Examples algebras over operads

Example 12.2.11. For specific instances of \(\Oo \), the \(\Oo \)-algebras often have specialized names:

• For \(\Oo = \Assoc \) these are called associative algebras;
• For \(\Oo = \Comm \) they are commutative algebras;
• For \(\Oo = \Ee _k\) they are called \(\Ee _k\)-algebras;
• For \(\Oo = \oMod \), an \(\Oo \)-algebra is called a module. Restricting along the inclusion \(\Comm \hookrightarrow \oMod \) gives the underlying commutative algebra of the module, generically denoted \(A\). Restricting along the inclusion \(\Triv \hookrightarrow \oMod \) gives the underlying object, generically denoted \(M\).
• Similarly, \(\Oo = \oLMod \) encodes left modules: pairs \((A,M)\) where \(A\) is an associative algebra rather than a commutative one.