Recall that an ordinary commutative ring may be equivalently described as a commutative algebra object in the symmetric monoidal category \((\Ab ,\otimes )\) of abelian groups. Given two abelian groups \(A\) and \(B\), their tensor product \(A \otimes B\) is characterized by the property that group homomorphisms \(A\otimes B\to C\) correspond to bilinear maps \(A\times B\to C\). We now use Day convolution to construct analogous tensor products on \(\CGrp (C)\) and \(\CMon (C)\) for any presentably symmetric monoidal \(\infty \)-category \(C\). This gives general notions of commutative ring and semiring objects.
Proposition 16.4.1. Let \(C\) be a presentably symmetric monoidal \(\infty \)-category. The inclusions \[ \CGrp (C)\hookrightarrow \CMon (C)\hookrightarrow \Fun (\Span (\Fin ),C) \] admit left adjoints. Moreover, the full subcategories \(\CMon (C)\) and \(\CGrp (C)\) inherit symmetric monoidal structures for which these left adjoints are symmetric monoidal Bousfield localizations of the Day convolution monoidal structure. Their tensor products preserve small colimits separately in both variables.
The free commutative monoid functor \[ F_C^{\CMon }\colon C\longrightarrow \CMon (C), \] left adjoint to evaluation at \(\lra {1}\in \Span (\Fin )\), is symmetric monoidal.
Proof. The existence of the two left adjoints is the structured-object case of presentability recorded in Proposition 22.2.7.
The category \(\Span (\Fin )\) is small, so Corollary 16.2.8, Proposition 22.5.5 equip \(\Fun (\Span (\Fin ),C)\) with its presentably symmetric monoidal Day convolution structure. It remains to apply the internal-hom criterion of Lemma 14.5.6. By Proposition 22.5.3, the category \(C\) is closed, and the Day convolution internal hom is computed by the end formula \[ \iHom _{\Day }(F,G)(S) \simeq \int _{T\in \Span (\Fin )} \iHom _C\bigl (F(T),G(S\times T)\bigr ). \] For every finite set \(T\), the functor \(-\times T\) preserves finite coproducts of finite sets, which are the finite products in \(\Span (\Fin )\). Consequently, if \(G\) preserves finite products, then so does the functor \(S\mapsto G(S\times T)\). Internal homs and ends preserve limits, so \(\iHom _{\Day }(F,G)\) again lies in \(\CMon (C)\). If \(G\) is grouplike, the shear map of \(\iHom _{\Day }(F,G)\) is obtained by applying the limit-preserving functor \(\iHom _{\Day }(F,-)\) to the shear isomorphism of \(G\), so \(\iHom _{\Day }(F,G)\) is grouplike as well. Thus both local subcategories are closed under internal hom with arbitrary \(F\), and Lemma 14.5.6 gives the claimed symmetric monoidal Bousfield localizations.
Part (3) of Proposition 14.5.3 shows that the localized tensor products preserve small colimits separately in both variables.
It remains to consider the free commutative monoid functor. Let \(\iota \colon *\to \Span (\Fin )\) select the monoidal unit \(\lra {1}\) for the cartesian product of finite sets. Left Kan extension along \(\iota \) is left adjoint to evaluation at \(\lra {1}\), \[ \iota _!\colon C\simeq \Fun (*,C)\rightleftarrows \Fun (\Span (\Fin ),C)\noloc \iota ^*. \] The functor \(\iota \) is symmetric monoidal, and Lemma 16.2.11 shows that \(\iota _!\) is symmetric monoidal. The composite of \(\iota _!\) with the symmetric monoidal reflection onto \(\CMon (C)\) is left adjoint to evaluation at \(\lra {1}\) on commutative monoids, hence is \(F_C^{\CMon }\). It is therefore symmetric monoidal. □
Proposition 16.4.2 (Commutative monoids and groups under adjunctions). Let \[ L\colon C\rightleftarrows D\noloc R \] be an adjunction between presentably symmetric monoidal \(\infty \)-categories, and assume that \(L\) is symmetric monoidal. Postcomposition with \(R\) induces functors on commutative monoids and groups which admit symmetric monoidal left adjoints: \[ \widetilde L\colon \CMon (C)\rightleftarrows \CMon (D)\noloc R_* \qquad \text {and}\qquad \widetilde L\colon \CGrp (C)\rightleftarrows \CGrp (D)\noloc R_*. \] If \(L\) preserves finite products, then the two functors \(\widetilde L\) are given by postcomposition with \(L\).
Proof. The right adjoint \(R\) preserves finite products, so postcomposition with \(R\) preserves commutative monoids. It also preserves grouplike commutative monoids, since it carries shear isomorphisms to shear isomorphisms. This gives the two functors denoted by \(R_*\) in the statement.
Let \(E_C\) and \(E_D\) denote either the commutative-monoid categories or the commutative-group categories of \(C\) and \(D\), respectively. Write \(Q_C\) and \(Q_D\) for the corresponding reflections of Proposition 16.4.1, and \(i_C\) and \(i_D\) for their fully faithful right adjoints. Postcomposition gives an adjunction \[ L^{\Span }\colon \Fun (\Span (\Fin ),C) \rightleftarrows \Fun (\Span (\Fin ),D)\noloc R^{\Span }. \] The left adjoint of \(R_*\) is the composite \[ \widetilde L:=Q_DL^{\Span }i_C. \] The adjunctions \(Q_D\dashv i_D\) and \(L^{\Span }\dashv R^{\Span }\), together with the full faithfulness of \(i_C\), immediately give \(\widetilde L\dashv R_*\).
