Entropy as a Topological Operad Derivation
We share a small connection between information theory, algebra, and topology—namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-09-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/23/9/1195 |
| _version_ | 1797519442015420416 |
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| author | Tai-Danae Bradley |
| author_facet | Tai-Danae Bradley |
| author_sort | Tai-Danae Bradley |
| collection | DOAJ |
| description | We share a small connection between information theory, algebra, and topology—namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the main example. We then give a general definition for a derivation of an operad in any category with values in an abelian bimodule over the operad. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in 1956 and a recent variation given by Leinster. |
| first_indexed | 2024-03-10T07:41:50Z |
| format | Article |
| id | doaj.art-b325cf46df1045f7aa9820597b0df8ec |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T07:41:50Z |
| publishDate | 2021-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-b325cf46df1045f7aa9820597b0df8ec2023-11-22T12:58:07ZengMDPI AGEntropy1099-43002021-09-01239119510.3390/e23091195Entropy as a Topological Operad DerivationTai-Danae Bradley0Sandbox@Alphabet, Mountain View, CA 94043, USAWe share a small connection between information theory, algebra, and topology—namely, a correspondence between Shannon entropy and derivations of the operad of topological simplices. We begin with a brief review of operads and their representations with topological simplices and the real line as the main example. We then give a general definition for a derivation of an operad in any category with values in an abelian bimodule over the operad. The main result is that Shannon entropy defines a derivation of the operad of topological simplices, and that for every derivation of this operad there exists a point at which it is given by a constant multiple of Shannon entropy. We show this is compatible with, and relies heavily on, a well-known characterization of entropy given by Faddeev in 1956 and a recent variation given by Leinster.https://www.mdpi.com/1099-4300/23/9/1195Shannon entropytopologyoperad |
| spellingShingle | Tai-Danae Bradley Entropy as a Topological Operad Derivation Entropy Shannon entropy topology operad |
| title | Entropy as a Topological Operad Derivation |
| title_full | Entropy as a Topological Operad Derivation |
| title_fullStr | Entropy as a Topological Operad Derivation |
| title_full_unstemmed | Entropy as a Topological Operad Derivation |
| title_short | Entropy as a Topological Operad Derivation |
| title_sort | entropy as a topological operad derivation |
| topic | Shannon entropy topology operad |
| url | https://www.mdpi.com/1099-4300/23/9/1195 |
| work_keys_str_mv | AT taidanaebradley entropyasatopologicaloperadderivation |