Recognition principle for loop animae

Theorem 5.2.1 (Recognition principle for loop spaces, cf. Stasheff (1963), May (1972), Boardman and Vogt (1973), Lurie (2009), Theorem 7.2.2.11). Let \(M\) be a monoid in \(\An \) and let \(X\) be a pointed anima.

(1)

The unit map \(M \to \bOmega \bB M\) is an isomorphism if and only if \(M\) is a group in \(\An \);

(2)

The pointed anima \(\bB M\) is connected;

(3)

The counit \(\bB \bOmega X \to X\) is an inclusion of the connected component of the basepoint of \(X\).