In Section 18.1, we introduced various classes of \(\infty \)-operads with additional algebraic structure: pointed, semiadditive, additive, and stable. The goal of this section is to show that each of the four fully faithful inclusions \[ \Op _{\infty }^{\pt } \hookrightarrow \Op _{\infty }^*, \qquad \Op _{\infty }^{\sadd } \hookrightarrow \Op _{\infty }^{\mathrm {prod}}, \qquad \Op _{\infty }^{\add } \hookrightarrow \Op _{\infty }^{\mathrm {prod}} \qquadtext { and } \Op _{\infty }^{\st } \hookrightarrow \Op _{\infty }^{\lex } \] admits a right adjoint. These right adjoints can be thought of as cofreely adding the respective algebraic structure to an \(\infty \)-operad. The constructions rely on the Day convolution machinery of Chapter 16 and its extension to arbitrary operadic targets in Section 18.3.
We treat the case of spectrum objects in detail; the pointed, semiadditive, and additive cases follow the same pattern and are treated more briefly at the end of the section.
18.4.1 Spectrum objects
Construction 18.4.1. Let \(\Oo \) be an \(\infty \)-operad that admits finite operadic limits. Consider the \(\infty \)-category \(\An _*^{\fin }\) equipped with the smash product monoidal structure from Definition 16.3.5. We define \[ \oSp (\Oo ) \quad \subseteq \quad \oDay (\Mm _{\An _*^{\fin }}, \Oo ) \] to be the full suboperad whose colors are those functors \(F\colon \An _*^{\fin } \to \Oo _{\lra {1}}\) that are reduced and excisive. Note that \[ \oSp (\Oo )_{\lra {1}} \simeq \Sp ^{\exc }(\Oo _{\lra {1}}) \] by construction; through Proposition 16.5.10, we identify this underlying \(\infty \)-category with the abstract stabilization \(\Sp (\Oo _{\lra {1}})\). The monoidal unit \(S^0\) defines an operad map \(\Comm \to \Mm _{\An _*^{\fin }}\), and restricting the evaluation map along it defines a morphism of \(\infty \)-operads \[ \Omega ^\infty \colon \oSp (\Oo ) \to \Oo . \]
Lemma 18.4.2. Let \(D\) be a small symmetric monoidal \(\infty \)-category and equip \(P := \PSh (D)\) with the Day convolution symmetric monoidal structure. Then \(\oSp (\Mm _P)\) is a stable \(\infty \)-operad.
Proof. Put \(A := \An _*^{\fin }\). The Day convolution structure on \(P\) identifies \(\Mm _P\) with \(\oDay (\Mm _{D\catop },\Mm _{\An })\). Thus, using the universal property of Day convolution twice, \(\oDay (\Mm _A,\Mm _P)\) identifies with \[ \oDay (\Mm _{D\catop },\oDay (\Mm _A,\Mm _{\An })). \] Under this identification, \(\oSp (\Mm _P)\) is obtained by restricting the inner \(\oDay (\Mm _A,\Mm _{\An })\) to reduced excisive functors, using Lemma 18.3.2. By Theorem 16.6.1 and the construction of the symmetric monoidal localization there, this restriction is precisely the multimorphism operad \(\Mm _{\Sp }\). Hence \[ \oSp (\Mm _P) \simeq \oDay (\Mm _{D\catop },\Mm _{\Sp }). \] The latter is the multimorphism operad of \(\Fun (D\catop ,\Sp )\) with Day convolution. Since \(\Sp \) is stable and its tensor product preserves colimits in each variable, so does the Day tensor product: for a fixed functor in all but one variable, it is computed as a left Kan extension of a pointwise tensor product, and both operations preserve colimits in the remaining variable. It follows that \(\oSp (\Mm _P)\) is stable. □
Proposition 18.4.3. Let \(\Oo \) be an \(\infty \)-operad that admits finite operadic limits. Then \(\oSp (\Oo )\) is stable.
