Closed oriented two-dimensional TQFTs are commutative Frobenius algebras

Let $T:{\mathsf{C}\mathsf{o}\mathsf{b}}_2^{\mathrm{c}\mathrm{l}}\to {\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathrm{f}\mathrm{d}}(k)$ be a strong symmetric monoidal functor. Put $C=T(S^1)$, a finite-dimensional vector space. Apply $T$ to the pair of pants $S^1\bigsqcup S^1\to S^1$, the disk $\mathrm{\varnothing }\to S^1$, and their height reversals. Transport their images along the strong monoidal structure maps to obtain ${\mu }_C:C\otimes C\to C$, ${\eta }_C:k\to C$, ${\mathrm{\Delta }}_C:C\to C\otimes C$, and ${ϵ}_C:C\to k$. The associativity, unit, coassociativity, counit, and both Frobenius surface equalities in [F4] give those Frobenius equations. For commutativity, the symmetry cylinder is sent to the vector-space flip by the symmetry-preservation equation in [F3]; applying $T$ to the pair-of-pants swap equality in [F4] therefore gives ${\mu }_C={\mu }_C{\tau }_{C,C}$. Thus $C$ is a finite-dimensional commutative Frobenius algebra.

Conversely let $(C,{\mu }_C,{\eta }_C,{\mathrm{\Delta }}_C,{ϵ}_C)$ be a finite-dimensional commutative Frobenius algebra. Assign to a word of $m$ circles the vector space $C^{\otimes m}$, with $C^{\otimes 0}=k$. Assign to a connected morphism with the indicated genus and boundaries the formula in the statement, inserting the canonical associators, unitors, and boundary permutations; assign to a disconnected morphism the ordered tensor product of the connected formulas. Use the formal reduction in the proof cited in [F5], rather than inferring algebraic evaluation from geometric equality: contract a spanning tree of the closed generator graph by spider fusion, then contract each remaining edge by self-gluing. Fusion follows from Frobenius and the snake and merges two genus-zero spiders; each self-gluing inserts one $H$. The number of remaining edges is the genus, and the unit and counit handle empty boundary lists. Thus every decomposition reduces through closed Frobenius equations to the formula assigned above; the marked boundary symmetries are retained. Consequently these data are independent of the chosen surface decomposition. If two morphisms are glued, their chosen decompositions concatenate; evaluating the concatenated generator word is the composite of the two linear maps. Reducing the glued surface to its unique normal word preserves that value by the Frobenius equations. Hence identities, composition, tensor product and symmetry are preserved. The block-permutation cylinders evaluate to the corresponding vector-space factor permutations; under the canonical tensor comparison maps this is exactly the symmetry-preservation equation in [F3]. The canonical associators and unitors of vector spaces give the remaining strong monoidal coherence maps, so this defines a strong symmetric monoidal functor.

If $M$ is connected with $m$ incoming circles, $n$ outgoing circles, and genus $g$, the closed normal-form theorem [F5] reduces it to $m$-fold multiplication, $g$ copies of the handle $H={\mu }_C{\mathrm{\Delta }}_C$, and $n$-fold comultiplication. The empty input and output reductions are the unit disk and counit disk; a cylinder gives the identity convention at arity one. Therefore $T(M)$ is the stated formula. For a disconnected surface, the components are a disjoint union, so the strong monoidal structure and [F3] give the tensor product formula.

A map $f:C\to C^{\mathrm{\prime }}$ preserving all four Frobenius maps defines a component $f^{\otimes m}$ on every $m$-circle word, with the unit component at $k$. It is natural for each of the four generating surfaces by the four preservation equations; it is natural for a composite by induction on the number of generator factors, and for disjoint union by the tensor product rule. Thus it is a monoidal natural transformation. Conversely a monoidal natural transformation between two TQFTs restricts at the circle to a linear map $f:C\to C^{\mathrm{\prime }}$. Naturality at the pair of pants and the two disks gives the multiplication, unit, comultiplication, and counit preservation equations, and monoidal compatibility forces its component on every circle word to be $f^{\otimes m}$. The two constructions therefore give inverse maps on every hom-set.

Write $E$ for extraction of the commutative Frobenius algebra and $R$ for the construction from that algebra. Step 2.2 proves that $E$ is fully faithful. Step 1.2 constructs a functor from each algebra, so every object in the algebra category is in the image of $E$; on objects and generating maps, $ER$ is the identity after the fixed tensor-structure identifications. For any TQFT $T$, the canonical comparison maps from its monoidal structure identify $RE(T)$ with $T$ on each circle word; naturality on the four generators from step 1.1 and the normal-form reduction in step 2.1 prove naturality on every surface. These comparison maps are monoidal by [F3], so $RE$ is naturally isomorphic to the identity. Hence $E$ and $R$ are quasi-inverse equivalences of categories.