Let \(\Oo \) be a colored operad and let \(C\) be a symmetric monoidal 1-category. Recall that an \(\Oo \)-algebra in \(C\) is defined as an operad map \(\Oo \to \Mm _C\) into the multimorphism operad of \(C\), whose multimorphisms are given by \[ \Mm _C(\{x_i\}_{i \in I};y) \quad := \quad \Hom _C(\bigotimes _{i \in I} x_i,y); \] see Definition 12.2.10. The goal of this section is to define the analogous notion of algebras over operads in the \(\infty \)-categorical setting.
14.2.1 Symmetric monoidal \(\infty \)-categories as \(\infty \)-operads
We start by constructing the multimorphism operad \(\Mm _C\) of a symmetric monoidal \(\infty \)-category \(C\). Recall from Definition 8.1.1 that a symmetric monoidal \(\infty \)-category is a commutative monoid in \(\Cat _{\infty }\), i.e. a finite-product-preserving functor \(C\colon \Span (\Fin ) \to \Cat _{\infty }\). These naturally form an \(\infty \)-category that we denote by \[ \Cat ^{\otimes }_{\infty } \quad := \quad \CMon (\Cat _{\infty }). \]
Since the opposite-category involution \((-)\catop \colon \Cat _{\infty }\to \Cat _{\infty }\) preserves finite products, postcomposition with it defines an involution on \(\Cat ^{\otimes }_{\infty }\). For a symmetric monoidal \(\infty \)-category \((C,\otimes )\), its image under this involution is called the opposite symmetric monoidal \(\infty \)-category and is denoted \((C,\otimes )\catop \).
Given symmetric monoidal \(\infty \)-categories \(C\) and \(D\), write \(\Map ^{\otimes }(C,D)\) for their hom anima in \(\Cat ^{\otimes }_{\infty }\).
Example 14.2.1 (Terminal category). The terminal \(\infty \)-category \(*\) admits a unique symmetric monoidal structure, given by the constant functor \(\Span (\Fin ) \to \Cat _{\infty }, S \mapsto *\).
Example 14.2.2 (Pointwise monoidal structure). If \(C\) is a symmetric monoidal \(\infty \)-category and \(I\) is an \(\infty \)-category, then \(\Fun (I,C)\) admits a symmetric monoidal structure, called the pointwise monoidal structure. It is given by the composite \[ \Span (\Fin ) \xrightarrow {(C,\otimes )} \Cat _{\infty } \xrightarrow {\Fun (I,-)} \Cat _{\infty }. \] This is functorial in both \(I\) and \(C\), resulting in a functor \[ \Fun (-,-)\colon \Cat _{\infty }\catop \times \Cat _{\infty }^{\otimes } \to \Cat _{\infty }^{\otimes }. \]
The symmetric monoidal functors and their monoidal natural transformations themselves form an \(\infty \)-category:
Definition 14.2.3. Let \(C\) and \(D\) be symmetric monoidal \(\infty \)-categories. We define \(\Fun ^{\otimes }(C,D)\) to be the \(\infty \)-category whose associated complete Segal anima is \[ N(\Fun ^{\otimes }(C,D))_n := \Map ^{\otimes }(C,\Fun ([n],D)), \] where \(\Fun ([n],D)\) carries the pointwise symmetric monoidal structure. This is a complete Segal anima because \(\Map ^{\otimes }(C,-)\) preserves limits; see Proposition 1.8.6.
Given a symmetric monoidal \(\infty \)-category \(C\colon \Span (\Fin ) \to \Cat _{\infty }\), we may unstraighten this to a cocartesian fibration \(p_C\colon C^{\otimes } \to \Span (\Fin )\). The next result implies that this is always an \(\infty \)-operad.
