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  "body": "\n## Summary\n\nUniversal coalgebra is the general study of coalgebras for an endofunctor of the category of sets.\nSuch a coalgebra is a set of states $X$ with a map $X \\to F(X)$.\nThe map tells what can be observed of a state and which states come next.\nIt is the dual of [[universal algebra|W82R6N0GM9]]: algebras build elements from smaller ones, coalgebras unfold states into their behaviour.\nStreams, automata, labelled transition systems and many other state-based systems are coalgebras, and the theory treats their behaviour and their bisimulations in one framework.\nThe general notion over any category is the [[F-coalgebra|6BS0300CMV]].\n\n## Explanations\n\n### L1 human\n\nThink of a machine with hidden states that you can only watch from outside.\nAt each state it shows you something and then moves to a next state.\nA coalgebra records exactly this: for each state, what you see and where you go.\nFor example, a stream of numbers is a coalgebra for $F(X) = \\N \\times X$: each state gives a head $n \\in \\N$ and a next state.\nTwo states are treated as equal when no experiment can tell them apart.\n",
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