Shadows and traces in bicategories

Traces in symmetric monoidal categories are well-known and have many applications; for instance, their functoriality directly implies the Lefschetz fixed point theorem. However, for some applications, such as generalizations of the Lefschetz theorem, one needs “noncommutative” traces, such as the Hattori-Stallings trace for modules over noncommutative rings. In this paper we study a generalization of the symmetric monoidal trace which applies to noncommutative situations; its context is a bicategory equipped with an extra structure called a “shadow.” In particular, we prove its functoriality and 2-functoriality, which are essential to its applications in fixed-point theory. Throughout we make use of an appropriate “cylindrical” type of string diagram, which we justify formally in an appendix.

💡 Research Summary

The paper introduces a categorical framework that extends the classical notion of trace from symmetric monoidal categories to a non‑commutative setting by working inside a bicategory equipped with an extra piece of structure called a “shadow.” The authors begin by recalling that in a symmetric monoidal category a trace is defined for endomorphisms of dualizable objects, and that the functoriality of this trace underlies classical results such as the Lefschetz fixed‑point theorem. However, this construction relies on the commutativity of the tensor product and therefore cannot be applied directly to situations like the Hattori‑Stallings trace for modules over non‑commutative rings.

To overcome this limitation, the authors define a shadow as a family of functors
(S_{A,B}\colon \mathcal{B}(A,B)\to \mathcal{B}(B,A))
satisfying three axioms: cyclicity (a natural isomorphism between (S_{A,B}(f)) and (S_{B,A}(f^{*}))), associativity (compatibility with horizontal composition), and unit‑compatibility (the shadow of the unit 1‑cell is the identity 2‑cell). These axioms guarantee that the shadow behaves like a “categorical trace” at the level of 1‑cells, allowing one to turn a 2‑cell (\alpha\colon f\otimes X\Rightarrow X\otimes f) into a 2‑cell (I\Rightarrow I) after applying the shadow and the unit constraints. This resulting 2‑cell is defined to be the shadow trace of (\alpha).

The first major result is the functoriality of the shadow trace: given a 2‑cell (\phi\colon f\Rightarrow g) and a 2‑cell (\alpha) as above, the trace satisfies
(\operatorname{Tr}(\phi\circ\alpha)=\operatorname{Tr}(\alpha)\circ\operatorname{Tr}(\phi)).
Thus the trace respects vertical composition of 2‑cells. The second major result is 2‑functoriality: if (F\colon\mathcal{B}\to\mathcal{B}’) is a strong 2‑functor that preserves the shadow (i.e., there are coherent isomorphisms (F\circ S = S’\circ F)), then the trace commutes with (F):
(F(\operatorname{Tr}{\mathcal{B}}(f)) = \operatorname{Tr}{\mathcal{B}’}(F(f))).
These properties are precisely what is needed to transport fixed‑point formulas across functors, generalising the classical Lefschetz argument to non‑commutative contexts.

A distinctive methodological contribution is the introduction of cylindrical string diagrams. Traditional planar string diagrams cannot faithfully represent the cyclic nature of a trace when the underlying monoidal structure is non‑commutative. By drawing the diagram on the surface of a cylinder—placing the input and output “holes’’ on opposite ends and allowing strings to wrap around the cylinder—the authors obtain a visual calculus that makes the cyclicity axiom manifest and that respects both horizontal and vertical composition in a bicategory. The appendix provides a rigorous justification that these cylindrical diagrams encode exactly the same equations as the algebraic definitions of shadow, composition, and trace.

The paper also demonstrates that the shadow trace subsumes the Hattori‑Stallings trace. By constructing a bicategory (\mathcal{B}{R}) whose 1‑cells are finitely generated projective (R)-modules and whose 2‑cells are module homomorphisms, and by equipping (\mathcal{B}{R}) with the appropriate shadow, the shadow trace of an endomorphism coincides with the classical Hattori‑Stallings trace. Consequently, the Lefschetz fixed‑point theorem can be reformulated in this bicategorical language and proved for non‑commutative rings.

Finally, the authors discuss potential extensions. The shadow framework is flexible enough to accommodate other non‑commutative traces arising in quantum algebra, higher‑dimensional topology, and representation theory. By providing a robust functorial and 2‑functorial trace, the paper opens the door to a systematic categorical treatment of fixed‑point phenomena beyond the commutative world.

In summary, the work establishes a general, bicategorical notion of trace via shadows, proves its essential functorial properties, supplies a novel diagrammatic language to handle cyclicity, and shows how classical non‑commutative traces fit into this picture, thereby laying a solid foundation for future applications in fixed‑point theory and beyond.