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?



Intermediate notions of bilinearity in higher algebra
Let me clarify a bit what I meant in [my comment](https://mathoverflow.net/questions/398187/intermediate-notions-of-bilinearity-in-higher-algebra#comment1020095_398187) 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](https://mathoverflow.net/questions/398187/intermediate-notions-of-bilinearity-in-higher-algebra#comment1020102_398187) when instead one goes to "lax/colax" notion of bilinearity.


