I'm reading the book Tensor Categories by Etingof, Gelaki, Nikshych, and Ostrik.
In Section 2.6 they discuss the monoidal functors between two categories $\mathcal{C}_{G_1}^{\omega_1}(A)$ and $\mathcal{C}_{G_2}^{\omega_2}(A)$, where $G_i$ are finite groups, $A$ a fixed abelian group and $\omega_i \in Z^3(G_i,A)$.
The monoidal category $\mathcal{C}_G^\omega$ consists of the following datum:
- Objects are given by the fixed finite group $G$, i.e. $\mathrm{ob}(\mathcal{C}_G^\omega) := G$.
- Morphisms: $\mathrm{Hom}(g,h) = \emptyset $ if $g\neq h$ and $\mathrm{Hom}(g,g) = A$ for the fixed abelian group $A$. (In other words, as a category $\mathcal{C}_G^\omega = \mathrm{B}A^{\amalg G} $).
- Tensor products on objects is given by multiplication of $G$ and on morphisms by multiplication in $A$.
- Finally, $\omega \in Z^3(G,A)$ corresponds to the associators.
I think in addition to $f:G_1\rightarrow G_2$ (monoidal on objects) and $\mu:G_1 \times G_1\rightarrow A$ (monoidality isomorphism), a monoidal functor is also specified by an endomorphism $g:A\to A$ that determines how the functor maps morphisms.
Also,
$$\omega_1 = f^\ast \omega_2 \cdot d_3(\mu)$$
(see (2.31) in [EGNO]) should be changed into
$$g_{*}\omega_1=f^{*}\omega_2\cdot d_3(\mu)\,.$$
Therefore, a monoidal functor from $\mathcal{C}_{G_1}^{\omega_1}$ to $\mathcal{C}_{G_2}^{\omega_2}$ should correspond to a triple $(f,g,\mu)$, instead the mere pair $(f,\mu)$.
Who's correct? Me, the book, or neither?
Best Answer
Yes, you are correct in that a monoidal functor $F:\mathcal{C}_{G_1}^{\omega_1}\rightarrow \mathcal{C}_{G_2}^{\omega_2}$ is determined by a triple $(f,g,\mu)$.
In EGNO they consider functors that act trivially on the coefficient group $A$, i.e. $g = \mathrm{id}$ in your example. They explain that in their corrections here.