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Cofiber and triangle presentations of K zero
PDF p. 12: Bundle classes in spectral K zero
Statement
Let C be a small stable infinity-category. Its cofiber Grothendieck group, generated by equivalence classes of objects with [Y]=[X]+[Z] for every cofiber sequence X→Y→Z, is canonically the triangle Grothendieck group of its homotopy category hC, generated by isomorphism classes with the same relation for every distinguished triangle.
A stable infinity-category is a pointed infinity-category with finite limits and colimits in which pushout squares are pullback squares. Its homotopy category hC has the same objects and morphisms given by connected components of mapping spaces; its distinguished triangles are, by the standard induced triangulated structure, the triangles isomorphic to X→Y→Z→X[1] arising from cofiber sequences. Smallness ensures that the presentations use sets.
Facts & Assumptions
Given: A small stable infinity-category C with its induced triangulated homotopy category.
Goal: Identify the two presented groups canonically.
By the stated construction of hC, equivalences in C correspond to isomorphisms in hC, and distinguished triangles are exactly triangles isomorphic to cofiber triangles.
Proof
The identity on objects gives a bijection between equivalence classes in C and isomorphism classes in hC by [F1]. It therefore identifies the free abelian groups on those classes.
Under this identification each cofiber relation is a distinguished-triangle relation. Conversely, every distinguished triangle is isomorphic to a cofiber triangle, so its relation agrees with one of the cofiber relations. The two relation subgroups are equal.
The identity on the common free group therefore induces mutually inverse quotient maps. This is the asserted canonical identification.