Lemma 14.2.4. Let \(p_{\Oo }\colon \Oo ^{\otimes } \to \Span (\Fin )\) be a cocartesian fibration. Then \((\Oo ^{\otimes },p_{\Oo })\) is an \(\infty \)-operad if and only if the cocartesian straightening of \(p_{\Oo }\) \[ \Str ^{\cc }(p_{\Oo })\colon \Span (\Fin ) \to \Cat _{\infty } \] preserves finite products.
Proof. We check conditions (i)–(iii) of Proposition 14.1.9. Condition (i) is part of the assumption on \(p_{\Oo }\). Since the spans \(\rho _i\colon I \hookleftarrow \{i\} \xrightarrow {=} \{i\}\) exhibit \(I\) as the product of the one-element sets in \(\Span (\Fin )\), condition (ii) is equivalent to preservation of finite products by \(\Str ^{\cc }(p_{\Oo })\). Under this assumption, Reference ? of [Cisinski et al. (2026)] shows that the cocartesian lifts of the maps \(\rho _i\) exhibit every object over \(I\) as the product of its components, which is condition (iii). □
Definition 14.2.5. Given a symmetric monoidal \(\infty \)-category \(C\), we denote its cocartesian unstraightening by \[ C^{\otimes } \quad := \quad \Un ^{\cc }(C\colon \Span (\Fin ) \to \Cat _{\infty }). \] By the previous lemma, the resulting cocartesian fibration \(p_C\colon C^{\otimes } \to \Span (\Fin )\) defines an \(\infty \)-operad, which we will denote by \(\Mm _C := (C^{\otimes },p_C)\) and refer to as the multimorphism operad associated to \(C\). By functoriality of unstraightening, this defines a (non-full) inclusion \[ \Mm \colon \Cat _{\infty }^{\otimes } \hookrightarrow \Op _{\infty }. \] Morphisms in its essential image are studied in the next section.
Lemma 14.2.6. Given \(C \in \Cat _{\infty }^{\otimes }\) and objects \(x_1, \dots , x_n, y \in C^{\simeq }\), there is a natural equivalence \[ \Mm _C((x_1, \dots , x_n);y) \simeq \Hom _C(x_1\otimes \dots \otimes x_n,y). \]
Proof. Let \(\alpha \) denote the active span \(\lra {n}\xleftarrow {=}\lra {n}\to \lra {1}\). Cocartesian transport along \(\alpha \) sends \((x_1,\dots ,x_n)\) to \(x_1\otimes \dots \otimes x_n\). The universal property of this cocartesian lift gives \[ \Hom ^{\alpha }_{C^{\otimes }}((x_1,\dots ,x_n),y) \simeq \Hom _C(x_1\otimes \dots \otimes x_n,y), \] and the left-hand side is the indicated multimorphism anima. □
Observation 14.2.7. Cocartesian transport in \(C^{\otimes }\) is compatible with finite products. More precisely, suppose that \(\alpha _j\colon I_j\to J_j\) are morphisms in \(\Span (\Fin )\) and that \(\phi _j\colon X_j\to Y_j\) are \(p_C\)-cocartesian lifts of \(\alpha _j\). Since the straightening of \(p_C\) preserves finite products, the product \[ \prod _j\phi _j\colon \prod _jX_j\longrightarrow \prod _jY_j \] is a \(p_C\)-cocartesian lift of the product \(\prod _j\alpha _j\). In particular, cocartesian transport along a forward span is computed separately over the elements of its target.
We have the following useful criterion for checking that an \(\infty \)-operad corresponds to a symmetric monoidal \(\infty \)-category:
Lemma 14.2.8. Let \(\Oo \) be an \(\infty \)-operad. Assume that the following two conditions are satisfied:
- (1)
-
For every object \(\{x_i\}_{i \in I}\) of \(\Oo ^{\otimes }\) the functor \[ \Oo (\{x_i\}_{i \in I}; -)\colon \Oo _{\lra {1}} \to \An \] is corepresentable, in the sense that there exists a multimorphism \(\phi \colon \{x_i\}_{i \in I} \to y\) such that for every other color \(z \in \Oo ^{\simeq }\) precomposition with \(\phi \) induces an equivalence \[ \Hom _{\Oo _{\lra {1}}}(y,z) \iso \Oo (\{x_i\}_{i \in I}; z). \] This condition uniquely determines \(y\), and we will denote it by \(\bigotimes _{i \in I} x_i\).
