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Entropy as a Topological Operad Derivation.

Tai-Danae Bradley¹

¹Sandbox@Alphabet, Mountain View, CA 94043, USA.

Entropy (Basel, Switzerland)

|September 28, 2021

Summary

We found a link between information theory and algebra/topology, showing Shannon entropy corresponds to derivations of topological simplices. This reveals a fundamental connection between entropy and algebraic structures.

Area of Science:

Mathematics
Information Theory
Algebraic Topology

Background:

Operads are algebraic structures used to study composition.
Topological simplices and the real line serve as key examples in operad representations.
Derivations of operads are essential for understanding their structure and properties.

Purpose of the Study:

To establish a novel connection between information theory and algebraic topology.
To explore the relationship between Shannon entropy and derivations of the operad of topological simplices.
To generalize the concept of derivations in operads.

Main Methods:

Reviewing operads and their representations.
Defining derivations of operads in a general categorical setting.

Keywords:

Shannon entropy operad topology

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Applying these definitions to the operad of topological simplices.

Main Results:

Shannon entropy is shown to be a derivation of the operad of topological simplices.
Every derivation of this operad is equivalent to a multiple of Shannon entropy at a specific point.
This finding is consistent with established characterizations of entropy by Faddeev and Leinster.

Conclusions:

A significant link exists between information theory (Shannon entropy) and algebraic topology (operad derivations).
This correspondence provides a new perspective on the mathematical underpinnings of entropy.
The results generalize and unify existing theories on entropy characterization.