nLab Quillen Q-construction

Idea

The Quillen Q-construction (Quillen 72) is a tool for producing the algebraic K-theory of a Quillen exact category $\mathcal{C}$. 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).

Related concepts

• Waldhausen S-construction

References

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

• Rune Haugseng, the Q-construction for stable $\infty$-Categories, 2010 (can be found here)

• Clark Barwick, On the Q construction for exact quasicategories (arXiv:1301.4725)

• Clark Barwick, On exact infinity-categories and the Theorem of the Heart, arXiv:1212.5232.

See also

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