Fixed point theory and trace for bicategories

The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is the identification of these different invariants. We define a generalization of the trace in symmetric monoidal categories to a trace in bicategories with shadows. We show the invariants used in the converse of the Lefschetz fixed point theorem are examples of this trace and that the functoriality of the trace provides some of the necessary identifications. The methods used here do not use simplicial techniques and so generalize readily to other contexts.

💡 Research Summary

The paper “Fixed point theory and trace for bicategories” revisits the classical Lefschetz fixed‑point theorem and its converse from a purely categorical perspective, replacing the traditional simplicial and homological machinery with a generalized notion of trace that lives in bicategories equipped with a shadow. The authors begin by recalling that in a symmetric monoidal category the trace of an endomorphism yields the Lefschetz number, and that the equality between the Lefschetz number and the fixed‑point index follows from the functoriality of this trace. While this observation is well‑known, the usual proofs of the converse—namely that a non‑zero Lefschetz number forces the existence of a fixed point—rely on intricate chain‑complex arguments, cross‑product constructions, and delicate homological algebra.

The central contribution of the work is the construction of a “shadow‑trace” in any bicategory 𝔅 that admits a shadow functor 𝔖: 𝔅 → 𝒞, where 𝒞 is a symmetric monoidal category. A shadow assigns to each 1‑cell f: A → A an object 𝔖(f) in 𝒞 together with coherent isomorphisms that relate the shadows of composable 1‑cells. The authors define the trace of f to be the canonical morphism tr(f): 𝔖(f) → I (the monoidal unit) obtained by “closing” the shadow using the bicategorical composition and the monoidal structure of 𝒞. They prove that this trace satisfies a strong functoriality property: (i) it is invariant under 2‑cell isomorphisms, and (ii) it respects horizontal composition, i.e. tr(g ∘ f) = tr(g) ∘ tr(f) after appropriate identifications.

With this machinery in place, the Lefschetz number of a map f on a compact ENR (or any space for which a suitable bicategorical model exists) is identified as tr(f) in the shadow‑trace setting. Simultaneously, the fixed‑point index, traditionally defined via local degree or intersection theory, is shown to be the same shadow‑trace applied to the same 1‑cell. The identification is no longer a separate theorem but a direct consequence of the functoriality of the shadow‑trace: the two classical invariants are simply two interpretations of the same categorical morphism.

The paper then leverages this identification to give a concise proof of the converse Lefschetz theorem. Assuming tr(f) ≠ 0, the functoriality of the shadow‑trace forces the shadow object 𝔖(f) to admit a non‑trivial map to the unit, which, by the coherence data of the shadow, translates into the existence of a non‑zero fixed‑point index. Since the index detects actual fixed points, one concludes that f must have a fixed point. This argument avoids any appeal to simplicial approximations, cellular chain complexes, or the classical Lefschetz–Hopf trace formula.

Beyond the core theorem, the authors discuss several extensions. Because the construction only requires a bicategory with a shadow, it applies verbatim to bicategories of spans, correspondences, or bimodules, and therefore to contexts such as stable homotopy theory, derived algebraic geometry, and even quantum algebra where “fixed‑point‑like” phenomena appear. They also outline how the same framework could accommodate more refined invariants—Nielsen numbers, Reidemeister traces, and equivariant Lefschetz numbers—by enriching the shadow with extra group‑action data.

In summary, the paper provides a powerful, conceptually clean unification of Lefschetz numbers and fixed‑point indices via a bicategorical trace with shadows. By eliminating the need for simplicial techniques, the authors open the door to applying fixed‑point theory in a wide variety of categorical settings, offering a new toolbox for both topologists and category theorists interested in trace methods, duality, and fixed‑point phenomena.