ScalingStacks

Given a map of operads \(\mathbb E_0 \rightarrow\mathcal O\), equivalently an \(A\in \mathrm{Alg}_{\mathbb E_0}(\mathcal O)\), the above pushout induces an isomorphism of spaces \[ \mathrm{Hom}_{\mathrm{Op}_{\mathbb E_0/}}(\mathbb A_2,\mathcal O) \simeq \mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal O)}(A,A;A) \times_{\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_0}(\mathcal O)}(A;A)^2} \{(\mathrm{id}_A,\mathrm{id}_A)\}\]