ScalingStacks

A.8.2 Key examples[00IR]

Perhaps the most important family of examples of \(\infty\)-operads is the sequence \(\mathbb E_0 \rightarrow\mathbb E_1 \rightarrow\cdots \rightarrow\mathbb E_\infty\). These are single-colored, with the space \(\mathbb E_k(n)\) of \(n\)-ary operations given by (the underlying space of) the topological space of configurations of \(n\) disjoint points in \(\mathbb R^k\).⁵⁴ The above maps are induced by the standard embeddings \(\mathbb R^0 \hookrightarrow\mathbb R^1 \hookrightarrow\cdots \hookrightarrow\mathbb R^\infty\). We note that \(\mathbb E_1\) and \(\mathbb E_\infty\) are respectively the \(\infty\)-operads underlying the colored operads that parametrize associative and commutative algebras (and in particular, their spaces of multimorphisms are discrete). Hence, we also write \({\textup{Assoc}}\coloneqq \mathbb E_1\) and \(\text{Comm}\coloneqq \mathbb E_\infty\) and respectively refer to these as the associative and commutative \(\infty\)-operads. In fact, \(\text{Comm}\) is simply the identity functor \(\text{Comm}\coloneqq \mathbb E_\infty \simeq \mathrm{Fin}_* \xrightarrow{\mathrm{id}} \mathrm{Fin}_*\), and defines a terminal object of \(\mathrm{Op}\).

Another illustrative example is the \(\infty\)-operad \(\mathrm{LM}\) associated to the two-colored operad parametrizing pairs of an associative algebra object along with a left module over it. We will return to \(\mathrm{LM}\) in subsection A.9.

It will occasionally be useful for us to refer to the single-colored \(\infty\)-operad \(\mathrm{Triv}\), which has no \(n\)-ary multimorphisms for \(n \not= 1\) and the only \(1\)-ary morphism is the identity morphism. Given an \(\infty\)-operad \(\mathcal O\), we write \(\mathcal O_\mathrm{Triv}\coloneqq \mathcal O\times_\text{Comm}\mathrm{Triv}\).