Rank-one trivial modular functor

The category $\mathcal{C}$ is $\mathbb{C}$-linear and abelian. Each morphism space is finite dimensional; every object is a finite direct sum of the single simple object $\mathbb{C}$; every object is projective because every surjection of vector spaces splits; and a vector space of dimension $d$ has a composition series of length $d$. Thus $\mathcal{C}$ is finite in the sense of [F1]. The contravariant equivalence $D(X)=X^{\vee }$ has the canonical biduality ${\eta }_X:X\to X^{\vee \vee }$, and $D({\eta }_X)\circ {\eta }_{DX}={\mathrm{i}\mathrm{d}}_{DX}$. Under the Deligne equivalence $\mathcal{C}⊠\mathcal{C}\simeq \mathcal{C}$, the functor $\kappa$ sends $X⊠Y$ to $X\otimes Y$, and $\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{v}$ sends a vector space $W$ to the two-factor object $W⊠\mathbb{C}$. The displayed map $x\otimes y\mapsto (f\mapsto f(x)(y))$ identifies this pairing with $\mathrm{Hom}(X,DY{)}^{\ast }$; tensor flip corresponds to the duality transpose and canonical biduality. Thus it supplies the required pairing-duality compatibility. Its object $\mathrm{\Delta }=\mathbb{C}⊠\mathbb{C}$ is fixed by the factor-swap symmetry. The first triangle composite sends $X$ canonically through $\mathbb{C}\otimes (\mathbb{C}\otimes X)$ to $X$; the second sends it through $(X\otimes \mathbb{C})\otimes \mathbb{C}$ to $X$. These are the standard natural associativity and unit isomorphisms, so both triangle isomorphisms hold. The pairing is therefore symmetric and nondegenerate in the sense of [F1].

Tensor product is a right-exact functor in each finite-dimensional vector-space variable. For the identity extension $Q=\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}$, each homotopy fiber is contractible and the genus-zero section is the identity, so every disk-insertion functor ${\lambda }_{\varphi }$ is an equivalence between contractible groupoids. For a parametrization-preserving mapping class, assigning the identity on each tensor product is natural and preserves composition because $\mathrm{i}\mathrm{d}\circ \mathrm{i}\mathrm{d}=\mathrm{i}\mathrm{d}$. For a disjoint union, the assigned map is the canonical associator and symmetry of tensor products; the empty surface receives $\mathbb{C}$, the tensor unit. Capping an added boundary labeled by the distinguished vacuum object $\mathbb{C}$ uses $X\otimes \mathbb{C}\cong X$, so vacuum insertion holds.

Since $\mathcal{C}$ has exactly one simple isomorphism class, its coevaluation sum in [F2] has one term ${\mathbb{C}}^{\vee }⊠\mathbb{C}\cong \mathbb{C}⊠\mathbb{C}$. Inserting it at either pair of oppositely oriented boundaries leaves the external tensor product unchanged up to its canonical associator and unit maps. Thus the separating and nonseparating sewing maps are isomorphisms, and all intermediate-label sums have one term.

Every composite graph operation reduces to inserting or removing tensor units, rebracketing, and permuting tensor factors. On a pure tensor, each canonical associator or unitor preserves the ordered list of vector entries after rebracketing or deleting a unit, and each symmetry permutes that list by the prescribed boundary permutation. Pure tensors span every finite tensor product, so any two composites with the same graph operation agree as linear maps. The identity mapping-class action preserves this equality. The torus generators are particular mapping classes and therefore each acts on the one-dimensional state space as $[1]$. This verifies the modular-algebra axioms and the claimed $S,T$ matrices.