E1 operad equivalent associative

Remark 12.1.8. For \(k = 1\), ordering the little intervals from left to right defines an operad morphism \(\Ee _1 \to \Assoc \). For every \(n \geq 0\), its fibers are the contractible components of \(\Ee _1(n)\), so the map \(\Ee _1(n) \to \Assoc (n)\) is a homotopy equivalence. In particular, after passing from sets and topological spaces to animae (as we will do when working with \(\infty \)-operads) there is no longer a distinction between \(\Ee _1\) and \(\Assoc \). In more classical setups, for example when working with model categories, the homotopically correct operad to work with is often \(\Ee _1\). This is the reason why in older literature you will often find the phrase ‘\(\Ee _1\)-algebras’ in places where we would say ‘associative algebra’.