# Deformation Theory I (draft) by Maxim Kontsevich, Yan Soibelman

By Maxim Kontsevich, Yan Soibelman

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N}. ,n} = V ⊗ · · · ⊗ V (n times) acts the symmetric group Σn . The action of σ ∈ Σn we will denote by Pσ . -4. 4). 2. It is clear from the definitions that one can perform acyclic tensor calculus in an arbitrary tensor category. 5 can be defined by the same formulas. Finally, we would like to mention that one can drop the requirement for C to be k-linear. In that case we arrive to the notion of symmetric monoidal category considered in the Appendix. Most of the properties and constructions of this Chapter can be generalized to the case of symmetric monoidal categories.

Using last Proposition we can offer a more conceptual point of view on formal pointed manifolds. Indeed, we see that a cocommutative non-unital coalgebra B gives rise to an ind-object in the category of finite-dimensional cocommutative non-counital coalgebras in V ectk . If we write B = − lim → I Bi , where Bi are finite-dimensional cocommutative coalgebras, then we have a covariant functor ∗ ∗ FB : Artink → Sets such that FB (R) = − lim → I HomCoalgk (Bi , R ). The functor FB commutes with finite projective limits.

The tensor product ⊗i∈I Xi is defined if the labeling set I is a subset of R2 . Let I1 and I2 be two n-element subsets of R2 . Any homotopy class of paths between I1 and I2 in the space {n − element subsets of R2 } should give a canonical isomorphism between corresponding tensor products. In particular, for any object X on its n-th tensor power corresponding to the subset {(1, 0, (2, 0), . , (n, 0} ⊂ R2 acts the braid group Bn (nad not Σn as for symmetric monoidal categories). We will not discuss here in details the notion of a braided monoidal category because it will be not used further.