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The following is a straightforward generalization of the notion of dualizable object in a symmetric monoidal category given in Duality, Trace and Transfer by Albrecht Dold and Dieter Puppe to non-symmetric monoidal categories. Throughout this question $(\mathcal{C}, \otimes, I)$ denotes a monoidal category.

Definition: Let $X$ be an object in $\mathcal{C}$. A right dual of $X$, if it exists, is a pair $(X',\, \varepsilon: X' \otimes X \to I)$ such that for every object $Y \in \mathcal{C}$ the map $$ \begin{array}{rcl} \mathcal{C}(Y,X') & \to & \mathcal{C}(Y \otimes X, I) \\ f & \mapsto & \varepsilon \circ (X \otimes f) \end{array} $$ is a bijection. If such a right dual exists we say that $X$ is weakly right dualizable. The notion of a left dual and weakly left dualizablity is defined... dually. If $\mathcal{C}$ is symmetric monoidal, then the two notions coincide, and we simply speak of the dual of $X$, and say that $X$ is weakly dualizable.

Call $\mathcal{C}$ left-closed (resp. right-closed) if for every object $Y \in \mathcal{C}$ the functor $-\otimes Y$ (resp. $Y \otimes -$) admits a left-adjoint $F_l$ (resp. $F_r$), then the left dual (resp. right dual) of $X$ is given by $F_l(X,I)$ (resp. $F_r(X,I)$). If $\mathcal{C}$ is symmetric monoidal, then $F: = F_l = F_r$, and the dual of $X$ is given by $F(X,I)$. In this way we obtain many examples of weakly dualizable objects, as any object in closed symmetric monoidal is weakly dualizable, so e.g. every modules over a ring is weakly dualizable.

Question: What are examples of weakly dualizable objects in non-closed monoidal categories. I would be particularly curious about examples of weakly dualizable objects in such categories that are non-symmetric.

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    $\begingroup$ This isn't the right notion: consider how it degenerates if $\otimes$ is the coproduct and $C$ has a strictly initial object. You want $\epsilon:X'\otimes X\to I$ as well as $\eta:I\to X\otimes X'$, even just for a one-sided dual: the point in symmetric categories is that $X$ and $X'$ are then each others' left and right duals; in the nonsymmetric case this (together with the triangle identities) says $X'$ is right dual to $X$ and equivalently $X$ is left dual to $X'$. $\endgroup$ – Kevin Carlson May 9 '18 at 20:10
  • $\begingroup$ I agree with Kevin; this is the wrong definition. $\endgroup$ – Qiaochu Yuan May 11 '18 at 0:56
  • $\begingroup$ @KevinCarlson: I don't think your objection applies: Assume $X$ admits a dual (in the sense of the post), then for $X$ to be dualizable in the modern sense means that the extra condition is satisfied that the coevaluation map exists and that the triangle identities are satisfied, as the (putative) coevaluation is determined by the evaluation (because the unit of an adjunction is determined by the counit). Thus the notion of dualizability in your sense likewise degenerates in the situation you describe. $\endgroup$ – Adrian Clough May 11 '18 at 14:27
  • $\begingroup$ In any case, I am mainly curious to figure out what may have motivated Dold and Puppe to give the definition in this generality (in symmetric monoidal categories). In Duality, Trace and Transfer "weakly dualizable" (in the above sense) is called "dualizable" and "dualizable" is "strong dualizable", which suggests that to them the notion of being weakly dualizable is the more fundamental notion. However, they don't give any examples of weakly dualizable objects which don't live in closed symmetric monoidal categories, so I am curious whether such examples exist (which are interesting). $\endgroup$ – Adrian Clough May 11 '18 at 14:31
  • $\begingroup$ Of course, the notion of weakly dualizable was all but abandoned in the subsequence literature (notably LMS), so maybe it simply wasn't clear yet when writing Duality, Trace and Transfer that this notion isn't particularly useful, as this seems to have been the first systematic treatment of duality. $\endgroup$ – Adrian Clough May 11 '18 at 14:38

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