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– Objects are oriented triangulated closed one-dimensional manifolds. Thus objects
are disjoint unions of circles with a finite number n of marked points. Examples of
objects are thus oriented standard circles S 1 ⊂ C with n marked points at roots of
unity and inherited orientations for the intervals. We suppress the latter information.
– Morphisms are two-dimensional, oriented triangulated manifolds with boundary, up
to equivalence which preserves the boundary triangulation, but not necessarily the
triangulation in the interior. More precisely, we have orientation and triangulation
preserving smooth maps
φ B :
M × [0, ɛ) ∐ N × [0, ɛ) → ∂B
that parametrize a small neighborhood of the boundary respecting triangulations.
– Since we can find triangulated cylinders relating the circles S 1 n and S 1 m for all n, m ∈
Z, which are mutually inverses, we still have one isomorphism class of objects.
There is an obvious monoidal functor
Cob tr (2) → Cob(2)
which forgets the triangulation. It turns out that this functor is full and essentially surjective:
any one-dimensional and two-dimensional smooth manifold admits a triangulation.
It is also faithful and thus an equivalence of categories.
• We now construct a two-dimensional topological field theory that is even more local than
our previous construction: we start by associating vector spaces to the objects S 1 n. Our
first input datum is thus
– A K-vector space V . We assume, for simplicity, from the very beginning that this
vector space is finite-dimensional.
Tentatively, we assign to a circle S 1 n with n (positively oriented) intervals the vector space
˜Z tr (S 1 n) := V ⊗n .
This cannot possibly be the true value of our functor Z tr since the latter has to assign
isomorphic vector spaces to the isomorphic objects S 1 n.
• We next have to construct linear maps for triangulated cobordisms. We keep the idea
that we should assign the same vector space V to one-dimensional structures, i.e. to all
intervals, and thus to edges of the triangulation. We have the idea that we build up the
surface by gluing triangles and thus by identifying edges of different triangles. For this
reason, we assign a copy of V to each positively oriented pair (triangle, edge).
We now need to get rid of all vector spaces associated to inner edges. They appear twice,
with opposite relative orientation. Finally, we need to make disappear the triangles. This
leads to the following data:
– A non-degenerate symmetric bilinear pairing μ : V ⊗ V → K.
– A tensor t ∈ V ⊗ V ⊗ V that is invariant under the natural action of the group Z 3
of cyclic permutations on V ⊗3 .
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