Fusion and monodromy in the Temperley-Lieb category

1- Beautifully written.

2- Strategies for lengthy steps and derivations explained clearly.

1- None that I can think of.

In this paper, authors consider a category (denoted $\widetilde{\mathsf{TL}}$) which is designed to capture all the Temperley-Lieb algebras in a single framework.
They endow this category with the structure of a braided tensor category with a twist.
Then they introduce a category of modules for $\widetilde{\mathsf{TL}}$ and
provide a ribbon structure on this category.
These are the main results of the paper.
Analogous ribbon structures are quite important from many points of view, especially they appear frequently in many CFT considerations where fusion is intimately related to the operator product expansions.
Authors also discuss relations to integrability of statistical models.

The paper is beautifully written and is a delight to read.
The results are deep, interesting and new yet the paper could be read by a graduate student with a rudimentary knowledge of category theory and basic algebra.
The steps are explained clearly and enough motivation is provided.

I have only found one mathematical issue with the paper (point 1 below). I wholeheartedly recommend its publication after this issue is resolved.
Further, I am requesting a few other changes, almost all of them are completely minor, such as typos etc.

1. The main mathematical issue is thus:
Let $\mathcal{C}$ be a rigid tensor category. Then, given any morphism $f:X\rightarrow Y$ in $\mathcal{C}$, there is a left and right dual to $f$,
one of which is given by:
$$f^\ast: Y^\ast
\xrightarrow{1\otimes i_X} Y^\ast\otimes(X\otimes X^\ast)
\xrightarrow{1\otimes (f\otimes 1)} Y^\ast\otimes(Y\otimes X^\ast)
\xrightarrow{\mathrm{\alpha^{-1}}} (Y^\ast\otimes Y)\otimes X^\ast\xrightarrow{e_Y\otimes 1}
X^\ast,$$
and the other one is similar.
For a twist to yield a ribbon structure it is required to be compatible with the rigidity, i.e., $(\theta_X)^\ast = \theta_{X^\ast}$ for all objects $X$ in $\mathcal{C}$ (see Definition 8.10.1 from the book by Etingof, Gelaki, Nikshych, Ostrik). This condition needs to be checked for Proposition 3.7. It may be an easy verifaction, but it must be carried out.


2. Is the category $\widetilde{\mathsf{TL}}$ abelian? Please make a remark accordingly. The rigidity of $\mathrm{Mod}_{\widetilde{{\mathsf{TL}}}}$ is discussed in Section 3. Is $\widetilde{\mathsf{TL}}$ also rigid? Ribbon?


3. The definition of natural (iso)morphisms is recalled in the opening paragraph of Section 2.2. It is better to move this in Section 2.1, where natural isomorphisms first enter, in the definition of the monoidal category.


4. Last line of Lemma 2.2: it should be $\prod_{i=s}^1t_i$. The $t_i$ part is missing.


5. For the braiding in $\widetilde{\mathsf{TL}}$. Once one has checked that $\eta_{\cdot,\cdot}$ provides the braiding, is it beneficial to introduce a diagram for it, by representing it with crossing lines (either under or over crossing, make a choice)?
With these diagrams, truth of some results, esp. Lemmas 2.4 - 2.5, 2.9 etc., may also be explained pictorially: morphisms with isotopic diagrams are equal.
I haven't checked if such diagrams are incompatible with anything, and it may be potentially dangerous to introduce them; but it will be good if the authors could ponder on this point. They may chose to not make any changes.


6. On the fourth line on page 12, there are too many closing parentheses.


7. In the very last line of proof of Proposition 2.10, it appears that $\mathsf{S}_{n,0}$ are used, but they are defined only in the later section.


8. Near equation (29), the action of the functor on morphisms should also be defined. Similarly for equation (31).


9. For the footnote 1 on page 14, perhaps it is better to append another clause, ``as the latter contain only semi-simple modules, _among other requirements_.''


10. On page 17, 3rd line, ``often the left and right partners coincide''. I think they are isomorphic if the category is ribbon.


11. In proposition 3.10, perhaps it is better to avoid using the letter $\alpha$ for that constant, since $\alpha$ already denotes associativity and similarly, later on page 21, maybe avoid using $\eta$ $(=\eta_{\mathsf{TL}_2,\mathsf{TL}_2} \circ \eta_{\mathsf{TL}_2,\mathsf{TL}_2})$ to denote monodromy to avoid confusion with braiding.