ScalingStacks

A.9 Module \(\infty\)-categories[00J4]

A.9.1 Left, right, and bimodules for associative algebras[00J5]

We briefly review the theory of module \(\infty\)-categories from [Lur17, § 4]. There are \(\infty\)-operads \(\mathrm{LM}, \mathrm{RM}\) and \(\mathrm{BM}\) parametrizing pairs \((a, {}_{a}m)\) of an associative algebra object \(a\) along with a left module \(m\) over it, pairs \((a, m_{a})\) of an associative algebra with a right module, and triples \((a, b, {}_{a}m_b)\) of associative algebras \(a, b\) and a bimodule \(m\) between them, respectively. Forgetting the module \(m\) gives rise to operad maps \(\mathbb E_1 \rightarrow\mathrm{LM}\), \(\mathbb E_1 \rightarrow\mathrm{RM}\) and \(\mathbb E_1 \sqcup \mathbb E_1 \rightarrow\mathrm{BM}\).

An \(\mathrm{LM}\)-monoidal \(\infty\)-category \(\mathcal M\) amounts to a monoidal \(\infty\)-category \(\mathcal M_a\) together with a left module \(\infty\)-category \(\mathcal M_m\) over it. An \(\mathrm{LM}\)-algebra in such an \(\mathrm{LM}\)-monoidal \(\infty\)-category therefore consists of an \(\mathbb E_1\)-algebra in \(\mathcal M_a\) together with a left module in \(\mathcal M_m\). We denote the \(\infty\)-category of \(\mathrm{LM}\)-algebras in \(\mathcal M\) by \(\mathrm{LMod}(\mathcal M)\). Furthermore, pre-composing with the map \(\mathbb E_1 \rightarrow\mathrm{LM}\) induces a functor \(\mathrm{LMod}(\mathcal M) \rightarrow\mathrm{Alg}(\mathcal M_a)\). For an algebra \(A \in \mathrm{Alg}(\mathcal M_a)\), we denote the fiber of this functor at \(A\) by \(\mathrm{LMod}_A(\mathcal M_m)\), and call it the \(\infty\)-category of \(A\)-modules in \(\mathcal M_m\). If \(\mathcal M\) is a presentably \(\mathrm{LM}\)-monoidal \(\infty\)-category, then \(\mathrm{LMod}_A(\mathcal M_m)\) is also presentable [Lur17, Cor. 4.2.3.7].

Any monoidal \(\infty\)-category \(\mathcal C\) can be considered a \(\mathrm{LM}\)-monoidal \(\infty\)-category by setting \(\mathcal M_m = \mathcal M_a\) with its canonical left module action. In this case, \(\mathrm{LMod}_A(\mathcal C)\) carries a canonical right action by \(\mathcal C\)[Lur17, § 4.3.2], if \(\mathcal C\) is further presentably monoidal this exhibits \(\mathrm{LMod}_A(\mathcal C)\) as an object in \(\mathrm{RMod}_\mathcal C(\mathrm{Pr}^\mathrm{L})\).

We use analogous notation for \(\mathrm{RM}\) and \(\mathrm{BM}\)-monoidal \(\infty\)-categories and algebras; for example, in a \(\mathrm{BM}\)-monoidal \(\infty\)-category consisting of two monoidal \(\infty\)-categories \(\mathcal M_a\) and \(\mathcal M_b\) and an \(\mathcal M_a\)–\(\mathcal M_b\) bimodule \(\infty\)-category \(\mathcal M_m\), the fiber of \(\mathrm{BMod}(\mathcal M) \rightarrow\mathrm{Alg}_{\mathbb E_1}(\mathcal M_a) \times \mathrm{Alg}_{\mathbb E_1}(\mathcal M_b)\) at an algebra \(A\) and \(B\) is denoted \({}_{A}\mathrm{BMod}_{B}(\mathcal M)\) and is presentable if \(\mathcal M\) is a presentably \(\mathrm{BM}\)-monoidal \(\infty\)-category.

A.9.2 The bar construction and relative tensor product of bimodules[00J6]

We describe the relative tensor product of bimodules in the presentably monoidal case, though the theory works much more generally.

Given bimodules \({}_{A}M_B\) and \({}_{B}N_C\) between algebras \(A,B,C\) in a monoidal \(\infty\)-category \(\mathcal M\) the bar construction defines a simplicial object \({\textup{Bar}}(M,B,N)\in {}_{A} \mathrm{BMod}_C(\mathcal C)\) with \({\textup{Bar}}(M, B, N)_n \coloneqq M \otimes B^{\otimes n} N\) with face and degeneracy maps given by multiplication, actions and the unit. The relative tensor product ([Lur17, Prop. 4.4.2.14]) \(M\otimes_B N\) is defined as the geometric realization of \({\textup{Bar}}(M, B, N)\) in \({}_{A} \mathrm{BMod}_C(\mathcal C)\). If \(\mathcal M\) is a presentably monoidal \(\infty\)-category, this defines a functor in \(\mathrm{Pr}^\mathrm{L}\): \[-\otimes_B - \colon {}_A\mathrm{BMod}_B(\mathcal C) \otimes {}_B\mathrm{Mod}_C(\mathcal C) \rightarrow{}_A\mathrm{BMod}_C(\mathcal C).\] For \(A=B=C\), this induces a presentably monoidal structure on \({}_{A}\mathrm{BMod}_A(\mathcal C)\).

A.9.3 Module categories of commutative algebras[00J7]

If \(\mathcal C\) is a symmetric monoidal \(\infty\)-category and \(A\) a commutative algebra in \(\mathcal C\), there is an equivalence \(\mathrm{LMod}_A(\mathcal C) \simeq \mathrm{RMod}_A(\mathcal C)\) treating the given left action as a right action and vice versa. For this reason, we denote the \(\infty\)-category of modules of a commutative algebra simply by \(\mathrm{Mod}_A(\mathcal C)\) and refer to it as the \(\infty\)-category of \(A\)-modules. Moreover, treating an \(A\)-module as a bimodule induces a functor \(\mathrm{Mod}_A(\mathcal C) \rightarrow{}_A\mathrm{BMod}_A(\mathcal C)\). If \(\mathcal C\) is presentably symmetric monoidal, the relativ tensor product \(-\otimes_A -\) defines a presentably monoidal structure on \({}_A\mathrm{BMod}_A(\mathcal C)\). This lifts to a presentably symmetric monoidal structure on \(\mathrm{Mod}_A\) [Lur17, Thm. 4.5.2.1].