nLab Quillen Q-construction



Additive and abelian categories



Special and general types

Special notions


Extra structure





The Quillen Q-construction (Quillen 72) is a tool for producing the algebraic K-theory of a Quillen exact category 𝒞\mathcal{C}. The Quillen Q-construction is generalized (Waldhausen 83, section 1.9) by the Waldhausen S-construction which applies more generally to Waldhausen categories. (However, the Quillen Dévissage theorem? does not generalize to Waldhausen categories.)

Both these constructions appear in stable homotopy theory as special cases of the concept of algebraic K-theory of a stable (∞,1)-category (Haugseng 10, Barwick 13).


The construction is due to

  • Daniel Quillen, Higher algebraic K-theory I, Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 85–147. Lecture

    Notes in Math., Vol. 341

The generalization to the Waldhausen S-construction is due to

  • Friedhelm Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983), Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419 (pdf)

Refinement of the construction to stable (∞,1)-categories and exact (infinity,1)-categories? is discussed in

See also

  • Motivation/interpretation for Quillen’s Q-construction?, MO/1006.

Last revised on November 11, 2019 at 18:53:47. See the history of this page for a list of all contributions to it.