- (2)
-
For every morphism \(f\colon I \to J\) of finite sets, the canonical comparison map \[ \bigotimes _{i \in I} x_i \to \bigotimes _{j \in J} \bigotimes _{i \in f^{-1}(j)} x_i \] is an equivalence.
Then \(p_{\Oo }\colon \Oo ^{\otimes } \to \Span (\Fin )\) is a cocartesian fibration, corresponding to a symmetric monoidal structure on the underlying \(\infty \)-category \(\Oo ^{\otimes }_{\lra {1}}\).
Proof. We need to produce cocartesian lifts of morphisms in \(\Span (\Fin )\). Cocartesian lifts of backwards morphisms exist by the defining property of \(\infty \)-operads, so it suffices to produce cocartesian lifts for active spans \(\alpha = (I \xleftarrow {=} I \xrightarrow {f} J)\). Let \(X = \{x_i\}_{i \in I}\) be an object in \(\Oo ^{\otimes }_I\). Applying the assumption on \(\Oo \) to each \(X_j := \{x_i\}_{i \in f^{-1}(j)}\), we may find multimorphisms \(\phi _j\colon \{x_i\}_{i \in f^{-1}(j)} \to y_j\) in \(\Oo \) corepresenting the functor \(\Oo (\{x_i\}_{i \in f^{-1}(j)}; -)\colon \Oo _{\lra {1}} \to \An \). The multimorphisms \(\phi _j\) define a morphism \(\phi \colon X \to Y := \{y_j\}_{j \in J}\) in \(\Oo ^{\otimes }\). We wish to show that \(\phi \) is a cocartesian lift of \(\alpha \).
To this end, consider any other span \(\beta = (J \xleftarrow {g} K \xrightarrow {h} L)\) of finite sets, and consider an object \(Z = \{z_l\}_{l \in L}\) of \(\Oo ^{\otimes }_L\). We need to show that precomposition with \(\phi \) induces an equivalence \[ - \circ \phi \colon \Hom _{\Oo ^{\otimes }}^{\beta }(Y,Z) \iso \Hom _{\Oo ^{\otimes }}^{\beta \circ \alpha }(X,Z). \] Since \(Z\) is a product in \(\Oo ^{\otimes }\) of the one-element tuples \((z_l)\), we may assume that \(L = \lra {1}\), so that there is only a single color \(z := z_1\). Using the inert morphisms \(\{y_j\} \to \{y_{g(k)}\}_{k \in K}\) and \(\{x_i\}_{(i,k) \in I \times _J K}\), we must equivalently show that composition with the map \(\phi \) induces an equivalence \[ - \circ \phi \colon \Oo (\{y_{g(k)}\}_{k \in K}; z) \iso \Oo (\{x_i\}_{(i,k) \in I \times _J K}; z). \] Now, applying condition (1) another time to the families \(\{y_{g(k)}\}_{k \in K}\) and \(\{x_i\}_{(i,k) \in I \times _J K}\), both sides are corepresented by objects \(\bigotimes _{k \in K} y_{g(k)}\) and \(\bigotimes _{(i,k) \in I \times _J K} x_i\), respectively. But since \(y_{g(k)} = \bigotimes _{i \in f^{-1}(g(k))} x_i\), the claim now follows from the equivalence \[ \bigotimes _{(i,k) \in I \times _J K} x_i \iso \bigotimes _{k \in K} \bigotimes _{i \in f^{-1}(g(k))} x_i \] obtained by applying condition (2) to the projection map \(I \times _J K \to K\). Under the two corepresenting equivalences, the map induced by composition with \(\phi \) is precisely precomposition with this comparison map, by the definition of the latter through operadic composition with the multimorphisms \(\phi _j\). Thus \(\phi \) is cocartesian.
