In "Lectures on tensor categories and modular functors" by Bakalov, Kirillov, the definition of a $(d+1)$-dimensional TFT $\tau$ is given in section 4.2.
Let $k$ be a field. The very last axiom they state is called "Normalization"
$\tau(S^d) = k$ and $\tau($ the unit ball in $\mathbb{R}^{d+1} ) = 1\in k$
But that can't be right. For example, it would imply that the Frobenius algebra underlying any $(1+1)$ TFT has dimension 1.
So, should this part of the normalization axiom actually just be
$\tau(S^{d+1}) = 1 \in k$ ?
So if their axiom is indeed not correct, then I also don't understand what a modular functor is, as they define at the beginning of chapter 5. It has the same normalization axiom, is it correct in this case? And what it is used for?
They say that any (d+1) TFT gives a $d$-dimensional modular functor, basically by forgetting. I cannot reconcile these things in my head.