ScalingStacks

Proof. Suppose $G$ is not conservative. Then to prove the lemma, it suffices to exhibit a presentable $\infty$-category $\mathcal{D}$ also cotensored over $\mathcal{C}$ and a nontrivial right adjoint $j:\mathcal{D}\rightarrow\mathcal{M}^{\prime}$ cotensored over $\mathcal{C}$ such that $j\circ G$ is trivial.

Define $\mathcal{D}$ to be the full $\infty$-subcategory of $\mathcal{M}^{\prime}$ of $G$-acyclic objects, that is, objects $m\in\mathcal{M}$ such that $G(m)$ is trivial. Our first task is to show that $\mathcal{D}$ is indeed presentable.

Observe that $\mathcal{D}$ is equivalent to the fiber product $\mathcal{D}\simeq 0\times_{\mathcal{M}}\mathcal{M}^{\prime}$, where the limit is computed in the $\infty$-category $\rm Cat_{\infty}$ of $\infty$-categories. Recall by [L2, Proposition 5.5.3.13], the natural functor $\mathcal{P}r^{\rm L}\rightarrow{\rm Cat}_{\infty}$ preserves limits. Furthermore, the forgetful functor $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})\to\mathcal{P}r^{\rm L}$ also preserves limits since it has a left adjoint (given by induction).

Since the functor $G$ preserves colimits and is $\mathcal{C}$-linear, we may regard it as a morphism in $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})$. Thus $\mathcal{D}$ can be computed as a limit in $\mathrm{Mod}_{\mathcal{C}}(\mathcal{P}r^{\rm L})$, and so can be regarded as an object of $\mathcal{P}r^{\rm L}$. In other words, $\mathcal{D}$ is presentable and furthermore tensored over $\mathcal{C}$. Finally, since $\mathcal{D}$ is tensored over $\mathcal{C}$, it is automatically cotensored as well.

Now it remains to show that the inclusion $j:\mathcal{D}\rightarrow\mathcal{M}^{\prime}$ is indeed a right adjoint and cotensored over $\mathcal{C}$. Since $j$ preserves all limits and colimits (and in particular $\kappa$-filtered colimits), the adjoint functor theorem applies. Finally, since $G$ is cotensored over $\mathcal{C}$, $j$ is as well. ∎

Proof. Observe that for any $\mathcal{D}$ tensored over $\mathcal{C}$, the pullback

$$\Fun^{\rm L}_{\mathcal{C}}(\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M},\mathcal{D})\rightarrow\Fun^{\rm L}_{\mathcal{C}}(\mathcal{M},\mathcal{D})$$

induced by the induction $F:\mathcal{M}\to\mathrm{Mod}_{A}\otimes_{\mathcal{C}}\mathcal{M}$ is conservative. In other words, if a functor out of $\mathrm{Mod}_{A}\times\mathcal{M}$ (which preserves colimits in each variable) is trivial when restricted to $\mathcal{M}$, then it is necessarily trivial.

Consequently, switching to opposite categories, we have that the corresponding functor

$$\Fun^{\rm R}_{\mathcal{C}}(\mathcal{D},\mathrm{Mod}_{A}\otimes\mathcal{M})\rightarrow\Fun^{\rm R}_{\mathcal{C}}(\mathcal{D},\mathcal{M})$$

induced by the forgetful functor $G:\mathrm{Mod}_{A}\otimes_{\mathcal{C}}\mathcal{M}\to\mathcal{M}$ is conservative.

Now we can apply Lemma 4.2 with $\mathcal{M}^{\prime}=\mathrm{Mod}_{A}\otimes\mathcal{M}$ to obtain that $G$ is conservative. ∎

Proof of Proposition 4.1. We first prove that $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M}$ is equivalent to $\mathrm{Mod}_{A}(\mathcal{M})$ by the natural evaluation functor. Consider the adjunction

where $F(-)=A\otimes-$ is the induction, and $G$ is the forgetful functor.

The above adjunction induces an adjunction

and thus a functor to modules over the monad $T=(G\otimes{\rm id})\circ(F\otimes{\rm id})$ acting on $\mathcal{M}$. The functor underlying $T$ is given by tensoring with $A$, so we also have an equivalence $\mathrm{Mod}_{T}(\mathcal{M})\simeq\mathrm{Mod}_{A}(\mathcal{M})$.

By its universal characterization, the functor $G\otimes{\rm id}$ is colimit preserving. Note as well that $G$ and hence $G\otimes{\rm id}$ is also $\mathcal{C}$-linear (or in other words, the adjunction satisfies an analogue of the projection formula). Thus it follows from Lemma 4.3 that $G\otimes{\rm id}$ is also conservative. Thus $G\otimes{\rm id}$ satisfies the monadic Barr-Beck conditions, and we obtain the desired equivalence $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathcal{M}\simeq\mathrm{Mod}_{A}(\mathcal{M})$.

Next, we can apply this to the instance where $\mathcal{M}$ is the $\infty$-category of left modules over another associative algebra $A^{\prime}$ to conclude that there is a natural equivalence $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\prime}}(\mathcal{C})\simeq\mathrm{Mod}_{A}(\mathrm{Mod}_{A^{\prime}}(\mathcal{C}))$. We now have a chain of adjunctions

in which the composite $G^{\prime}\circ G^{\prime\prime}$ is colimit preserving and conservative, and hence satisfies the monadic Barr-Beck conditions.

