ScalingStacks

Example 7.2.7.

Using corollary 7.2.6, we may factor the canonical map \(\mathbb E_0 \rightarrow\mathbb E_1\) through an operad map \(\mathbb A_2 \rightarrow\mathbb E_1\) which remembers of an \(\mathbb E_1\)-algebra only the binary multiplication and its unitality structure. This is the first step in the well-known filtration \(\mathbb E_0 = \mathbb A_1 \rightarrow\mathbb A_2 \rightarrow\ldots \rightarrow\mathbb A_{\infty} = \mathbb E_{1}\) of the \(\mathbb E_1\)-operad by the unital \(\mathbb A_n\) operads, encoding higher coherent associativity (see [Lur17, § 4.1.4] for a non-unital version of this filtration in the setting of \(\infty\)-operads). In this paper, we will only need the first stage \(\mathbb E_0 \rightarrow\mathbb A_2\rightarrow\mathbb E_1\).