Proof. Choose a universe in which \(D := \Env (\Oo )\) is small, let \(P := \PSh (D)\) be the corresponding presheaf category, and suppress the universe decoration from the notation. Let \[ \iota \colon \Oo \hookrightarrow \Mm _P \] be the multiplicative Yoneda embedding of Proposition 18.2.3. By Lemma 18.3.2, the operad \(\oSp (\Oo )\) is the full suboperad of \(\oSp (\Mm _P)\) spanned by those reduced excisive functors \(\An _*^{\fin } \to P\) whose values lie in the essential image of \(\Oo _{\lra {1}} \to P\).
By Corollary 18.2.4, the essential image of \(\Oo _{\lra {1}} \to P\) is closed under finite limits. Since \(\iota \) is fully faithful and preserves finite limits, it also reflects them. Thus reducedness and excisiveness can be checked either in \(\Oo _{\lra {1}}\) or after applying \(\iota \). Hence the essential image of the fully faithful functor \[ \Sp ^{\exc }(\Oo _{\lra {1}}) \to \Sp ^{\exc }(P) \] is closed under finite limits, since limits of reduced excisive functors are computed pointwise. The source and target of this functor are stable by Proposition 16.5.7, and a finite-limit-preserving functor between stable \(\infty \)-categories is exact. It follows that this essential image is closed under finite colimits as well. Since \(\oSp (\Mm _P)\) is stable by Lemma 18.4.2, the claim follows from Lemma 18.1.8. □
18.4.2 The end calculation
The remaining point is to show that if \(\Oo \) is already stable, then evaluation at \(S^0\) is an equivalence of \(\infty \)-operads between \(\oSp (\Oo )\) and \(\Oo \). The underlying categorical statement is Lemma 16.5.8; what is missing is full faithfulness on multimorphism animae. This is where the end formula from Theorem 18.3.5 enters.
Lemma 18.4.4. Let \(T\colon (\An _*^{\fin })\catop \times \An _*^{\fin } \to \An \) be a functor with the following properties:
- (1)
-
For every \(L \in \An _*^{\fin }\), the functor \(T(-,L)\colon (\An _*^{\fin })\catop \to \An \) sends finite colimits in \(\An _*^{\fin }\) to limits.
- (2)
-
For every \(K \in \An _*^{\fin }\), the functor \(T(K,-)\colon \An _*^{\fin } \to \An \) is reduced and excisive.
Then the projection from the end to the component at \(S^0\), \[ \int _{K \in \An _*^{\fin }} T(K,K) \to T(S^0,S^0), \] is an equivalence.
Proof. Let \(A := \An _*^{\fin }\) and let \(j\colon A \hookrightarrow \An _*\) be the inclusion. Also let \[ W\colon \An _* \to \PSh (A), \qquad X \mapsto \Hom _{\An _*}(-,X)\vert _A \] be the restricted Yoneda embedding from Lemma 22.3.9. By currying, the functor \(T\) determines a functor \[ \widetilde {T}\colon A \to \PSh (A), \qquad L \mapsto T(-,L). \] By assumption (1) and Lemma 22.3.9, the functor \(\widetilde {T}\) factors uniquely as \(W\circ T'\) for a functor \(T'\colon A \to \An _*\). Since \(W\) is fully faithful and preserves and reflects limits, and since limits in presheaf categories are computed pointwise, assumption (2) implies that \(T'\) is reduced and excisive.
Let \(Y_A\colon A \to \PSh (A)\) denote the Yoneda embedding. Using the end formula for natural transformations, the Yoneda lemma, and the identity \(W\circ j\simeq Y_A\), we obtain equivalences \[ \int _{K \in A}T(K,K) \simeq \Nat (Y_A,\widetilde {T}) \simeq \Nat (j,T'). \] Since \(T'\) is reduced and excisive, Lemma 16.5.14 gives \[ \Nat (j,T') \simeq \Nat (P_1j,T') \simeq \Hom _{\An _*}(S^0,T'(S^0)). \] Finally, by the definition of \(W\) we have \[ \Hom _{\An _*}(S^0,T'(S^0)) \simeq W(T'(S^0))(S^0) \simeq T(S^0,S^0). \] Tracing through the construction identifies this composite with projection from the end to the component at \(S^0\). □
Proposition 18.4.5. If \(\Oo \) is stable, then the evaluation map \[ \Omega ^\infty \colon \oSp (\Oo ) \to \Oo \] is an equivalence of \(\infty \)-operads.
