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  "body": "\n## Statement\n\nLet $\\mathcal{O}$ be an [[$E_\\infty$ operad|T78YG90VVT]], and let $X$ be an $\\mathcal{O}$-algebra in spaces, with the homotopy type of a CW complex.\nSuppose $X$ is grouplike: the monoid $\\pi_0(X)$, with the product induced by any binary operation of $\\mathcal{O}$, is a group.\nThen $X$ is weakly equivalent to an [[infinite loop space|A7VB2NSGJ2]].\nMore precisely, there is a connective $\\Omega$-spectrum $E$ with $E_0$ [[weakly equivalent|4Q3WY1RX91]] to $X$, and this equivalence is compatible with the $\\mathcal{O}$-structures.\n\n## Remarks\n\nMay proved the principle in 1972 by building the spectrum from iterated bar constructions.\nThe key input is his approximation theorem.\nIt says that, for a connected space $Y$, the free algebra on $Y$ over the little $n$-cubes operad is weakly equivalent to $\\Omega^n \\Sigma^n Y$.\nBoardman and Vogt gave another proof with their theory of homotopy-everything spaces.\nThe finite version holds too: a grouplike algebra over the little $n$-cubes operad is weakly equivalent to an $n$-fold [[loop space|2JFF5F3EGF]].\nIf $X$ is not grouplike, the zeroth space of the spectrum is instead the group completion of $X$.\n\nAt work: the disjoint union $\\coprod_{n \\ge 0} B\\Sigma_n$ is an algebra over the Barratt–Eccles operad, but $\\pi_0 = \\N$ is not a group.\nIts group completion is $\\Omega^\\infty \\Sigma^\\infty S^0 = QS^0$, which is the [[Barratt–Priddy–Quillen theorem|ZD7BNADK4H]].\n\n## Explanations\n\n### L1 human\n\nInfinite loop spaces carry a multiplication that is commutative and associative up to all higher homotopies.\nThe recognition principle says that this property characterizes them.\nAny space with such a coherent multiplication, in which every element has an inverse up to homotopy, is an infinite loop space.\nSo to show that a space comes from a cohomology theory, it is enough to find the multiplicative structure.\n\n### L2 human\n\nThe principle is an \"infinite loop space machine\": it turns $E_\\infty$ spaces into spectra.\nIt applies to classifying spaces of symmetric monoidal categories, which is how algebraic K-theory spectra are built.\nIt also shows that spaces such as $BO$, $BU$ and $BTop$ are infinite loop spaces, which gives new cohomology theories and operations.\nDifferent machines, by May, Segal and Boardman–Vogt, give equivalent spectra by the uniqueness theorem of May and Thomason.\n",
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