Question

Let $D$ be a monoidal category (without the structure of a symmetric monoidal category), with unit object $Id$, and let $L$ be an invertible object in $D$, so that $L$ is dualizable and the pairing between $L$ and its dual $L^{*}$ is an isomorphism. (The situation I have in mind is that $D$ is the category of endofunctors of some category $C$ with composition giving the monoidal functor, and then $L$ is an autoequivalence of $C$.)

I want to put a monoid structure on $\coprod_{i \geq 0} D(Id, L^{\otimes i})$ by tensoring together two morphisms $\alpha: Id \rightarrow L^{\otimes i}$ and $\beta: Id \rightarrow L^{\otimes j}$ and then using the unit constraint on the domain $Id \otimes Id$ to get a morphism $\alpha \otimes \beta: Id \rightarrow L^{\otimes i +j }$.

Questions:

1. Does this construction make sense? I am a little worried that I have to fuss with the associativity constraints to really make sense of `$L^{\otimes i}$ '.

2. Presuming that the construction makes sense, is the resulting product commutative? To try to check this, I would (and did) attempt the following: Given a product of two morphisms $\alpha \otimes \beta: Id \rightarrow L^{\otimes i} \otimes L^{\otimes j}$, tensor it on the right with $L^{\otimes -j}$ and on the left with $L^{\otimes j}$. This produces a morphism $L^{\otimes j} \otimes Id \otimes Id \otimes L^{\otimes -j} \rightarrow L^{\otimes j} \otimes L^{\otimes i} \otimes L^{\otimes j} \otimes L^{\otimes -j}$. Now use the unit and pairing on the domain and the pairing on the codomain to identify this with a morphism $Id \rightarrow L^{\otimes j} \otimes L^{\otimes i}$. Is this morphism $\beta \otimes \alpha$? And could this show that $\alpha \otimes \beta= \beta \otimes \alpha$, once the appropriate identifications of domain and codomain have been used?

Answer 1

EDIT: As Peter points out in the comments, my answer to number 2 is incorrect. When $D$ is braided, you can "pull out the twist" using naturality of the braiding, but for general monoidal categories, this morphism will not be equal to $\beta \otimes \alpha$.

However, there is an alternative procedure to the one you describe that always produces $\beta \otimes \alpha$, as shown in this even more crudely drawn diagram:

This map corresponds to tensoring by $L^{\otimes - j}$ in the middle rather than on the right. In either case, $\alpha \otimes \beta \neq \beta \otimes \alpha$.

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1. Yes, by coherence. For instance, if you take a strict monoidal category $D'$ equivalent to $D$, then any bracketing of $L^{\otimes i}$ in $D$ identifies with the canonical $L^{\otimes i}$ in $D'$ in a manner compatible with taking homs and tensor products. This guarantees that all ways you choose to bracket your $L^{\otimes i}$ and apply associators produce the same result, so your monoid is well-defined.
2. The result of your manipulations is $\beta \otimes \alpha$. This can be checked most easily using string diagrams. Here is my rather crude attempt to illustrate this: The $\beta$ can be "pulled around" to the left side of $\alpha$. However, this morphism is not the same as $\alpha \otimes \beta$ in general: this diagram is not isotopic to the one for $\alpha \otimes \beta$, so the result fails in the "free" case.

Note that the answers to both parts of your question are forms of coherence. The first is perhaps most easily seen as a case of "every monoidal category is equivalent to a strict one," while the second is most easily seen as a case of "isotopic string diagrams represent the same morphism."