We have now produced cocartesian lifts of all active spans. Together with the inert lifts and the factorization of Remark 14.1.12, these make \(p_{\Oo }\) a cocartesian fibration. Since \(\Oo \) is already an \(\infty \)-operad, Lemma 14.2.4 shows that its straightening preserves finite products, and hence defines the asserted symmetric monoidal structure. □
14.2.2 Algebras over \(\infty \)-operads
Definition 14.2.9. Let \(\Oo \) be an \(\infty \)-operad and let \(C\) be a symmetric monoidal \(\infty \)-category. An \(\Oo \)-algebra in \(C\) is an operad morphism \(\Oo \to \Mm _C\). We write \[ \Alg _{\Oo }(C) := \Fun _{\Op _{\infty }}(\Oo , \Mm _C) \] for the \(\infty \)-category of \(\Oo \)-algebras in \(C\).
Example 14.2.10 (Commutative/associative algebras). Algebras over the operads \(\Comm \) and \(\Assoc \) are referred to as commutative algebras and associative algebras, respectively, and their \(\infty \)-categories are denoted by \[ \CAlg (C) := \Alg _{\Comm }(C) \qquadtext { and } \Alg (C) := \Alg _{\Assoc }(C). \]
Example 14.2.11 (Trivial algebras). Let \(D\) be an \(\infty \)-category. Applying Lemma 14.1.20 to the multimorphism operad \(\Mm _C\), we see that algebras over the trivial operad \(\Triv _D\) correspond to functors \(D \to C\): \[ \Alg _{\Triv _D}(C) \iso \Fun (D,C). \] In particular, taking \(D = *\) shows that a \(\Triv \)-algebra is simply an object of \(C\): \[ \Alg _{\Triv }(C) \iso C. \] Taking \(D = \emptyset \) shows that the \(\infty \)-category of \(\Triv _{\emptyset }\)-algebras is terminal.
Lemma 14.2.12 (Algebras in pointwise monoidal structure). Let \(C\) be a symmetric monoidal \(\infty \)-category, let \(I\) be an \(\infty \)-category, and let \(\Oo \) be an \(\infty \)-operad. Then an \(\Oo \)-algebra in \(\Fun (I,C)\) with the pointwise monoidal structure is the same data as a functor \(I \to \Alg _{\Oo }(C)\): there is an equivalence of \(\infty \)-categories \[ \Alg _{\Oo }(\Fun (I,C)) \iso \Fun (I,\Alg _{\Oo }(C)). \]
Proof. The pointwise symmetric monoidal structure of Example 14.2.2 is classified by the composite \[ \Span (\Fin )\xrightarrow {\Str ^{\cc }(p_C)} \Cat _{\infty }\xrightarrow {\Fun (I,-)}\Cat _{\infty }. \] By Proposition 23.2.4, its total category is \(\Span (\Fin )\times _{\Fun (I,\Span (\Fin ))} \Fun (I,C^{\otimes })\). Consequently, before imposing the finite-product condition, a functor over \(\Span (\Fin )\) from \(\Oo ^{\otimes }\) to this total category amounts to a functor \[ \Oo ^{\otimes }\longrightarrow \Fun (I,C^{\otimes }) \] whose composite with \(\Fun (I,p_C)\) is the constant \(I\)-diagram on \(p_{\Oo }\). Currying identifies these with functors \[ I\longrightarrow \Fun _{/\Span (\Fin )}(\Oo ^{\otimes },C^{\otimes }). \] Under this identification, preservation of finite products is pointwise in \(I\): the functor into the pointwise multimorphism operad is an operad map precisely when the corresponding functor \(\Oo ^{\otimes }\to C^{\otimes }\) is an operad map at every object of \(I\). Thus the equivalence restricts to the asserted equivalence of algebra categories. □
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