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We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exactcategories, with a particular focus on classes of examples of$\mathbb{F}_1$-linear nature. Our main results are analogues of theorems ofQuillen and Schlichting, relating the $K$-theory or Grothendieck-Witt theoriesof proto-exact categories defined using the (hermitian) $Q$-construction andgroup completion.

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