Possible monoidal structures on a given tensor category naturally form an object of an algebro- geometric nature (the moduli space). The tangent space to the moduli space of tensor structures is computed by the third cohomology of a certain complex, the deformation complex of the tensor category. The tangent cones to this moduli space are controlled by a degree 2-bracket on the deformation cohomology. This graded Lie bracket is compatible with the ∪-product on the deformation cohomology making it a 3-algebra.
This is in parallel with the deformation theory of associative algebras, where the role of the tan- gent cohomology is played by Hochschild cohomology. The tangent cones to the moduli space of associative algebras are given by a degree 2-bracket, the Gerstenhaber bracket. The Gerstenhaber bracket is a Lie bracket and is compatible with the ∪-product on the Hochschild cohomology making it a 2-algebra. According to the celebrated Delinge’s conjecture this 2-algebra structure lifts to an action of the E2-operad on the Hochschild complex.
The analogous statement is true for the deformation cohomology of a tensor category - the 3-algebra structure lifts to an action of the E3-operad on the deformation complex. The proof is in some sense easier than existing proofs of the Delinge’s conjecture and is based on the internal structure of the deformation complex of a tensor category. As well as Hochschild complex the deformation complex is a cosimplicial complex. The special feature of the deformation complex is that it is a cosimplicial complex of algebras. This together with a lattice paths model of the E3-operad provides an explicit E3-algebra structure on the deformation complex.