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The free monoid on a set is the type of lists over that set. The universal property says fold is the unique homomorphism from lists to any monoid. This explains why lists, multisets, and polynomials appear everywhere.
Many structures come in pairs: forward/reverse AD, push/pull iteration, encode/decode. Recognizing duality lets you transfer theorems and insights between domains.
A composable ecosystem of tools for manipulating nested data structures. From a simple helper function to a full data algebra, guided by purity, pedagogy, and the principle of least power.