Typed colored knowledgeable Frobenius data classify open-closed sewing

Let $T$ be a strong symmetric monoidal functor. Set $C=T(Circle)$ and $A_{ab}=T(I_{ab})$. Transport the images of the ten elementary surfaces along the structure isomorphisms of $T$ to obtain the maps ${\mu }_C,{\eta }_C,{\mathrm{\Delta }}_C,{ϵ}_C$, the indexed maps ${\mu }_{abc},{\eta }_a,{\mathrm{\Delta }}_{abc},{ϵ}_a$, and the zipper/cozipper maps $i_a,i_a^{\ast }$. Each vector space is finite-dimensional because $T$ lands in ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathrm{f}\mathrm{d}}(k)$.

Conversely, let $D$ be S-colored data. Assign $C$ to a circle, $A_{ab}$ to $I_{ab}$, the tensor product of the assigned spaces to each ordered word, and $k$ to the empty word. Use the canonical associators, unitors, and flips of vector spaces for the strong symmetric monoidal structure. By [F4], every surface has a finite elementary decomposition; evaluate it by composing and tensoring the maps of $D$ along that decomposition. Apply the formal reduction proved in the supplier proof cited in [F4]. After invertible bending, its first-event replacements remove one open coproduct at a time without creating another; centrality and the cozipper module identity then remove zipper leaves. The surviving open product trees retain exactly the marked cyclic endpoint sequences. Closed spider fusion along a spanning tree and self-gluing of each remaining edge give the genus/window word. These are finite equations in the formal presentation, using only [F2] and [F3], and hence hold under evaluation in $D$. Two decompositions of the same surface have the same marked invariants and therefore the same final normal word, including the slot permutations. Both evaluations are consequently equal. The assignment is therefore independent of the decomposition and of the representative surface.

By [F2], every defining relation is a boundary-relative equality of elementary surfaces. Applying $T$ to the closed and open pair-of-pants, disk, zipper, and cozipper relations gives each equation 1–7 of the data definition, including cyclic trace, centrality, adjunction, and the exact colored Cardy map $A_{bb}\to A_{aa}$. In particular, the pair-of-pants swap gives commutativity because symmetry cylinders are carried to vector-space flips by [F1]. The snakes and indexed trace in [F3] show that each $A_{ab}$ pairs perfectly with $A_{ba}$; no extra nondegeneracy or semisimplicity hypothesis is needed. Thus extraction sends $T$ to an object of the stated data category.

The identity cylinder evaluates to the identity map by the unit and counit equations. If two surfaces are composable, concatenate their finite decompositions along the matching parameterized black boundary; the composite word evaluates to the composite of their linear maps. By [F4] it reduces to the normal word of the glued surface without changing that evaluation, so composition is preserved. Ordered disjoint union concatenates generator words, and the symmetry cylinder evaluates to the vector-space factor permutation. Under the canonical tensor comparison maps this is exactly the symmetry-preservation equation in [F1]; hence tensor, symmetry, and interchange are preserved. The canonical reassociation and unit maps satisfy the coherence equations in [F1], proving that the construction is a strong symmetric monoidal functor.

A data homomorphism $(f_C,(f_{ab}))$ defines on each object word the tensor product of its component maps, with the prescribed unit component on $k$. The equations defining a data homomorphism are exactly naturality on the ten generating surfaces. Naturality on an arbitrary surface follows by induction over its finite decomposition: it is preserved under composition by the induction hypothesis and under disjoint union by tensoring the naturality squares. Conversely, a monoidal natural transformation restricts at each circle and interval to maps $f_C,f_{ab}$; its naturality squares on the ten generators give every structure-preservation equation, while monoidal compatibility determines all word components. Therefore extraction is fully faithful.

Let $E$ be extraction and $R$ be the reconstruction above. On a data object, $ER$ recovers exactly its named vector spaces and generator maps. On a functor $T$, the components of the canonical monoidal structure maps of $T$ identify $RE(T)$ with $T$ on every word, including the empty word. They are natural on the ten generators by the definitions in step 1.1; [F4] extends their naturality to every surface. They form a monoidal natural isomorphism by [F1]. Thus $ER$ is the identity and $RE$ is naturally isomorphic to the identity, which proves the equivalence in [F5]. When $S$ is a singleton, the indexed maps reduce to $A=A_{\ast \ast }$ and the same construction is the usual knowledgeable pair. No step identifies $C$ with a center: only the explicitly stated equations are used.