One Eilenberg Theorem to Rule Them All

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Claim: a single, general Eilenberg-style theorem that unifies more than a dozen prior correspondences and produces new instances (infinite words, weighted languages, trees, cost functions). Method: categorical algebra using monads and dualities, construction of a profinite T-algebra (codensity monad) and a required unary presentation parameter that encodes derivatives. Result: bijection between language varieties and pseudo-varieties of finite T-algebras, a modular proof that recovers classical Eilenberg and yields local and profinite equation versions, and it demonstrably works across the listed examples.

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1. 0:00Intro
2. 1:12Overview
3. 2:50Eilenberg's Variety Theorem (1976)
4. 4:16Other Eilenberg-Type Theorems
5. 5:55Big Picture
6. 8:12Languages
7. 10:47Algebraic recognition
8. 11:53Profinite Words
9. 13:10General Duality
10. 16:50General Variety Theorem (2016)
11. 18:30Derivatives: Monoid Case
12. 19:40Derivatives: General Case
13. 22:04Applications
14. 23:17Conclusions and Further Work