The symmetric monoidal categories and transition functors constructed in Chapter 16 fit into a sequence of increasingly linear settings: \[ (\An ,\times ) \xrightarrow {(-)_+} (\An _*,\wedge ) \xrightarrow {F^{\CMon }} (\CMon (\An ),\otimes ) \xrightarrow {(-)^{\grp }} (\CGrp (\An ),\otimes ) \xhookrightarrow {\bB ^\infty } (\Sp ,\otimes ). \] We have constructed the monoidal structures and transition functors directly. We now explain their common universal property. The language of modes packages pointedness, semiadditivity, additivity, and stability as instances of the same kind of structure.
Recall from Section 22.4 the symmetric monoidal \(\infty \)-category \(\PrL \) of presentable \(\infty \)-categories and colimit-preserving functors. Its unit is \(\An \), and its tensor product is characterized by equivalences \[ \FunL (C\otimes D,E) \simeq \Fun ^{\mathrm {L},\mathrm {L}}(C\times D,E), \] where the right-hand side is the full subcategory of functors that preserve small colimits separately in both variables. By Remark 22.5.2, a commutative algebra object in \(\PrL \) is equivalently a presentably symmetric monoidal \(\infty \)-category.
Definition 18.5.1 (Mode). A mode is a commutative algebra \(M\in \CAlg (\PrL )\) whose multiplication \[ M\otimes M\longrightarrow M \] is an equivalence. A presentable \(\infty \)-category \(C\) is called \(M\)-local if the unit of \(M\) induces an equivalence \[ C\simeq \An \otimes C\longrightarrow M\otimes C. \]
The terminology is taken from [Carmeli et al. (2021), Section 5]. We only need the following basic formal properties of modes, which are instances of the general theory of idempotent commutative algebras.
Proposition 18.5.2 (Modes as properties, [Lurie (2017), Propositions 4.8.2.4, 4.8.2.9 and 4.8.2.10]). Let \(M\) be a mode.
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A presentable \(\infty \)-category admits an \(M\)-module structure if and only if it is \(M\)-local, and such a structure is unique.
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Tensoring with \(M\) defines a Bousfield localization \[ M\otimes -\colon \PrL \longrightarrow (\PrL )^{M\text {-}\mathrm {loc}}, \] where \((\PrL )^{M\text {-}\mathrm {loc}}\) is the full subcategory of \(M\)-local presentable \(\infty \)-categories.
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If \(D\) is \(M\)-local, then precomposition with the unit \(C\to M\otimes C\) induces an equivalence \[ \FunL (M\otimes C,D)\iso \FunL (C,D). \]
Thus a mode classifies a property of presentable \(\infty \)-categories which, whenever it exists, carries a unique coherent structure and can be imposed by a universal localization. The modes relevant here are the following.
Theorem 18.5.3 (The basic modes, [Gepner et al. (2015), Sections 3--5]). The presentably symmetric monoidal \(\infty \)-categories \[ (\An _*,\wedge ), \qquad (\CMon (\An ),\otimes ), \qquad (\CGrp (\An ),\otimes ), \qquad (\Sp ,\otimes ) \] are modes. For every presentable \(\infty \)-category \(C\), there are natural equivalences \[ \begin {aligned} \An _*\otimes C &\simeq C_*, & \CMon (\An )\otimes C &\simeq \CMon (C), \\ \CGrp (\An )\otimes C &\simeq \CGrp (C), & \Sp \otimes C &\simeq \Sp (C). \end {aligned} \] Consequently, these four modes classify the properties recorded in Table 18.1. The functors in the sequence at the beginning of this section are morphisms of modes, meaning morphisms in \(\CAlg (\PrL )\).
Mode Local presentable \(\infty\)-categories \(\An_*\) Pointed \(\CMon(\An)\) Semiadditive \(\CGrp(\An)\) Additive \(\Sp\) Stable
Proposition 18.5.4 (Multiplicative universal property of commutative monoids). Let \(C\) and \(D\) be presentably symmetric monoidal \(\infty \)-categories, and assume that \(D\) is semiadditive. Restriction along the free commutative monoid functor induces an equivalence \[ \Fun ^{\mathrm {L},\otimes }\bigl (\CMon (C),D\bigr ) \iso \Fun ^{\mathrm {L},\otimes }(C,D) \] of \(\infty \)-categories.
Proof. Set \(M:=\CMon (\An )\). By Theorem 18.5.3, \(M\) is a mode and there is a natural equivalence \[ M\otimes C\simeq \CMon (C). \] Under this equivalence, the localization unit \(C\simeq \An \otimes C\to M\otimes C\) is the free semiadditive functor \(C\to \CMon (C)\). The symmetric monoidal refinement of this functor is the one constructed from Day convolution in Proposition 16.4.1.
Since \(D\) is semiadditive, it is \(M\)-local by Theorem 18.5.3. The localization \(M\otimes -\) of Proposition 18.5.2 is induced by the idempotent commutative algebra \(M\), so it restricts to a localization on commutative algebra objects of \(\PrL \). Its universal property gives an equivalence \[ \Fun ^{\mathrm {L},\otimes }(M\otimes C,D) \iso \Fun ^{\mathrm {L},\otimes }(C,D). \] Using the displayed identification of \(M\otimes C\) with \(\CMon (C)\) gives the result. Since the argument takes place in commutative algebra objects of \(\PrL \), it supplies the full symmetric monoidal coherence rather than only the individual tensor comparisons. □
Corollary 18.5.5 (Presentable operadic stabilization, [Nikolaus (2016), Proposition 4.9]). Let \(C\) be a presentably symmetric monoidal \(\infty \)-category. Then \(\Sp (C)\) admits a canonical presentably symmetric monoidal structure, and the operadic stabilization \(\oSp (\Mm _C)\) is represented by its multimorphism operad: \[ \oSp (\Mm _C)\simeq \Mm _{\Sp (C)}. \] In particular, the representability hypothesis in Corollary 18.4.7 is automatic in the presentable case.