Proof. On underlying \(\infty \)-categories this is precisely Lemma 16.5.8. By Lemma 14.1.8, it remains to prove that \(\Omega ^\infty \) is fully faithful.
Let \(I\) be a finite set, and let \(\{F_i\colon \An _*^{\fin } \to \Oo _{\lra {1}}\}_{i \in I}\) and \(G\colon \An _*^{\fin } \to \Oo _{\lra {1}}\) be reduced excisive functors. By Theorem 18.3.5, the source of the induced map on multimorphism animae is \[ \int _{\{K_i\}_{i \in I}\in (\An _*^{\fin })^I} \Oo (\{F_i(K_i)\}_{i \in I};G(\bigwedge \nolimits _{i \in I}K_i)). \] By naturality of Theorem 18.3.5 in the evaluation operad map and the identification \(\bigwedge _{i\in I}S^0 \simeq S^0\), the map induced by \(\Omega ^\infty \) is the projection from this end to the component where all \(K_i\) are equal to \(S^0\).
We prove a slightly stronger statement by induction on the cardinality of \(I\): the same projection is an equivalence after adjoining any fixed finite tuple of spectator colors \(z_1,\dots ,z_m\) to the inputs of every multimorphism anima. For \(I=\emptyset \) there is nothing to prove. Otherwise choose \(i_0 \in I\) and put \(I' := I\setminus \{i_0\}\). By Fubini for ends, the relevant end is the end over \(K \in \An _*^{\fin }\) of \(T(K,K)\), where for \(K,L\in \An _*^{\fin }\) we define \[ T(K,L):= \int _{\{K_i\}_{i \in I'}\in (\An _*^{\fin })^{I'}} \Oo (z_1,\dots ,z_m,F_{i_0}(K),\{F_i(K_i)\}_{i \in I'};G(L\wedge \bigwedge \nolimits _{i \in I'}K_i)). \] We claim that \(T\) satisfies the hypotheses of Lemma 18.4.4. Since \(\Oo \) is stable, a reduced excisive functor \(\An _*^{\fin } \to \Oo _{\lra {1}}\) preserves finite colimits: finite colimits in \(\An _*^{\fin }\) are generated by the initial object and pushouts, and a square in a stable \(\infty \)-category is a pushout if and only if it is a pullback. Therefore \(F_{i_0}\) sends finite colimits in \(\An _*^{\fin }\) to finite operadic colimits in \(\Oo \). Since finite operadic colimits in an input variable are detected by mapping out of them, \(T(-,L)\) sends finite colimits to limits. Similarly, the functor \(L \mapsto G(L \wedge \bigwedge _{i \in I'}K_i)\) is reduced and sends pushout squares to finite limits in \(\Oo _{\lra {1}}\), and these limits are operadic. Thus \(T(K,-)\) is reduced and excisive. Ends preserve the relevant limits throughout, and the fixed spectator colors do not affect either verification.
Applying Lemma 18.4.4 collapses the \(i_0\)-variable to \(S^0\). The stronger induction hypothesis, with \(F_{i_0}(S^0)\) adjoined to the spectator colors, now collapses the remaining variables and gives an equivalence from the displayed end to \[ \Oo (\{F_i(S^0)\}_{i \in I};G(S^0)), \] which is the multimorphism anima in \(\Oo \) between the images under \(\Omega ^\infty \). This proves full faithfulness. □
Theorem 18.4.6 (Stabilization of operads). Let \(\Oo \) be an \(\infty \)-operad that admits finite operadic limits. Then the \(\infty \)-operad \(\oSp (\Oo )\) is stable, and the resulting functor \[ \oSp \colon \Op _{\infty }^{\lex } \to \Op _{\infty }^{\st } \] is right adjoint to the inclusion functor.
