Loop spaces of the Q-construction

A. Neeman

Fundamenta Mathematicae (2000)

  • Volume: 164, Issue: 1, page 71-95
  • ISSN: 0016-2736

Abstract

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Giffen in [1], and Gillet-Grayson in [3], independently found a simplicial model for the loop space on Quillen's Q-construction. Their proofs work for exact categories. Here we generalise the results to the K-theory of triangulated categories. The old proofs do not generalise. Our new proof, aside from giving the generalised result, can also be viewed as an amusing new proof of the old theorems of Giffen and Gillet-Grayson.

How to cite

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Neeman, A.. "Loop spaces of the Q-construction." Fundamenta Mathematicae 164.1 (2000): 71-95. <http://eudml.org/doc/212449>.

@article{Neeman2000,
abstract = {Giffen in [1], and Gillet-Grayson in [3], independently found a simplicial model for the loop space on Quillen's Q-construction. Their proofs work for exact categories. Here we generalise the results to the K-theory of triangulated categories. The old proofs do not generalise. Our new proof, aside from giving the generalised result, can also be viewed as an amusing new proof of the old theorems of Giffen and Gillet-Grayson.},
author = {Neeman, A.},
journal = {Fundamenta Mathematicae},
keywords = {-construction; -theory for triangulated categories; loop space},
language = {eng},
number = {1},
pages = {71-95},
title = {Loop spaces of the Q-construction},
url = {http://eudml.org/doc/212449},
volume = {164},
year = {2000},
}

TY - JOUR
AU - Neeman, A.
TI - Loop spaces of the Q-construction
JO - Fundamenta Mathematicae
PY - 2000
VL - 164
IS - 1
SP - 71
EP - 95
AB - Giffen in [1], and Gillet-Grayson in [3], independently found a simplicial model for the loop space on Quillen's Q-construction. Their proofs work for exact categories. Here we generalise the results to the K-theory of triangulated categories. The old proofs do not generalise. Our new proof, aside from giving the generalised result, can also be viewed as an amusing new proof of the old theorems of Giffen and Gillet-Grayson.
LA - eng
KW - -construction; -theory for triangulated categories; loop space
UR - http://eudml.org/doc/212449
ER -

References

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  1. [1] C. H. Giffen, Loop spaces for the Q-construction, J. Pure Appl. Algebra 52 (1988), 1-30. Zbl0655.18008
  2. [2] C. H. Giffen, Unitary algebraic Witt- and K-theories, preprint. 
  3. [3] H. Gillet and D. R. Grayson, The loop space of the Q-construction, Illinois J. Math. 31 (1987), 574-597. Zbl0628.55011
  4. [4] D. R. Grayson, Higher algebraic K-theory II (after D. Quillen), in: Algebraic K-Theory, Lecture Notes in Math. 551, Springer, 1976, 217-240. 
  5. [5] D. R. Grayson, Exterior power operations on higher K-theory, K-Theory 3 (1989), 247-260. Zbl0701.18007
  6. [6] J J. F. Jardine, The multiple Q-construction, Canad. J. Math. 39 (1987), 1174-1209. Zbl0656.55007
  7. [7] A. Neeman, K-theory for triangulated categories I(A), Asian J. Math. 1 (1997), 330-417. Zbl0906.19002
  8. [8] A. Neeman, K-theory for triangulated categories I(B), ibid. 1 (1997), 435-529. Zbl0906.19003
  9. [9] A. Neeman, K-theory for triangulated categories II, ibid. 2 (1998), 1-125. Zbl0923.19002
  10. [10] A. Neeman, K-theory for triangulated categories III(A), ibid. 2 (1998), 495-594. Zbl0937.19001
  11. [11] A. Neeman, K-theory for triangulated categories III(B), ibid. 3 (1999), 555-606. 
  12. [12] D. Quillen, Higher algebraic K-theory I, in: Algebraic K-Theory I, Lecture Notes in Math. 341, Springer, 1973, 85-147. Zbl0292.18004
  13. [13] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293-312. Zbl0284.55016

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