By G.P. Joubert, R.P. Moussu, R.H. Roussarie

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2), we compose the correspondences, using pullbacks, and see that both compositions (the lower left one and the upper right one) are equal to the following correspondence: C(k) × C(l) × (LM )k+l−1 ← C(k) ◦i C(l)M → LM, where C(k) ◦i C(l)M := {(C1 , C2 , f ) | C1 ∈ C(k), C2 ∈ C(l), and f : C1 ◦i C2 → M is continuous} and the maps are obvious. 2. Verifying the statement at the homology level is a little subtler, as not any correspondence induces a morphism on homology. To verify we have such a morphism in this case, we follow the approach of Cohen and Jones [CJ02] γ e as described in Chapter 1.

We will be talking about a ﬁnite-dimensional retract of P, the framed little disks operad f D, which may be deﬁned as follows, see Getzler [Get94] and Markl-Shnider-Stasheﬀ [MSS02]. 7. , (1, 0), on the boundary of the “big,” unit disk, which should not be thought of as extra data, because we are talking about the standard plane R2 , with ﬁxed x and y coordinates. An identity element is the framed little disk coinciding with the big disk, together with framing. The symmetric group acts by relabeling the framed little disks, as usual.

A Conformal Field Theory (CFT ) is an algebra over the PROP P. We would like to consider variations on this theme. 2. A Cohomological Field Theory-I (CohFT-I ) is an algebra over the homology PROP H∗ (P). The Roman numeral one in the name is to distinguish this theory from a standard Cohomological Field Theory (CohFT ), which is an algebra over the PROP H∗ (M), where M is the PROP of moduli space of stable compact algebraic curves with punctures with respect to the operation of attaching curves at punctures.