Proof. The first claim is Proposition 18.4.3. We now prove the adjunction. Postcomposition with a map in \(\Op _{\infty }^{\lex }\) preserves reduced excisive functors, since its color functor preserves finite limits. Thus the construction of \(\oSp (\Oo )\) is functorial in \(\Oo \in \Op _{\infty }^{\lex }\), and evaluation at \(S^0\) gives a natural transformation \[ \Omega ^\infty \colon \oSp \to \id _{\Op _{\infty }^{\lex }}. \] By the dual of the recognition criterion for Bousfield localizations from Proposition 21.8.9, applied in \((\Op _{\infty }^{\lex })\catop \), it is enough to show that \(\Omega ^\infty _{\Oo }\) is an equivalence whenever \(\Oo \) is stable, and that \[ \oSp (\Omega ^\infty _{\Oo })\colon \oSp (\oSp (\Oo )) \to \oSp (\Oo ) \] is an equivalence for every \(\Oo \in \Op _{\infty }^{\lex }\). The first assertion is Proposition 18.4.5.
For the second assertion, observe that \(\oSp (\oSp (\Oo ))\) may be identified with the full suboperad of \(\oDay (\Mm _{\An _*^{\fin } \times \An _*^{\fin }},\Oo )\) spanned by bifunctors \(\An _*^{\fin } \times \An _*^{\fin } \to \Oo _{\lra {1}}\) which are reduced and excisive in each variable. Under this identification, the map \(\oSp (\Omega ^\infty _{\Oo })\) evaluates in one variable at \(S^0\), while the map \[ \Omega ^\infty _{\oSp (\Oo )}\colon \oSp (\oSp (\Oo )) \to \oSp (\Oo ) \] evaluates in the other variable at \(S^0\). These two maps are conjugate by the symmetry of \(\An _*^{\fin }\times \An _*^{\fin }\). Since \(\oSp (\Oo )\) is stable by the first part of the theorem, Proposition 18.4.5 shows that \(\Omega ^\infty _{\oSp (\Oo )}\) is an equivalence. Hence \(\oSp (\Omega ^\infty _{\Oo })\) is an equivalence as well. □
Recall from Definition 11.1.15 that a symmetric monoidal \(\infty \)-category \(C\) is called stably symmetric monoidal if \(C\) is stable and the tensor product \(- \otimes -\colon C \times C \to C\) is exact in both variables; equivalently, if \(\Mm _C\) is a stable \(\infty \)-operad.
Corollary 18.4.7. Let \(C\) be a symmetric monoidal \(\infty \)-category with finite limits, and assume that \(\oSp (\Mm _C)\) is represented by a symmetric monoidal structure on \(\Sp (C)\). Then \(\Omega ^{\infty }\colon \Sp (C) \to C\) admits a canonical lax symmetric monoidal structure. Furthermore, for every stably symmetric monoidal \(\infty \)-category \(D\), composition with \(\Omega ^{\infty }\) induces an equivalence of \(\infty \)-categories \[ \Omega ^{\infty } \circ - \colon \Fun ^{\otimes \dlax ,\lex }(D,\Sp (C)) \iso \Fun ^{\otimes \dlax ,\lex }(D,C), \] where the superscript \(\lex \) denotes the full subcategories of lax symmetric monoidal functors whose underlying functors preserve finite limits. On the left-hand side this is equivalently the condition that the underlying functor be exact.
Proof. Applying the adjunction of Theorem 18.4.6 to finite-operadic-limit-preserving maps \(\Mm _D\to \Mm _C\) gives the asserted equivalence on maximal animae. The enhancement to an equivalence of \(\infty \)-categories, including non-invertible lax monoidal natural transformations, is [Nikolaus (2016), Corollary 4.13]; it is obtained by applying the functoriality of operadic stabilization to the corresponding operad-map categories. □
Remark 18.4.8. The construction \(\oSp (\Mm _C)\) always produces an \(\infty \)-operad whose underlying \(\infty \)-category is \(\Sp (C)\), but this operad need not be the multimorphism operad of a symmetric monoidal \(\infty \)-category. Thus the representability hypothesis in Corollary 18.4.7 is genuine. It does hold when \(C\) is presentably symmetric monoidal; this will follow from the mode-theoretic construction in Corollary 18.5.5.
18.4.3 Pointing and additivization
The pointed, semiadditive and additive cases work similarly.
