1. Context
One page 31 of Weakly distributive categories Cockett and Seely define tensor inverses in monoidal categories.
Let me unpack their definition in different language:
Let $(C, \otimes, I, a, l,r)$ be a monoidal category. They say that an object $A$ in $C$ has tensor inverse $(A^{-1}, (s^L)^{-1}: I \rightarrow A \otimes A^{-1}, s^R: A^{-1} \otimes A \rightarrow I)$ if the tuple $(A, A^{-1},(s^L)^{-1}, s^R)$ is an adjoint equivalence in the one-object 2-category incarnation of $C$ except that only one zig-zag-identity holds. Namely, their definition is precisely that of an adjoint equivalence except that they do not require that the following diagram commutes
$$\require{AMScd}
\begin{CD}
I \otimes A @>{(s^L)^{-1}\otimes id_A}>> (A \otimes A^{-1}) \otimes A @>{a_{A,A^{-1},A}}>> A \otimes (A^{-1} \otimes A)\\
@V{l_A}VV @. @V{id_A \otimes s^R}VV\\
A @>r_A^{-1}>>A \otimes I @= A \otimes I\\
\end{CD}$$
2. Question
- Do these "one-sided adjoint equivalences" have a name in the literature?
Best Answer
There is no need for new terminology. In Catégories Tannakiennes Saavedra Rivano proved that for adjoint equivalences one zig-zag identity follows from the other. A proof using string diagrams can be found on p.11 of this paper by Baez and Lauda.