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Added:
The inclusion of the 2-category of monoidal categories into the 2-category of rigid monoidal categories admits a left 2-adjoint functor $L$.
Furthermore, the unit of the adjunction is a strong monoidal fully faithful functor, i.e., any monoidal category $C$ admits a fully faithful strong monoidal functor $C\to L(C)$, where $L(C)$ is a rigid monoidal category.
See Theorems 1 and 2 in Delpeuch \cite{Delpeuch}.
Thanks!
Looking at the entry now, I have reorganized a little: Moved the previous section “Remarks” and your new section on string diagrams from after the Examples-section to subsections inside the Definition-section, where they seem to better belong.
While I was at it (and unrelated to your edit), I looked through the old Examples-section (here) and made some adjustments on wording and hyperlinking.
I think the examples on the category of finite-dimensional vector spaces (here) was missing the finite-dimensionality clause in a couple of crucial places.
and I noitce that the Section “Endofunctor categories” is somewhat misleading in its title and never quite gets around to naming the actual rigid monoidal subcategory of the full endofunctor category that is an example. I have made some adjustments, but this still deserves attention.
In the old section “Free rigid monoidal catgeories” (here) I have
fixed the sentence about which 2-category is embedded in which one,
fixed the intended hyperlink to adjoint 2-functor
added more hyperlinks overall
and added the publication data to the reference
I fact, looking closer at the paragraph on the example of “endofunctor categories”, I found it awkward in how it spoke about adjunction (co)units in a rounabout way without naming them and using ill-typed notation for them.
So I have taken the liberty of deleting this and replacing it simply by a paragraph saying straight away that adjoint endofunctors are the dualizable objects with (co)evaluation being their (co)unit.
Even so, the example still does not fit well here at rigid monoidal category. This is really an example of dualizable objects…. and now that I write this I decide to delete it here entirely and move it to there.
If anyone feels strongly otherwise, please say and we can try to find out what the intent of the example here was.
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