Definition 18.4.9. If \(\Oo \) is an \(\infty \)-operad with an operadic terminal object, equip \([1]\) with the symmetric monoidal structure given by the minimum, as used in the proof of Lemma 16.3.3, and denote by \[ \Oo _* \quad \subseteq \quad \oDay (\Mm _{[1]}, \Oo ) \] the full suboperad whose colors are those functors \(F\colon [1] \to \Oo _{\lra {1}}\) satisfying \(F(0) \simeq *\). If in addition \(\Oo \) has finite operadic products, equip \(\Span (\Fin )\) with the symmetric monoidal structure induced by the cartesian product of finite sets, and denote by \[ \oCGrp (\Oo ) \quad \subseteq \quad \oCMon (\Oo ) \quad \subseteq \quad \oDay (\Mm _{\Span (\Fin )}, \Oo ) \] the full suboperads whose colors are those functors \(F\colon \Span (\Fin ) \to \Oo _{\lra {1}}\) that are commutative groups or commutative monoids in \(\Oo _{\lra {1}}\), respectively.
Notice that this is not the cartesian monoidal structure on \(\Span (\Fin )\): its categorical products are disjoint unions by Lemma 13.3.8. The cartesian product of finite sets is the monoidal structure used in the Day convolution description of Proposition 16.4.1.
It is immediate from the definitions that the underlying \(\infty \)-categories of these \(\infty \)-operads are given by \[ (\Oo _*)_{\lra {1}} \simeq (\Oo _{\lra {1}})_*, \qquad \oCMon (\Oo )_{\lra {1}} \simeq \CMon (\Oo _{\lra {1}}) \qquadtext { and } \oCGrp (\Oo )_{\lra {1}} \simeq \CGrp (\Oo _{\lra {1}}), \] justifying the notation.
Lemma 18.4.10 (Day convolution on pointed objects). Let \(C\) be an \(\infty \)-category with finite products and finite colimits such that its cartesian product preserves finite colimits separately in both variables. Then there is a natural equivalence \[ (\Mm _{(C,\times )})_* \simeq \Mm _{(C_*,\wedge )}. \]
Proof. The operad \((\Mm _{(C,\times )})_*\) is the full suboperad of the Day convolution operad on \(\Ar (C)\) spanned by the arrows with source the terminal object. The monoidal localization \[ \cofib \colon \Ar (C)^{\square }\longrightarrow (C_*,\wedge ) \] of Lemma 16.3.4 identifies this full suboperad with the multimorphism operad of its local objects. □
Lemma 18.4.11. If \(\Oo \) is pointed, then evaluation at the target defines an equivalence of \(\infty \)-operads \[ \ev _1\colon \Oo _* \to \Oo . \]
Proof. On underlying \(\infty \)-categories, evaluation identifies the category of arrows from the zero object with \(\Oo _{\lra {1}}\). It remains to prove full faithfulness on multimorphism animae.
Let \(\{F_i\}_{i\in I}\) and \(G\) be colors of \(\Oo _*\). By Theorem 18.3.5, their multimorphism anima is an end over \([1]^I\). We collapse its variables one at a time. After applying Fubini, the variable to be collapsed appears as the end of a bifunctor \(T\colon [1]\catop \times [1]\to \An \), allowing arbitrary fixed spectator colors. This end is the pullback \[ T(0,0)\times _{T(0,1)}T(1,1). \] Both \(T(0,0)\) and \(T(0,1)\) are contractible because \(F_i(0)\) is the zero object and hence operadic initial. Thus this pullback is equivalent to \(T(1,1)\). Fubini for ends and induction, with the values \(F_i(1)\) already obtained treated as spectator colors, identify the original end with \[ \Oo (\{F_i(1)\}_{i\in I};G(1)). \] By naturality of the end formula, this equivalence is the map induced by evaluation at \(1\). □
For the semiadditive and additive cases below, the coherence of the universal properties is supplied by Nikolaus (2016), Theorem 5.7; we include the argument that identifies the resulting operads and their underlying algebraic structure. The pointed case will be proved internally.
