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We present a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arise naturally in algebraic combinatorics. We explore several applications of this interplay. For instance, we provide a new interpretation of the Leray number of a clique complex in terms of non-commutative algebra. R\'esum\'e. Nous pr\'esentons une magnifique interaction entre la topologie combinatoire et l'alg\'ebre homologique d'une classe de mono\"ides qui figurentarXiv:1311.6680v1 fatcat:vpwz7uzzwfc2tkx65cl65evdxm