcal{C}$-modules
  5. Cross-effects of $\mathcal{C}$-modules
  6. Polynomial and analytic functors
  7. Construction of the polynomial filtration
  8. Coradical filtration of coaugmented coalgebras
  9. Coradical filtration
  10. Coradical filtration: the case of a field $\mathds{k}$
  11. Examples of coradical filtrations
  12. The polynomial bialgebra
  13. The shuffle bialgebra
  14. Finite-dimensional bialgebras
  15. Isomorphism of polynomial and coradical filtrations
  16. ...and 17 more sections

Key Result

Theorem 1

(Theorem 202207212109) The evaluation on ${\mathsf{F}_1}$ induces an equivalence:

Theorems & Definitions (146)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 3.4
  • ...and 136 more