Since \(L\) is a left adjoint, it preserves small colimits, so Proposition 16.2.10 shows that \(L^{\Span }\) is symmetric monoidal for the ambient Day convolution structures. Moreover, the two functors \[ \widetilde LQ_C \qquad \text {and}\qquad Q_DL^{\Span } \] are left adjoint to the same functor \(R^{\Span }i_D=i_CR_*\). They are therefore naturally isomorphic. Since \(Q_DL^{\Span }\) is symmetric monoidal, the universal property of the symmetric monoidal Bousfield localization \(Q_C\) supplies \(\widetilde L\) with a unique symmetric monoidal refinement. Its right adjoint \(R_*\) acquires the canonical lax symmetric monoidal structure of Proposition 14.3.6.
If \(L\) preserves finite products, postcomposition with \(L\) preserves commutative monoids and groups. The object \(L^{\Span }i_C(A)\) is then already local for every \(A\), so the localization map \[ L^{\Span }i_C(A)\longrightarrow i_DQ_DL^{\Span }i_C(A) \] is an isomorphism. Thus \(\widetilde L\) is given by postcomposition with \(L\), as claimed. □
Lemma 16.4.3 (Bilinear maps). Let \(C\) be a presentably symmetric monoidal \(\infty \)-category. For \(A,B,M\in \CMon (C)\) there is a natural equivalence \[ \Hom _{\CMon (C)}(A\otimes B,M) \simeq \Nat _{\Span (\Fin )^2} \bigl (A(-)\otimes _C B(-),M(-\times -)\bigr ). \] The same formula holds in \(\CGrp (C)\) for commutative group objects. We refer to objects of the anima on the right as bilinear maps from \(A\) and \(B\) to \(M\).
Proof. The tensor product \(A\otimes B\) is the reflection of the ambient Day convolution \(A\otimes _{\Day }B\). Since \(M\) is local, maps from this reflection to \(M\) are the same as maps from \(A\otimes _{\Day }B\) to \(M\). The universal property of Day convolution, in the form of Proposition 16.2.6, identifies the latter anima with the displayed anima of natural transformations. The argument for commutative groups is identical. □
Example 16.4.4 (Abelian groups). Take \(C=\Set \) with its cartesian monoidal structure. Then \(\CGrp (\Set )=\Ab \), and the symmetric monoidal structure supplied by Proposition 16.4.1 is the usual tensor product of abelian groups. Indeed, regard abelian groups \(A\), \(B\) and \(C\) as product-preserving functors \(\Span (\Fin )\to \Set \), so that \(A(S)\cong A^S\), and similarly for \(B\) and \(C\). By the universal property of Day convolution and the fact that \(C\) is local, maps from the localized Day convolution of \(A\) and \(B\) to \(C\) correspond to families of maps \[ A(S)\times B(T)\longrightarrow C(S\times T) \] that are functorial in the finite sets \(S\) and \(T\) and their spans.
Evaluating such a family at \(S=T=*\) gives a map \(\beta \colon A\times B\to C\). Compatibility with the spans encoding addition shows that \(\beta \) is additive in each variable. Conversely, a bilinear map \(\beta \colon A\times B\to C\) determines the family \[ \bigl ((a_s)_{s\in S},(b_t)_{t\in T}\bigr ) \longmapsto \bigl (\beta (a_s,b_t)\bigr )_{(s,t)\in S\times T}, \] and bilinearity is exactly what is needed for compatibility with composition of spans. We therefore obtain natural bijections \[ \Hom _{\CGrp (\Set )}(A\otimes B,C) \cong \{\text {Bilinear maps $A\times B\to C$}\} \cong \Hom _{\Ab }(A\otimes _{\Z }B,C). \] By Yoneda, the tensor product furnished by the Day convolution localization is the usual tensor product \(A\otimes _{\Z }B\).
Proposition 16.4.5 (Path components). The adjunction
is symmetric monoidal on the left and lax symmetric monoidal on the right, where \(\Ab \) is equipped with its usual tensor product.
Proof. The functor \(\pi _0\colon \An \to \Set \) is a symmetric monoidal left adjoint for the cartesian monoidal structures and preserves finite products. Thus Proposition 16.4.2 applies. Finally, Example 16.4.4 identifies the symmetric monoidal structure on \(\CGrp (\Set )=\Ab \) with the usual tensor product. □
For a presentably symmetric monoidal \(\infty \)-category \(D\), we now set \[ \CRig (D):=\CAlg (\CMon (D)) \qquad \text {and}\qquad \CRing (D):=\CAlg (\CGrp (D)), \] where \(\CMon (D)\) and \(\CGrp (D)\) carry the symmetric monoidal structures of Proposition 16.4.1. We refer to their objects as commutative semiring objects and commutative ring objects in \(D\), respectively. For \(D=\Set \), these recover the 1-categories of commutative semirings and commutative rings.
Proposition 16.4.6 (Maximal subgroupoids). Passing to maximal subgroupoids defines a canonical lax symmetric monoidal functor \[ (-)^{\simeq }\colon \CMon (\Cat _{\infty })\longrightarrow \CMon (\An ), \] where both categories carry the symmetric monoidal structures of Proposition 16.4.1.
Proof. The inclusion \(\An \hookrightarrow \Cat _{\infty }\) is a symmetric monoidal left adjoint for the cartesian monoidal structures, preserves finite products, and has right adjoint \((-)^{\simeq }\). The result is therefore the commutative-monoid instance of Proposition 16.4.2. □
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