Furthermore, the above adjunction naturally extends to a diagram in which the cycle of left adjoints (denoted by bowed arrows), and hence also the cycle of right adjoints (denoted by straight arrows), commute

Here $F(-)=A_{1}\otimes A_{2}\otimes-$ is the induction, $G$ is the forgetful functor, $f$ is the natural functor factoring through $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\prime}}(\mathcal{C})$, and $g$ is its right adjoint. From this diagram, we obtain a morphism of monads

Now the underlying functors of the monads $GF(-)$ and $G^{\prime}G^{\prime\prime}F^{\prime\prime}F^{\prime}(-)$ are both equivalent to the tensor $A\otimes A^{\prime}\otimes(-)$, so the above morphism of monads is an equivalence. Thus we obtain the promised equivalence $\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\prime}}(\mathcal{C})\simeq\mathrm{Mod}_{A}(\mathrm{Mod}_{A^{\prime}}(\mathcal{C}))\simeq\mathrm{Mod}_{A\otimes A^{\prime}}(\mathcal{C})$.

Finally, we show that the $\infty$-category of left $A$-modules $\mathrm{Mod}_{A}(\mathcal{C})$ is a dualizable $\mathcal{C}$-module by directly exhibiting the $\infty$-category of right $A$-modules $\mathrm{Mod}_{A^{\rm op}}(\mathcal{C})$ as its dual. The trace map is given by the two-sided bar construction

$$\tau:\mathrm{Mod}_{A}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A^{\rm op}}(\mathcal{C})\to\mathcal{C}\qquad M,N\mapsto M\otimes_{A}N$$

The unit map is given by the induction

$$u:\mathcal{C}\to\mathrm{Mod}_{A^{\rm op}}(\mathcal{C})\otimes_{\mathcal{C}}\mathrm{Mod}_{A}(\mathcal{C})\simeq\mathrm{Mod}_{A^{\rm op}\otimes A}(\mathcal{C})\qquad c\mapsto A\otimes c$$

where we regard $A\otimes c$ as an $A$-bimodule.

One can verify directly that the composition

is equivalent to the identity. First, $({\rm id}\otimes u)(M)$ is equivalent to $A\otimes M$ regarded as an $A\otimes A^{\rm op}\otimes A$-module, and second, $(\tau\otimes{\rm id})(A\otimes M)$ is equivalent to $A\otimes_{A}M\simeq M$. ∎

Proof. For $\mathcal{C}_{1},\mathcal{C}_{2}\in{st}$, we define their tensor product by

$$\mathcal{C}_{1}\otimes\mathcal{C}_{2}=(\operatorname{Ind}(\mathcal{C}_{1})\otimes\operatorname{Ind}(\mathcal{C}_{2}))^{c}$$

where the tensor product of the right hand side is calculated in the $\infty$-category $\mathcal{P}r^{\rm L}$ of presentable $\infty$-categories (with morphisms left adjoints), and the superscript ^c denotes the full $\infty$-subcategory of compact objects of a presentable $\infty$-category. Since $\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2}$ is idempotent complete and retracts of compact objects are compact, $(\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2})^{c}$ is idempotent complete as well. Thus the tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ is indeed an object of ${st}$.

For $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{D}\in{st}$, let $\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})$ be the full $\infty$-subcategory of functors $\mathcal{C}_{1}\times\mathcal{C}_{2}\to\mathcal{D}$ that preserve finite colimits in $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ separately. We claim that For $\mathcal{C}_{1},\mathcal{C}_{2}\in{st}$, the tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ corepresents the functor $\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},-)$ in the sense that for any $\mathcal{D}\in{st}$, there is a canonical equivalence

$$\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})\simeq\Fun_{{st}}(\mathcal{C}_{1}\otimes\mathcal{C}_{2},\mathcal{D}).$$

As a consequence, the associativity and symmetry of the tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ will immediately follow from the analogous properties of $\Fun^{\prime}$.

For $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{D}\in\mathcal{P}r^{\rm L}$, let $\Fun^{L\times L}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})$ be the full $\infty$-subcategory of functors $\mathcal{C}_{1}\times\mathcal{C}_{2}\to\mathcal{D}$ that preserve colimits in $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ separately. To prove the claim, observe that the inclusion $\mathcal{D}\rightarrow\operatorname{Ind}\mathcal{D}$ induces a fully faithful functor

$$\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})\rightarrow\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})\simeq\Fun^{L\times L}(\operatorname{Ind}\mathcal{C}_{1}\times\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})$$

Its essential image consists of functors that preserve compact objects. By definition of the monoidal structure on the $\infty$-category $\mathcal{P}r^{\rm L}$ of presentable $\infty$-categories, we have a further equivalence

$$\Fun^{L\times L}(\operatorname{Ind}\mathcal{C}_{1}\times\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})\simeq\Fun^{\rm L}(\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D}).$$

Since the compact objects of $\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2}$ are generated by finite colimits of external products of compacts objects, we obtain an equivalence between $\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})$ and the full $\infty$-subcategory of $\Fun^{\rm L}(\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2},\operatorname{Ind}\mathcal{D})$ consisting of functors that preserve compact objects. In other words, we have the asserted equivalence that characterizes the tensor product

$$\Fun^{\prime}(\mathcal{C}_{1}\times\mathcal{C}_{2},\mathcal{D})\simeq\Fun_{{st}}((\operatorname{Ind}\mathcal{C}_{1}\otimes\operatorname{Ind}\mathcal{C}_{2})^{c},\mathcal{D}).$$

Finally, the assertion that the functor $\operatorname{Ind}:{st}\to\mathcal{P}r^{\rm L}$ is symmetric monoidal is immediate from the constructions and the natural equivalence $\operatorname{Ind}(\mathcal{C}^{c})\simeq\mathcal{C}$, for $\mathcal{C}\in\mathcal{P}r^{\rm L}$. ∎