Definition 14.4.5. Given an \(\Oo \)-monoidal \(\infty \)-category \(C\), we let \(p_C\colon C^{\otimes } \to \Oo ^{\otimes }\) denote its cocartesian unstraightening, which by the lemma defines an \(\infty \)-operad \(\Mm _{C/\Oo } = (C^{\otimes },p_{\Oo } \circ p_C)\). Since \(p_C\) is a morphism of \(\infty \)-operads, it turns \(\Mm _{C/\Oo }\) into an object of the slice category \((\Op _{\infty })_{/\Oo }\), producing a functor \[ \Mm _{-/\Oo }\colon \Mon _{\Oo }(\Cat _{\infty }) \to (\Op _{\infty })_{/\Oo }. \] If \(\Pp \to \Oo \) and \(\Qq \to \Oo \) are two \(\infty \)-operads over \(\Oo \), we denote by \(\Fun _{(\Op _{\infty })_{/\Oo }}(\Pp ,\Qq )\) the \(\infty \)-category of operad maps \(\Pp \to \Qq \) over \(\Oo \), defined as the following fiber:
As a special case, we define the \(\infty \)-category \(\Pp /\Oo \)-algebras in \(C\) as \[ \Alg _{\Pp /\Oo }(C) := \Fun _{(\Op _{\infty })_{/\Oo }}(\Pp ,\Mm _{C/\Oo }). \] When \(\Oo = \Assoc \), we usually drop \(\Oo \) from the notation and abusively denote the \(\Pp /\Assoc \)-algebras in \(C\) by \(\Alg _{\Pp }(C)\). When \(\Pp = \Assoc \), we also write this as \(\Alg (C)\).
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