One-object lax natural transformation

Question

A lax natural transformation $\alpha$ between two pseudofunctors $F,G: \mathcal{K} \to \mathcal{L}$ between bicategories $\mathcal{K}, \mathcal{L}$ consists of the following data:

• For every object $A \in \mathcal{K}$ a 1-cell $\alpha_A: FA \to GA$ in $\mathcal{L}$,
• For every 1-cell $f: A \to B$ in $\mathcal{K}$ a 2-cell $Gf \circ \alpha_A \Rightarrow \alpha_B \circ Ff$ in $\mathcal{L}$,

subject to axioms, see the references here.

Assume now that $\mathcal{K}, \mathcal{L}$ have one object. It is well known that they correspond to monoidal categories $M_{\mathcal{K}}$, $M_{\mathcal{L}}$ and that $F,G$ correspond to monoidal functors $\widetilde{F}, \widetilde{G}$. Under this correspondence, the data of $\alpha$ will consist of an object $A_\alpha$ in the monoidal category $M_\mathcal{L}$ together with a 1-cell for every object $X$ of $M_\mathcal{K}$:

$$\tilde{\alpha_X}: \widetilde{G} X \otimes A_\alpha \to A_\alpha \otimes \widetilde{F} X,$$ subject to axioms.

In case $A_\alpha$ is equal to the monoidal unit (this means that the original lax natural transformation is an icon), this becomes an ordinary monoidal natural transformation.

Now my question is: Does this structure have a name in monoidal category literature? Has it been studied?

Answer 1

The two monoidal functors $F,G: M_{\mathcal K} \to M_{\mathcal L}$ select a way to make $M_{\mathcal L}$ into an $M_{\mathcal K}$-bimodule category. The data $(A_\alpha, \tilde{\alpha})$ is called a half-braiding (more precisely, $\tilde{\alpha}$ is a half-braiding on $A_\alpha$) with respect to the bimodule structure. The category of all half-braidings is the relative Drinfeld centre of the bimodule. I could not find a textbook source for these words, but they are used by the community, and attested in the literature, e.g. How does the relative Drinfeld center interact with the relative Deligne tensor product?.