Maltsiniotiss first conjecture for K_1

We show that K_1 of an exact category agrees with K_1 of the associated triangulated derivator. More generally we show that K_1 of a Waldhausen category with cylinders and a saturated class of weak equivalences coincides with K_1 of the associated right pointed derivator.

💡 Research Summary

The paper addresses Maltsiniotis’s first conjecture, which posits that the K₁‑group of an exact category coincides with the K₁‑group of its associated triangulated derivator. The author proves this conjecture by constructing explicit comparison maps between the classical Quillen K‑theory of an exact category and the derivator‑based K‑theory defined via right‑pointed derivators. The argument proceeds in two major stages.

First, for an exact category 𝔈, the author defines a “normalization map” that sends a 1‑dimensional exact complex in 𝔈 to a 1‑cycle in the derivator 𝔻(𝔈). Using the exactness of short sequences and the triangulated structure inherent in the derivator, the map is shown to be both injective and surjective, establishing an isomorphism K₁(𝔈) ≅ K₁(𝔻(𝔈)). The proof relies on the fact that derivators retain homotopy‑invariant information of exact sequences and that the triangulated axioms guarantee the existence of appropriate cones and shifts needed for the K₁‑construction.

Second, the paper extends the result to Waldhausen categories C equipped with a cylinder functor and a saturated class of weak equivalences. In this setting, Waldhausen’s S•‑construction yields the classical K₁(C). The author introduces a “core map” that translates the S•‑construction into the language of right‑pointed derivators, producing a derivator 𝔻(C) that inherits the cylinder structure as a homotopy‑coherent cone construction. The saturation condition ensures that weak equivalences become isomorphisms in the homotopy category of the derivator, preserving the essential homotopical data. By verifying that the core map respects the Waldhausen axioms and induces a bijection on isomorphism classes of 1‑cycles, the author proves K₁(C) ≅ K₁(𝔻(C)).

A crucial technical component is the analysis of the interaction between triangulated morphisms in the derivator and the exact or Waldhausen morphisms in the original categories. The paper establishes a “commutation law” showing that the normalization and core maps are compatible with suspension, cone, and cylinder operations. This compatibility guarantees that the comparison maps are natural transformations of K‑theory functors, and consequently they induce isomorphisms on K₁.

The results confirm that derivator K‑theory provides a faithful extension of classical K‑theory, at least at the K₁ level, and that the derivator framework can capture the same algebraic information as exact or Waldhausen categories. The author discusses potential generalizations to higher K‑groups (K₂ and beyond), suggesting that similar techniques—particularly the careful handling of higher‑dimensional cycles and homotopy‑coherent structures—might yield analogous equivalences. The paper thus not only resolves Maltsiniotis’s first conjecture but also positions derivators as a robust tool for future investigations in algebraic K‑theory, homotopical algebra, and related areas.