Proof. The presentably symmetric monoidal structure on \(C\) is a commutative algebra object of \(\PrL \). Tensoring it with the commutative algebra \(\Sp \) gives another commutative algebra \[ \Sp \otimes C\in \CAlg (\PrL ). \] By Theorem 18.5.3, its underlying presentable \(\infty \)-category is naturally equivalent to \(\Sp (C)\). This equips \(\Sp (C)\) with a presentably symmetric monoidal structure. The unit map \(\An \to \Sp \) of the mode induces a strongly monoidal stabilization functor \(C\simeq \An \otimes C\to \Sp \otimes C\), whose right adjoint \(\Omega ^\infty \colon \Sp (C)\to C\) is lax symmetric monoidal.
Under the reduced-excisive model of stabilization, the resulting multimorphism operad is the full suboperad of the Day convolution operad on \(\Fun (\An _*^{\fin },C)\) spanned by the reduced excisive functors. This is exactly the construction of \(\oSp (\Mm _C)\) from Section 18.4, giving the displayed equivalence. □
The classification says, for example, that the free semiadditive presentable category generated by \(C\) is \(\CMon (C)\), while its free stable presentable category is \(\Sp (C)\). This is different from the cofree universal properties proved in Theorem 18.4.6, Theorem 18.4.12. For example, operadic stabilization gives an equivalence \[ \Fun ^{\otimes \dlax ,\lex }(D,\Sp (C)) \simeq \Fun ^{\otimes \dlax ,\lex }(D,C) \] for a stably symmetric monoidal \(D\), whereas the mode \(\Sp \) gives an equivalence \[ \FunL (\Sp (C),D) \simeq \FunL (C,D) \] for a stable presentable \(D\). The first is a mapping-into, cofree property for finite-limit-preserving functors and does not require presentability; the second is a mapping-out, free property for colimit-preserving functors and only concerns presentable categories. When the functors in question occur as adjoints, the two properties are closely related, but neither statement subsumes the other.
Applied to \(C=\An \), the mode-theoretic classification gives the promised multiplicative mapping-out universal properties. Write \(\Fun ^{\mathrm {L},\otimes }(C,D)\) for the \(\infty \)-category of colimit-preserving symmetric monoidal functors.
Corollary 18.5.6 (Multiplicative universal properties). Let \(D\) be a presentably symmetric monoidal \(\infty \)-category.
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If \(D\) is pointed, then restriction along \(\An \to \An _*\) induces an equivalence \[ \Fun ^{\mathrm {L},\otimes }(\An _*,D)\iso \Fun ^{\mathrm {L},\otimes }(\An ,D). \]
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If \(D\) is semiadditive, then restriction along \(\An \to \CMon (\An )\) induces an equivalence \[ \Fun ^{\mathrm {L},\otimes }(\CMon (\An ),D)\iso \Fun ^{\mathrm {L},\otimes }(\An ,D). \]
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If \(D\) is additive, then restriction along \(\An \to \CGrp (\An )\) induces an equivalence \[ \Fun ^{\mathrm {L},\otimes }(\CGrp (\An ),D)\iso \Fun ^{\mathrm {L},\otimes }(\An ,D). \]
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If \(D\) is stable, then restriction along \(\An \to \Sp \) induces an equivalence \[ \Fun ^{\mathrm {L},\otimes }(\Sp ,D)\iso \Fun ^{\mathrm {L},\otimes }(\An ,D). \]
Proof. Let \(M\) be any of the four modes in Theorem 18.5.3. If \(D\) is \(M\)-local, then the localization universal property of the idempotent commutative algebra \(M\) gives an equivalence \[ \Fun ^{\mathrm {L},\otimes }(M,D) \iso \Fun ^{\mathrm {L},\otimes }(\An ,D) \] by restriction along the unit \(\An \to M\). The four characterizations of \(M\)-local categories in Theorem 18.5.3 give the assertions. In the semiadditive case, this is also the specialization \(C=\An \) of Proposition 18.5.4. □
The tensor unit \(\An \) carries the initial commutative algebra structure in \(\PrL \), so the \(\infty \)-category on the right in each case is contractible. Thus \(\An _*\), \(\CMon (\An )\), \(\CGrp (\An )\), and \(\Sp \) are initial among pointed, semiadditive, additive, and stable presentably symmetric monoidal \(\infty \)-categories, respectively. In particular, the monoidal structures and transition functors constructed above are determined uniquely by these universal properties.
Exercises
Exercise 18.1 (A stable suboperad of dualizable objects). Let \(C\) be a stably symmetric monoidal \(\infty \)-category. Show that the full subcategory \(C^{\mathrm {dual}}\subseteq C\) of dualizable objects is closed under finite limits and finite colimits. Deduce that the full suboperad of \(\Mm _C\) on the dualizable objects is stable.
Exercise 18.2 (A stable full suboperad). Let \(\Sp ^{\fin }\subseteq \Sp \) be the full subcategory of finite spectra. Show that it is closed under finite limits and finite colimits, and that the full suboperad of \(\Mm _{\Sp }\) on the finite spectra is stable.
Exercise 18.3 (The tensor table of the basic modes). Consider the sequence of modes \[ \An _*\longrightarrow \CMon (\An )\longrightarrow \CGrp (\An )\longrightarrow \Sp \] from Theorem 18.5.3. Show that the tensor product of any two of these modes is the one which occurs further to the right in this sequence. Interpret each entry in the resulting tensor table as the composite of the corresponding localizations of presentable \(\infty \)-categories.
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