Conjectured involvement of the tricategory B_∞–Span^3 in constructing a larger lax functor for Hochschild cochains

Construct a larger lax functor from the bicategory of small differential graded (dg) categories whose Hom categories are the full derived categories D(A ⊗ B) (including all morphisms), sending each dg category A to its Hochschild cochain complex C(A), and determine whether the appropriate target structure is the tricategory B_∞–Span^3. In particular, establish the existence and precise form of this lax functor and verify the conjectured role of B_∞–Span^3 in this setting.

Background

The paper constructs lax functors from bicategories of small dg categories to bicategories built from B_∞-algebras (notably B_∞–Span^2), thereby establishing lax functoriality of Hochschild cochain complexes. In these constructions, the Hom categories are typically restricted to isomorphisms in derived categories to ensure the necessary coherence constraints.

The authors point out that attempting to use the entire derived categories D(A ⊗ B), with all morphisms rather than only isomorphisms, presents additional technical challenges (e.g., natural candidates for associativity constraints may no longer be 2-cells). Motivated by this, they suggest that a higher-categorical target may be needed and conjecture that the tricategory B_∞–Span^3 is involved in realizing a larger lax functor that still sends A to C(A).