Theorem 18.4.12 (Additivization of operads, [Nikolaus (2016), Theorem 5.7]). Let \(\Oo \) be an object in \(\Op _{\infty }^*\) or \(\Op _{\infty }^{\mathrm {prod}}\) respectively. Then the operads \(\Oo _*\), \(\oCMon (\Oo )\) and \(\oCGrp (\Oo )\) are pointed, semiadditive, and additive, respectively. Moreover, the resulting functors \[ (-)_*\colon \Op _{\infty }^* \to \Op _{\infty }^{\pt }, \qquad \oCMon \colon \Op _{\infty }^{\mathrm {prod}} \to \Op _{\infty }^{\sadd } \qquadtext { and } \oCGrp \colon \Op _{\infty }^{\mathrm {prod}} \to \Op _{\infty }^{\add } \] are right adjoint to the respective inclusion functors.
Proof. We first prove the pointed case. Choose a universe in which \(\Env (\Oo )\) is small and let \(P\) be the corresponding presheaf category. The multiplicative Yoneda embedding and Lemma 18.3.2 give full embeddings \[ \Oo _* \hookrightarrow (\Mm _{\Env (\Oo )})_* \hookrightarrow (\Mm _P)_* \] where the last target is understood in the chosen universe. By Lemma 18.4.10, this last operad is the multimorphism operad of the pointed presheaf category and is therefore pointed by Example 18.1.7. The suboperad \(\Oo _*\) contains its zero object, since the multiplicative Yoneda embedding preserves the operadic terminal object of \(\Oo \). It follows from Lemma 18.1.8(1) that \(\Oo _*\) is pointed.
Evaluation at the target gives a natural operad map \(\Oo _*\to \Oo \). If \(\Oo \) is pointed, this map is an equivalence by Lemma 18.4.11. Applying the pointed-object construction a second time gives the same conclusion because \(\Oo _*\) is pointed. The dual of Proposition 21.8.9, applied in \((\Op _{\infty }^*)\catop \), therefore identifies \((-)_*\) as right adjoint to the inclusion of pointed operads.
For commutative monoids and groups, choose a universe in which \(\Oo \) is small and apply the multiplicative Yoneda embedding to regard \(\Oo \) as a full suboperad of the presheaf operad on \(\Env (\Oo )\). In that presheaf category, Proposition 16.4.1 supplies the Day convolution operads of commutative monoids and commutative groups. Their underlying categories are semiadditive and additive by Proposition 5.3.20, and their tensor products preserve colimits separately in both variables. Hence their finite biproducts are operadic. These biproducts are computed pointwise. Since the multiplicative Yoneda image is closed under finite products, the full suboperads consisting of objects valued in that image are closed under these biproducts. They are precisely \(\oCMon (\Oo )\) and \(\oCGrp (\Oo )\), so Lemma 18.1.8(2) shows that the former is semiadditive and the latter additive.
If \(\Pp \) is semiadditive, every color of \(\Pp \) has a commutative-monoid structure supplied by its biproducts, compatibly with all multimorphisms because the biproducts are operadic. If \(\Pp \) is additive, these structures are grouplike. The cited theorem supplies the coherent lifts through \(\oCMon (\Oo )\to \Oo \) and \(\oCGrp (\Oo )\to \Oo \) and identifies them with the right adjoints to the respective inclusions. □
Corollary 18.4.13. Applied to the cartesian multimorphism operad \(\Mm _{(\An ,\times )}\), the four operadic constructions recover the symmetric monoidal structures of Chapter 16: \[ (\Mm _{(\An ,\times )})_* \simeq \Mm _{(\An _*,\wedge )},\qquad \oCMon (\Mm _{(\An ,\times )}) \simeq \Mm _{\CMon (\An )}, \] \[ \oCGrp (\Mm _{(\An ,\times )}) \simeq \Mm _{\CGrp (\An )},\qquad \oSp (\Mm _{(\An ,\times )}) \simeq \Mm _{\Sp }. \]
Proof. The pointed case is Lemma 18.4.10, the semiadditive and additive cases are the symmetric monoidal localizations of Proposition 16.4.1, and the stable case is Theorem 16.6.1. □
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