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Clones from comonoids

Ulrich Krähmer, Myriam Mahaman

Abstract

The fact that the cocommutative comonoids in a symmetric monoidal category form the best possible approximation by a cartesian category is revisited when the original category is only braided monoidal. This leads to the question when the endomorphism operad of a comonoid is a clone (a Lawvere theory). By giving an explicit example, we prove that this does not imply that the comonoid is cocommutative.

Clones from comonoids

Abstract

The fact that the cocommutative comonoids in a symmetric monoidal category form the best possible approximation by a cartesian category is revisited when the original category is only braided monoidal. This leads to the question when the endomorphism operad of a comonoid is a clone (a Lawvere theory). By giving an explicit example, we prove that this does not imply that the comonoid is cocommutative.

Paper Structure

This paper contains 17 sections, 34 theorems, 76 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Cocommutative comonoids and Cartesian categories
  3. Semi-cartesian and cartesian categories
  4. Categories of counital comagmas
  5. Cocommutative comonoids
  6. Recognition theorem
  7. Cartesian operads and clones
  8. Operads
  9. The category of finite cardinals
  10. The substitution product
  11. Cartesian operads
  12. Clones
  13. An example
  14. Motivation
  15. A deformation of $\mathbb{Z} [\sqrt{q}]$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\mathbf{C}$ be a braided monoidal category, $\operatorname{Com}(\mathbf{C})$ be the category of comonoids in $\mathbf{C}$, and $(X, \Delta _X, \varepsilon _X)$ be a comonoid. Then the monoidal subcategory of $\operatorname{Com} (\mathbf{C})$ formed by the tensor powers $X^{\otimes n}$ is cartes

Figures (4)

  • Figure 1: Selections
  • Figure 2: $- \cdot f: \mathbf{M}(A_1,A_1,A_2; B) \to \mathbf{M}(A_1,A_2;B)$
  • Figure 3: From a cartesian operad into a clone
  • Figure 4: Conditions for cocommutativity

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 74 more