Intermediate notions of bilinearity in higher algebra

Question

It is well-known that when passing to $\infty$-categories the notion of commutativity gets replaced by an infinite array of notions of commutativity: $\mathbb{E}_{1}$, $\mathbb{E}_{2}$, ..., $\mathbb{E}_{\infty}$. This is already apparent when passing from sets to categories and $2$-categories:

• For sets, we have monoids ($\mathbb{E}_{1}$) and commutative monoids ($\mathbb{E}_2=\cdots=\mathbb{E}_\infty$);
• For categories, we have monoidal ($\mathbb{E}_{1}$), braided ($\mathbb{E}_{2}$), and symmetric monoidal categories ($\mathbb{E}_{3}=\mathbb{E}_{4}=\cdots=\mathbb{E}_{\infty}$);
• For $2$-categories, we have monoidal ($\mathbb{E}_{1}$), braided ($\mathbb{E}_{2}$), sylleptic ($\mathbb{E}_{3}$), and symmetric monoidal categories ($\mathbb{E}_{4}=\mathbb{E}_{5}=\cdots=\mathbb{E}_{\infty}$).

A similar phenomenon happens to bilinearity:

• A morphism $f\colon A\times B\to C$ of commutative monoids is bilinear if, for each $a,a'\in A$ and each $b,b'\in B$, we have \begin{gather*} f(a,b+b') = f(a,b)+f(a,b'),\\ f(a+a',b) = f(a,b)+f(a',b),\\ f(1_A,b) = 1_C,\\ f(a,1_B) = 1_C. \end{gather*}

• For categories, these relations are replaced by morphisms: we say that a strong bilinear structure on a functor $F\colon\mathcal{C}\times\mathcal{D}\to\mathcal{E}$ of symmetric monoidal categories is a collection of isomorphisms \begin{align*} F^{\mathsf{bil}}_{A,B\otimes_{\mathcal{D}}B'} &\colon F(A,B\otimes_{\mathcal{D}}B') \longrightarrow F(A,B)\otimes_{\mathcal{E}}F(A,B'),\\ F^{\mathsf{bil}}_{A\otimes_{\mathcal{C}}A',B} &\colon F(A\otimes_{\mathcal{C}}A',B) \longrightarrow F(A,B)\otimes_{\mathcal{E}}F(A',B),\\ F^{\mathsf{bil}}_{\mathbf{1}_{\mathcal{C}},B} &\colon F(\mathbf{1}_{\mathcal{C}},B) \longrightarrow \mathbf{1}_{\mathcal{E}},\\ F^{\mathsf{bil}}_{A,\mathbf{1}_{\mathcal{D}}} &\colon F(A,\mathbf{1}_{\mathcal{D}}) \longrightarrow \mathbf{1}_{\mathcal{E}} \end{align*} satisfying coherence conditions.

Questions:

1. Is there a similar array of notions of bilinearity in higher algebra?
2. In particular, can we speak of "$\mathbb{B}_{k}$-morphisms of spectra"?
3. Tensor products can be characterised/defined as universal bilinear maps; do we also have intermediate tensor products corresponding to "$\mathbb{B}_{k}$"-bilinearity?

Answer 1

Let me clarify a bit what I meant in my comment on how the notion of bilinearity will depends on "how commutative" are $A$, $B$ and $C$, and this is one way to define a hierarchy of notion of bilinear maps in higher algebras.

The idea is that up to equivalence of $\infty$-categories, (and let's say in a cartesian monoidal $(\infty,1)$-category for simplicity), an $\mathbb{E}_{n+k}$-algebra is the same as an $\mathbb{E}_n$-algebra in the (cartesian monoidal) category of $\mathbb{E}_{k}$-algebra.

So if $A$ is an $\mathbb{E}_k$-algebra, $B$ is an $\mathbb{E}_n$-algebra and $C$ is an $\mathbb{E}_{n+k}$-algebras then I can define a $(n,k)$-bilinear map $A \times B \to C$ to be a morphism of $\mathbb{E}_k$-algebra from $A$ to the $\mathbb{E}_k$-algebra $\operatorname{Map}_{\mathbb{E}_n}(B,C)$.

Where by $\operatorname{Map}_{\mathbb{E}_n}(B,C)$ I mean the space of morphisms of $\mathbb{E}_n$-algebra morphism from $B$ to $C$, which has an $\mathbb{E}_k$-algebra structure induced by the fact that $C$ is an $\mathbb{E}_k$-algebra when seen as an object of the category of $\mathbb{E}_n$-algebra.

But this is very different from the notion you mention in your coment when instead one goes to "lax/colax" notion of bilinearity.