symmetric monoidal (∞,1)-category of spectra
The original Deligne conjecture on the structure of Hochschild cohomology of an associative algebra and more generally an A-∞ algebra states that Hochschild cohomology naturally has the structure of a BV-algebra and that moreover the full Hochschild cohomology complex has the structure of an algebra over an operad over the (framed) little disk operad. (The second statement implies the first since (BV-)Gerstenhaber algebras are algebras over an operad over the (framed) little disk operad.)
A general geometric (higher geometric) interpretation has been indicated in Ben-ZviFrancisNadler. They observed (see Hochschild cohomology for details) that the Hochschild homology of is naturally interpreted as the algebra of functions on the derived loop space . In the presence of good geometric ∞-function theory this naturally induces an action of the little disk operad on . Since a Gerstenhaber algebra is an algebra over the homology of the little disk operad, this immediately explains the existence of this structure.
More generally, analogs of the statement of the Deligne conjecture exist and work for -algebras: k-tuply monoidal n-categories. This is closely related to the statement (and proofs) of the delooping hypothesis.
Historically it was first found that there is the structure of a Gerstenhaber algebra on . By (Cohen) it was known that Gerstenhaber algebras arise as the homology of E2-algebras in chain complexs. In a letter in 1993 Deligne wondered whether the Gerstenhaber algebra structure on the Hochschild cohomology lifts to an E2-algebra-structure on the cochain complex .
In GerstenhaberVoronov (1994) a resolution of the Gerstenhaber algebra structure was given, but the relationship to -algebras remaind unclear.
In (Tamarkin (1998)) a genuine resolution in the model structure on operads of the Gerstenhaber operad was given and shown to act via the Gerstenhaber-Voronov construction on . This proved Deligne’s conjecture.
Various authors later further refined this result. A summary of this history can be found in (Hess).
Direct proofs of the Deligne conjecture have been given in.
A review is in
A transparent higher geometric interpretation in a suitably dualizable context is indicated in
Section 2.5 of
A construction in monoidal model categories is in