Theorem (Lurie, Higher Algebra)
Let (a presentably monoidal category). There’s an adjunction
Remarks (multiplicativity)
- Suppose now , then both sides of the adjunction lift to monoidal categories, and we claim the adjunction lifts to a lax-E_1-monoidal one.
- More generally, if , the adjunction lifts to a lax -monoidal one.
- Taking the limit , if is symmetric monoidal category, above lifts to a lax symmetric monoidal adjunction.
Consequence (Dunn Addivity) For now, say is symmetric monoidal. Applying , we find adjunctions But since is symmetric monoidal is again presentably symmetric monoidal, hence, we can substitute to get map Landing in algebras of categories “doubly-tensored” over . These two maps compose, and we can iterate the procedure.
To write this down cleanly let’s give these categories on the RHS a name:
Definition: the category of presentable -categories tensored -fold over is given by
}(Pr^L)$$ Where the base case of the inductive definition is specified by $Mod^1_V(Pr^L):= Mod_V(Pr^L)$. In this notation, we see that Dunn additivity provides a tower of djunctions $$Alg^{(k)}(V) \to Alg^{(k-1)}(Mod_V(Pr)) \to Alg^{(k-2)}(Mod^2_V(Pr))) \to \cdots $$ $$\cdots \to Alg^{(1)}(Mod^{k-1}_V(Pr)) \to Alg^{(0)}(Mod^{k}_V(Pr))$$ Explicitly, it takes an $E_k$ algebra $A$ through the sequence of transformations $$A \mapsto Mod_A(V) \mapsto Mod_{Mod_A(V)}(Pr) \cdots$$ **Example** Taking $V = Sp^\otimes$, this is a mode, as in [[recollection - mode theory]]. The above gives a sequence of right adjoints $$Alg^{k}_{\mathbb S} \to Alg^{k-1}(Pr_\mathbb S) \to Alg^{k-2}(LMod_{Pr_{\mathbb S}}(Pr_\mathbb S)) \to \cdots$$ encoding the transformations $$A \mapsto Mod(A) \mapsto Mod_{Mod_A}(Pr) \to \cdots$$