Coherent algebra package

Proposition 8.1.5 (Coherent algebra package). Let \(C\) be a cocomplete symmetric monoidal \(\infty \)-category whose tensor product preserves colimits separately in both variables.

(1)

There are \(\infty \)-categories \(\Alg (C)\) and \(\CAlg (C)\) of associative and commutative algebras in \(C\). Their objects have underlying objects of \(C\) equipped with coherently unital and associative multiplication maps; for commutative algebras the multiplication is coherently commutative as well.

(2)

For every \(A\in \Alg (C)\) there are \(\infty \)-categories \(\LMod _A(C)\) and \(\RMod _A(C)\) of left and right \(A\)-modules. The forgetful functors to \(C\) create every limit admitted by \(C\) and all colimits, and the free left \(A\)-module on \(X\in C\) has underlying object \(A\otimes X\). If \(C\) is stable, then \(\LMod _A(C)\) and \(\RMod _A(C)\) are stable.

(3)

There is a relative tensor product \[ -\otimes _A-\colon \RMod _A(C)\times \LMod _A(C)\longrightarrow C, \] which preserves colimits separately in both variables. It is unital: for every left \(A\)-module \(M\) and right \(A\)-module \(N\) there are natural isomorphisms \[ A\otimes _A M\iso M, \qquad N\otimes _A A\iso N. \]

(4)

Every morphism \(f\colon A\to B\) in \(\Alg (C)\) induces a restriction functor \[ f_*\colon \LMod _B(C)\longrightarrow \LMod _A(C), \] which admits the extension-of-scalars functor \[ f^*:=B\otimes _A-\colon \LMod _A(C)\longrightarrow \LMod _B(C) \] as a left adjoint. There is an analogous adjunction for right modules, whose left adjoint is \(-\otimes _A B\).

(5)

If \(R\in \CAlg (C)\), then left and right \(R\)-modules agree, and the resulting \(\infty \)-category \(\Mod _R(C)\) is symmetric monoidal under \(-\otimes _R-\). There are natural equivalences \[ \CAlg (\Mod _R(C))\simeq \CAlg (C)_{R/}. \]

(6)

Lax symmetric monoidal functors preserve algebra objects and their modules. In particular, a symmetric monoidal adjunction \(F\colon C\rightleftarrows D\noloc G\) induces adjunctions on algebras and on their module categories. If \(A\in \Alg (C)\) and the unit \(A\to GF(A)\) is an isomorphism, this gives an adjunction \[ \LMod _A(C)\rightleftarrows \LMod _{F(A)}(D) \] whose underlying functors are induced by \(F\) and \(G\). Right adjoints of symmetric monoidal functors are canonically lax symmetric monoidal.

(7)

If \(C\) is presentable, then \(\LMod _A(C)\) and \(\RMod _A(C)\) are presentable for every \(A\in \Alg (C)\).