Usual mathematical formulation of a 2d (closed) TQFT is as a functor from the category of 2-dim cobordisms between 1-dim manifolds to the category of vector spaces (satisfying various properties.)
For example, a pair of pants (a morphism from $S^1$ to $S^1 \times S^1$) is mapped to a linear map $f:V\to V\otimes V$; similarly a pair of pants in the other direction, a morphism from $S^1 \times S^1$ to $S^1$ is mapped to a linear map $g:V\otimes V\to V$. Then there are axioms (of the symmetric monoidal category) which say that $f$ and $g$ are essentially the same, reflecting the fact that both $f$ and $g$ came from the same pair of pants.
For me (as a quantum field theorist) all this seems very roundabout. The extra axioms are there to ensure what is obvious from the point of view of the two-dimensional field theory; the extra axioms were necessary because the boundaries are arbitrarily grouped into "the source" and "the target" of a morphism, by picking the direction of time inside the 2d surface. (It's called "the Hamiltonian formulation" in physics.)
I think you shouldn't introduce the time direction in the first place, or in the physics terminology, you should just use the "Lagrangian formulation".
In some sense, the idea of "morphism" itself implies an implicit choice of the direction of time. However, you shouldn't introduce the direction of time in an Euclidean quantum field theory. So, you shouldn't use the concept of morphism. The idea of "arrow" itself is so passé, it's a pre-relativity concept which put paramount importance to "time" as something distinct from "space".
So, I would just formulate a 2d TQFT as an association of $f_k:Sym^k V \to K $ to a Riemann surface having $k$ $S^1$ boundaries, and an axiom relating $f_{k}$ and $f_{l}$ to $f_{k+l-2}$.
Why is this not preferred in mathematics? Yes in the physics literature too, the transition from the Hamiltonian framework (pre Feynman) to the Lagrangian framework (post Feynman) took quite a long time…
Or is the higher-category theory (of which I don't know anything) exactly the "Lagrangian formulation" of the TQFT?
Best Answer
Mathematicians have sometimes defined TQFTs in the way Yuji suggests. Indeed, Getzler and Kapranov define the notion of "modular operad" for precisely this purpose (it formalizes the relations between $f_k$, $f_l$ and $f_{k+l-2}$, as well as between $f_k$ and $f_{k-2}$). Earlier, Kontsevich and Manin axiomatized Gromov-Witten invariants along these lines (without distinguisning between incoming and outgoing).
Perhaps the main reason that mathematicians use the language of symmetric monoidal categories is that this is very familiar to them. If you want to explain the idea of a TQFT to the average mathematician, it's easier to say "it's a functor" than to say "it's a collection of linear maps $f_k$ satisfying these relations..."
In addition, there are many very basic examples where the distinction between incoming and outgoing is really important. For example, if $A$ is any associative algebra, then the Hochschild cohomology $HH(A)$ of $A$ carries maps $HH(A)^{\otimes n} \to HH(A)$ indexed by Riemann surfaces of genus $0$, with $n$ incoming and one outgoing boundary components. However, $A$ needs to have a great deal of additional structure -- it needs to be a Calabi-Yau algebra -- in order for this to extend to a fully-fledged TQFT.
As for Yuji's last point, I wouldn't think of the higher-categorical formulation of TQFT as a version of the Lagrangian formalism. After all, for $0+1$ dimensional TQFTs, the higher-categorical formulation reduces to the usual Hamiltonian formalism.