I remember that in the beginning, there was an axiom for (n+1)-dimensional $(n+1)$-dimensional TQFT that said that the state space $V(\Sigma)$ assigned to an $n$-dimensional oriented manifold is spanned by the invariants of all $n+1$-dimensional oriented manifolds $M$ with $\partial M=\Sigma$. If we call the invariant of $M$, $Z(M)\in V(\Sigma)$, this just says that $V(\Sigma)$ is spanned by all $Z(M)$. Maybe it was in Segal's Swansea notes, maybe it was in an early version of Atiyah's axioms. It doesn't seem to have made it into "The Geometry and Physics of Knots"
There is a theorem that the category of Frobenius algebras is equivalent to the category of $1+1$-dimensional TQFT's. For instance, let $A=\mathbb{C}[x]/(x^3)$, with Frobenius map $\epsilon(1)=\epsilon(x)=0$ and $\epsilon(x^2)=-1$. This is the choice that Khovanov made to construct his $sl_3$-invariant of links. At this point, no one would deny that this gives rise to a TQFT.