Definition 12.1.10. Let \(\Oo \) be a non-colored operad and let \(C\) be a symmetric monoidal category. We define an \(\Oo \)-algebra in \(C\) as a pair \((A,f_A)\) consisting of an object \(A\) of \(C\) together with an operad morphism \(f_A\colon \Oo \to \oEnd _C(A)\). In particular, \(f_A\) consists of an \(n\)-ary operation \(P_A\colon A^{\otimes n} \to A\) for every \(P \in \Oo (n)\) satisfying the condition that composition of operations in \(\Oo \) corresponds to the composition of operations on \(A\).
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