# Entropy as a Topological Operad Derivation

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Background

**Theorem**

**1**

- 1.
- the functions F are continuous and satisfy$$F(p\circ ({q}^{0},\dots ,{q}^{n}))=F\left(p\right)+\sum _{i=0}^{n}{p}_{i}F\left({q}^{i}\right)$$where $n\ge 0$ and $p\in {\Delta}^{n}$ and ${q}^{i}\in {\Delta}^{{k}_{i}}$ with ${k}_{0}$, ${k}_{1},\dots ,{k}_{n}\ge 0$;
- 2.
- $F=cH$ for some $c\in \mathbb{R}.$

**Theorem**

**2.**

## 2. Background: Operads and Their Representations

**Definition**

**1.**

- (i)
- [associativity] For all $p\in \mathcal{O}\left(n\right)$ and $q\in \mathcal{O}\left(m\right)$ and $r\in \mathcal{O}\left(k\right)$,$$\begin{array}{c}\hfill \left(p{\circ}_{j}q\right){\circ}_{i}r=\left(\right)open="\{"\; close>\begin{array}{cc}\left(p{\circ}_{i}r\right){\circ}_{j+k-1}q\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}1\le i\le j-1\hfill \\ p{\circ}_{j}\left(q{\circ}_{i-j+1}r\right)\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}j\le i\le j+m-1\hfill \\ \left(p{\circ}_{i-m+1}r\right){\circ}_{j}q\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}i\ge j+m\hfill \end{array}\end{array}$$
- (ii)
- [identity] The operation $1\in \mathcal{O}\left(1\right)$ acts as an identity in the sense that$$1{\circ}_{1}p=p{\circ}_{i}1=p$$for all $p\in \mathcal{O}\left(n\right)$ and $1\le i\le n.$

**Example**

**1.**

**Example**

**2.**

**Definition**

**2.**

## 3. Derivations of the Operad of Simplices

**Definition**

**3.**

**Example**

**3.**

**Definition**

**4.**

- (i)
- [associativity, commutativity] there is a morphism ${\mu}_{n}:M\left(n\right)\times M\left(n\right)\to M\left(n\right)$ in $\mathsf{C}$ such that ${\mu}_{n}({\mu}_{n}(a,b),c)={\mu}_{n}(a,{\mu}_{n}(b,c))$ and ${\mu}_{n}(a,b)={\mu}_{n}(b,a)$ for all $a,b,c\in M\left(n\right)$,
- (ii)
- [identity] there is an element $1\in M\left(n\right)$ such that ${\mu}_{n}(1,a)=a={\mu}_{n}(a,1)$ for all $a\in M\left(n\right)$.

**Remark**

**1.**

**Definition**

**5.**

**Proposition**

**1.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Associativity in an operad. (

**Left**) First composing q with p and then r is the same as first composing r with p and then q. The order in which this is performed does not matter. (

**Right**) The same is true if r appears to the right, rather than the left, of $q.$ (

**Middle**) Likewise, r may first be composed with q and their composite may then be composed with p, or q may be first composed with p followed by $r.$ Again, the order does not matter.

**Figure 3.**(

**Left**) A picture of the composition $p{\circ}_{3}q$ when p is the probability distribution associated to a six-sided die and q is that of a fair coin toss. (

**Right**) The simultaneous composition of n probability distributions ${q}^{i}\in {\Delta}_{{k}_{i}}$ with a given $p\in {\Delta}_{n}$.

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Bradley, T.-D.
Entropy as a Topological Operad Derivation. *Entropy* **2021**, *23*, 1195.
https://doi.org/10.3390/e23091195

**AMA Style**

Bradley T-D.
Entropy as a Topological Operad Derivation. *Entropy*. 2021; 23(9):1195.
https://doi.org/10.3390/e23091195

**Chicago/Turabian Style**

Bradley, Tai-Danae.
2021. "Entropy as a Topological Operad Derivation" *Entropy* 23, no. 9: 1195.
https://doi.org/10.3390/e23091195