Group completion in the K-theory and Grothendieck–Witt theory of proto-exact categories

Algebraic K-theory and Grothendieck-Witt theory of monoid schemes

“…We work in the setting of proto-exact categories, a non-additive generalization of Quillen's exact categories introduced by Dyckerhoff and Kapranov [10]. This is a convenient setting for both K-theory and Grothendieck-Witt theory and was developed by the authors in [11]. The first main result of [11] is a Group Completion Theorem for the K-theory of uniquely split proto-exact categories.…”

“…This is a convenient setting for both K-theory and Grothendieck-Witt theory and was developed by the authors in [11]. The first main result of [11] is a Group Completion Theorem for the K-theory of uniquely split proto-exact categories. The second main result of [11] is a description of the Grothendieck-Witt space GW Q (A) of a uniquely split protoexact category with duality A satisfying additional mild assumptions, defined using the hermitian Q-construction, in terms of the group completion of the groupoid of hyperbolic forms and the monoidal groupoid of isotropically simple symmetric forms in A.…”

“…The first main result of [11] is a Group Completion Theorem for the K-theory of uniquely split proto-exact categories. The second main result of [11] is a description of the Grothendieck-Witt space GW Q (A) of a uniquely split protoexact category with duality A satisfying additional mild assumptions, defined using the hermitian Q-construction, in terms of the group completion of the groupoid of hyperbolic forms and the monoidal groupoid of isotropically simple symmetric forms in A. The second result can be seen as playing the role of the Group Completion Theorem in the Grothendieck-Witt theory of uniquely split proto-exact categories with duality.…”

“…Let A be a (not necessarily commutative) pointed monoid. The category A -proj of finitely generated projective left A-modules has a uniquely split proto-exact structure (Lemma 2.4), and so fits into the framework of [11]. The category A -proj is particularly simple when A is integral, 1 in which case all projective A-modules are free.…”

“…By applying the results of [11], we can use Theorem B to describe the weak homotopy type of GW Q (A -proj n ).…”