# A lemma in Tensor Categories (Etingof et al)

Lemma 8.10.5 in EGNO's Tensor Categories basically states

Let $$\mathcal{C}$$ be a tensor category over an algebraically closed field $$\mathbb{k}$$ with braiding $$c$$. For any nonzero simple object $$X$$ the composition \begin{align} t(X) := \operatorname{ev}_X \circ c_{X, X^\vee} \circ \operatorname{coev}_X \in \operatorname{End}_{\mathcal{C}}(\mathbf{1}) \end{align} is nonzero.

I feel very conflicted. On the one hand, the one line proof given in the book seems plausible:

Since $$X$$ is simple, the corresponding composition \begin{align} \operatorname{End}(\mathbf{1}) \to \operatorname{Hom}(\mathbf{1}, X\otimes X^\vee) \to \operatorname{End}(\mathbf{1}) \end{align} consists of nonzero maps between 1-dimensional spaces, and is thus non-zero.

On the other hand, suppose that the lemma holds and that $$X$$ is projective. Then $$P = X \otimes X^\vee$$ is projective. Set $$f = t(X)^{-1} \operatorname{coev}_X$$ and $$g = \operatorname{ev}_X \circ c_{X, X^\vee}$$. But then \begin{align} \mathbf{1} \xrightarrow{f} P \xrightarrow{g} \mathbf{1} = \operatorname{id}_{\mathbf{1}} \ , \end{align} so that $$\mathbf{1}$$, being a direct summand in a projective, is projective. But then $$\mathcal{C}$$ is semisimple. A contradiction to the existence of non-semisimple finite tensor categories with simple projective objects.

Note that in fact the general heuristic in this last part implies that in a non-semisimple (finite) tensor category there exists no nonzero endomorphism of the tensor unit factoring through a projective object. For this heuristic, see also the proof of Theorem 6.6.1 in the book.

So, where is the mistake?

Edit:

Here are two examples for non-semisimple finite tensor categories with simple projective objects:

Edit 2: The mistake is in the proof in the book. Namely, as I prove, the map $$\operatorname{Hom}(\mathbf{1}, X \otimes X^\vee) \to \operatorname{End}(\mathbf{1})$$ is zero if $$X$$ is projective.

• I don't know any examples of non-semisimple finite tensor categories with simple projective objects off the top of my head, but I think what you've written down is a proof by contradiction that no such thing can have a braiding, yes? Sep 15, 2020 at 8:07
• @QiaochuYuan: I added two examples. Unfortunately, one of the categories I list definitely is braided. So I have absolutely no idea what's going on :D Sep 15, 2020 at 8:17
• @QiaochuYuan: Actually, there also exists a version of the second example where the coproduct is modified so that the category also becomes factorizable. Of course this is not twist equivalent to my example. Sep 16, 2020 at 4:48
• Please mail Etingof, who keeps a list of errata on his website. Sep 16, 2020 at 9:32
• @darijgrinberg I just did, thanks Sep 16, 2020 at 9:54

The deceptively simple proof in the book indeed managed to deceive us.

How? It assumes that the linear map \begin{align} \operatorname{Hom}(\mathbf{1}, X^\vee \otimes X) &\to \operatorname{End}(\mathbf{1}) \newline f &\mapsto \operatorname{ev}_X \circ f \end{align} is non-zero, which is not true according to my proof above.