The oriented open-closed surface category with colored free boundary

Finite smooth cutoffs and partitions on the compact corner charts are obtained from [F7]: restrict ambient Euclidean bumps to each half-space or quadrant chart, using a strict support margin to extend by zero. Choose a finite family whose smaller chart neighborhoods cover the compact set, and divide its bumps by their positive sum there. Thus the interpolation below uses actual finite smooth partitions without applying a boundaryless manifold theorem to corners. A corner in a compact surface with faces is isolated, so compactness gives finitely many corners. Choose pairwise disjoint quadrant charts about them; their two coordinate directions give product collars of the incident faces that agree on the quadrant. Delete smaller quadrant neighborhoods and round the new boundary junctions inside those neighborhoods. The remaining ambient surface is a smooth manifold with boundary, so [F4] gives collars of its remaining original boundary arcs. Restrict those collars to the black and free arcs. On each overlap with a corner chart, interpolate their inward transverse vector fields with the coordinate normal field by a smooth partition of unity, taking the coordinate field on a smaller corner neighborhood. Positive inward normal components remain positive under this interpolation. Near a corner use the two coordinate fields for the two incident faces; they commute there and are tangent to the other face. Their smooth ambient local flows from [F8], restricted to inward times and a common sufficiently small width on each compact face, give face collars equal to the fixed product collars in the smaller quadrant. Away from corners the two faces do not meet, so no further compatibility is required. This finite construction gives the compatible face collars in the definition. A black face glues to an oppositely oriented black face of the same type, and the interval endpoint colors match by the stated rule; therefore the quotient retains an oriented surface with corners and the unglued free faces retain their labels.

Arbitrary compatible black collars can also be compared. Each collar is the flow collar of its normal-coordinate vector field near the face. These fields are inward transverse to that face and tangent to its incident free faces; interpolate the two fields linearly on a common neighborhood. Transversality stays positive, and tangency is preserved. Their flows give a smooth family of collars $c_t$ of a common smaller width, fixed on the face and preserving the free faces. On this neighborhood the velocity $W_t=({\mathrm{\partial }}_t{c}_t)\circ c_t^{-1}$ vanishes on the black face and is tangent to the free faces. Multiply it by a cutoff equal to one on a still smaller collar and zero outside the common neighborhood. The ambient time-dependent flow (obtained from [F8] on the product with the time coordinate) is a diffeomorphism carrying $c_0$ to $c_t$ on that smaller collar, fixing all black parameterizations and preserving the free faces. At the corners use ambient extensions of the product charts; tangency makes their restricted flows preserve each free face. Thus the resulting collar comparison descends across a glued seam. Shortening or increasing reparameterization of a collar is handled by the same construction. A diffeomorphism of either input transports its collar to a compatible collar on the other representative, so this comparison also proves independence of representatives.

For three composable representatives, both parenthesized composites identify the same three collared surfaces along the same two black boundary parameterizations. In product coordinates, both quotients are obtained by replacing the two seam collars by the same concatenated interval; a smooth increasing reparameterization of that interval gives a diffeomorphism fixed on the exterior black boundary. Hence composition is associative on equivalence classes. Gluing an identity cylinder merely lengthens a collar and is equivalent to the original surface by step 2.1.

Ordered disjoint union is associative with the empty surface as strict unit at the level of words and diffeomorphism classes, and interchange makes it a bifunctor. The cylinder that permutes components gives the symmetry. Its double application is the identity permutation. For any morphisms $f:X\to X^{\mathrm{\prime }}$ and $g:Y\to Y^{\mathrm{\prime }}$, the two composites in the symmetry-naturality square are the same disjoint union of $f$ and $g$ with the same block permutation of boundary components, so they represent the same relative surface, with the component identification equal to the identity in every corresponding black collar. Each of the two hexagon composites for three words is a cylinder for the same permutation of the three blocks, so those composites also agree; a permutation with an empty block is the identity. The unit and associator are identities in this strict category, so the monoidal pentagon and triangle also hold. Thus the component-permutation cylinders satisfy the braiding and symmetry axioms directly.

The condition ${\mathrm{\partial }}_{1}M=\mathrm{\varnothing }$ is preserved by gluing and disjoint union, so the circle-only objects and no-free-boundary morphisms form the stated closed symmetric monoidal subcategory. If $S\mathrm{\ne }\mathrm{\varnothing }$, choose $a\in S$: a genus-zero circle-to-circle surface with an additional free circular face colored $a$ lies in the full circle-object subcategory but not in the closed category. If $S=\mathrm{\varnothing }$, a coloring of a nonempty free boundary would require an element of the empty set, so every morphism has no free boundary and the full subcategory equals the closed category.

For the strong symmetric monoidal functors of [F5] and [F9], identity natural transformations satisfy the binary and unit compatibility equations of [F6]; if two transformations satisfy them, substitution in those equations shows their componentwise vertical composite also satisfies them. Their identity and associativity laws hold componentwise in the target category [F1]. Thus the morphisms between the functors in the definition form the stated category.