Define a category B whose objects are the oriented submanifolds of X, and whose vector space of morphisms from Y to Z is O_{YZ} = Ext^{∗}_{H∗(X)}(H^{∗}(Y), H^{∗}(Z)) – the cohomology, as usual, has complex coefficients, and H^{∗}(Y) and H^{∗}(Z) are regarded as H^{∗}(X)-modules by restriction. The composition of morphisms is given by the Yoneda composition of Ext groups. With this definition, however, it will not be true that O_{YZ} is dual to O_{ZY}. (To see this it is enough to consider the case when Y = Z is a point of X, and X is a product of odd-dimensional spheres; then O_{YZ} is a symmetric algebra, and is not self-dual as a vector space.)

We can do better by defining a cochain complex O’_{YZ} of morphisms by

O’_{YZ} = B_{Ω(X)}(Ω(Y), Ω(Z)) —– (1)

where Ω(X) denotes the usual de Rham complex of a manifold X, and B_{A}(B,C), for a differential graded algebra A and differential graded A- modules B and C, is the usual cobar resolution

Hom(B, C) → Hom(A ⊗ B, C) → Hom(A ⊗ A ⊗ B, C) → · · · —– (2)

in which the differential is given by

dƒ(a_{1} ⊗ · · · ⊗ a_{k} ⊗ b) = a_{1} ƒ(a_{2} ⊗ · · · ⊗ a_{k} ⊗ b) + ∑(-1)^{i} ƒ(a_{1} ⊗ · · · ⊗ a_{i}a_{i+1} ⊗ a_{k} ⊗ b) + (-1)^{k} ƒ(a_{1} ⊗ · · · ⊗ a_{k-1} ⊗ a_{k}b) —– (3)

whose cohomology is Ext_{A}(B,C). This is different from O_{YZ} = Ext^{∗}_{H∗(X)}(H^{∗}(Y), H^{∗}(Z)), but related to it by a spectral sequence whose E_{2}-term is O_{YZ} and which converges to H^{∗}(O’_{YZ}) = Ext_{Ω(X)}(Ω(Y), Ω(Z)). But more important is that H^{∗}(O’_{YZ}) is the homology of the space P_{YZ} of paths in X which begin in Y and end in Z. To be precise, H_{p}(O’_{YZ}) ≅ H_{p+dZ}(P_{YZ}), where dZ is the dimension of Z. On the cochain complexes the Yoneda composition is associative up to cochain homotopy, and defines a structure of an A_{∞} category B’. The corresponding composition of homology groups

H_{i}(P_{YZ}) × H_{j}(P_{ZW}) → H_{i+j−dZ}(P_{YW}) —— (4)

is the composition of the Gysin map associated to the inclusion of the codimension dZ submanifold M of pairs of composable paths in the product P_{YZ} × P_{ZW} with the concatenation map M → P_{YW}.

Now let’s attempt to fit the closed string cochain algebra C to this A_{∞} category. C is equivalent to the usual Hochschild complex of the differential graded algebra Ω(X), whose cohomology is the homology of the free loop space LX with its degrees shifted downwards by the dimension d_{X} of X, so that the cohomology H_{i}(C) is potentially non-zero for −d_{X} ≤ i < ∞. There is a map H_{i}(X) → H_{−i}(C) which embeds the ordinary cohomology ring of X to the Pontrjagin ring of the based loop space L_{0}X, based at any chosen point in X.

The structure is, however, not a cochain-level open and closed theory, as we have no trace maps inducing inner products on H^{∗}(O’_{YZ}). When one tries to define operators corresponding to cobordisms it turns out to be possible only when each connected component of the cobordism has non-empty outgoing boundary.