ScalingStacks

A.8.1 Basic notions[00IQ]

The notion of an \(\infty\)-operad is an \(\infty\)-categorical version of the theory of colored operads. A colored operad consists of a set \(\iota_0\underline{\mathcal O}\) of colors along with for every finite set \(\{X_i \in \iota_0 \underline{\mathcal O} \}_{i \in I}\) of colors and every color \(Y \in \iota_0 \underline{\mathcal O}\) a set \(\mathrm{Mul}_\mathcal O(\{X_i\}_{i \in I},Y)\) of multimorphisms from \(\{X_i\}_{i \in I}\) to \(Y\), which altogether must be equipped with a associative and unital composition law.⁵³ In particular, the unary multimorphisms (i.e. those with \(|I| = 1\)) define a category \(\underline{\mathcal O}\) of colors (whose set of objects is \(\iota_0 \underline{\mathcal O}\)).

We now give a hint of the main definition. An \(\infty\)-operad \(\mathcal O\) is an \(\infty\)-category \(\mathcal O^{\otimes}\) (called the \(\infty\)-category of operators of \(\mathcal O\)) equipped with a functor \(\mathcal O^{\otimes} \rightarrow\mathrm{Fin}_*\) to the category of finite pointed sets satisfying certain conditions. We immediately introduce the notation \(\underline{n}_+ \coloneqq \{ 1, 2, \ldots, n \}_+ \in \mathrm{Fin}_*\) for the indicated standard object, as well as the notation \(\underline{\mathcal O} \coloneqq \mathcal O^{\otimes}_{\underline{1}_+}\) for the indicated fiber. We refer to \(\underline{\mathcal O}\) as the \(\infty\)-category of colors of \(\mathcal O\) (or sometimes as its underlying \(\infty\)-category, for reasons that will be explained shortly). We will sometimes abuse notation and denote the underlying \(\infty\)-category \(\underline{\mathcal O}\) of an \(\infty\)-operad \(\mathcal O\) simply also by \(\mathcal O\). The crux of the definition of an \(\infty\)-operad is that \(\mathcal O^{\otimes}\) satisfies a sort of “fiberwise” Segal condition which implies that for every \(n \geq 0\) there is a natural equivalence \(\mathcal O^{\otimes}_{\underline{n}_+} \simeq \underline{\mathcal O}^{\times n}\), as well as an “internal” Segal condition which implies that for every pair of objects \(X \coloneqq (X_1,\ldots,X_m) \in \underline{\mathcal O}^{\times m} \simeq \mathcal O^{\otimes}_{\underline{m}_+}\) and \(Y \coloneqq (Y_1,\ldots,Y_n) \in \underline{\mathcal O}^{\times n} \simeq \mathcal O^{\otimes}_{\underline{n}_+}\), we have a natural equivalence \[\mathrm{Hom}_{\mathcal O^{\otimes}}(X,Y) \simeq \bigsqcup_{f \in \mathrm{Hom}_{\mathrm{Fin}_*}(\underline{m}_+,\underline{n}_+)} \prod_{i = 1}^n \mathrm{Hom}_{\mathcal O^{\otimes}}(\{X_j\}_{j \in f^{-1}(i)}, Y_i) ~.\] An ordinary colored operad \(\mathcal O'\) defines an \(\infty\)-operad \(\mathcal O\) with \(\mathrm{Hom}_{\mathcal O^{\otimes}}(\{X_i\}_{i \in I},Y) \coloneqq \mathrm{Mul}_{\mathcal O'}(\{X_i\}_{i \in I},Y)\). As a result, we also write \(\mathrm{Mul}_\mathcal O(\{X_i\}_{i \in I},Y) \coloneqq \mathrm{Hom}_{\mathcal O^{\otimes}}(\{X_i\}_{i \in I},Y)\) for the hom-spaces in an \(\infty\)-operad \(\mathcal O\) whose targets lies in \(\underline{\mathcal O}\), and refer to their points as multimorphisms. Altogether, \(\infty\)-operads assemble into a (non-full) subcategory \(\mathrm{Op}\subset (\mathrm{Cat}_\infty)_{/\mathrm{Fin}_*}\): in essence, morphisms of \(\infty\)-operads are required to respect the Segal condition equivalences. In fact, allowing all 2-morphisms in \((\mathrm{Cat}_\infty)_{/\mathrm{Fin}_*}\) endows \(\mathrm{Op}\) with the structure of an \((\infty,2)\)-category, whose hom-\((\infty,1)\)-categories we denote by \(\underline{\mathrm{Hom}}_\mathrm{Op}(-,-)\).

We say that an \(\infty\)-operad \(\mathcal O\) is single-colored if its \(\infty\)-category of colors \(\underline{\mathcal O}\) is contractible. In this case, we may write \(\ast \in \underline{\mathcal O}\) for the unique point, and we write \(\mathcal O(n) \coloneqq \mathrm{Mul}_\mathcal O(\{\ast\}_{i \in \{1,\ldots,n\}} , \ast)\) for the unique space of \(n\)-ary multimorphisms in \(\mathcal O\).