nLab
model structure on chain complexes

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for ∞-sheaves / ∞-stacks

Edit this sidebar

Contents

Idea

Under the Dold–Kan correspondence there is an equivalence of categories relating non-negatively graded chain complexes in an abelian category A to simplicial objects in A.

Ch +(A)≃A Δ op.Ch_+(A) \simeq A^{\Delta^{op}} \,.

At least if A is the category of abelian groups, so that A Δ op is the category of abelian simplicial groups it inherits naturally a homotopical category from the model structure on simplicial sets. This is a very canonical model structure in that it is no less than the presentation of the archetypical (infinity,1)-category Infinity-Grpd of infinity-groupoid.

The model structure on chain complexes transports this presentation of the special ∞–groupoids given by abelian simplicial groups along the Dold-Kan correspondence to chain complexes.

Analogous statements apply to the category of unbounded chain complexes and the canonical stable (infinity,1)-category Spec of spectra.

Spaltenstein wrote a famous paper

  • N. Spaltenstein, Resoutions of unbounded complexes, Compositio Mathematica, 65 no. 2 (1988), p. 121-154 (numdam)

on how to do homological algebra with unbounded complexes (in both sides) where he introduced notions like K-projective and K-injective complexes. Later,

  • V. Hinich, Homological algebra of homotopy algebras, Comm. Algebra, vol. 25 (1997), no. 10, 3291-3323 (pdf at author’s page)

shown that there is a model category structure on the category of unbounded chain complexes, reproduced Spaltenstein’s results from that perspective and extended them widely. See also

  • J. D. Christensen, Derived categories and projective classes, 2005 (hopf archive)

Zoran Škoda: It is misleading to put positive complexes in the center of this nlab entry, because this is just the easiest, cheapest part of the story. I do not think the results for positive, negative or bounded complexes were just easily generalized from unbounded as it is impression from the sentence above on “analogous results”. Both Spaltenstein and Hinich wrote breakthrough papers over 20 years after Quillen’s school understood the examples of one-side-unbounded and bounded cases.

This Spaltenstein-Hinich story which I just mentioned with writing out the references, would thus be more essential to precisely cover (and will be more essentially used) than making the long story out of trivial resort to Dold-Kan argument, and modern lingo on (infty,1).

History and references

Of course the above description of categories of chain complexes as (presentations of) special cases of (stable) (∞,1)-categories is exactly opposite to the historical development of these ideas.

While the homotopical treatment of weak equivalences of chain complexes (quasi-isomorphisms) in homological algebra is at the beginning of all studies of higher categories and a “folk theorem” ever since

  • A. Joyal, Letter to Alexander Grothendieck. April 11, 1984

it seems that the injective model structure on chain complexes has been made fully explicit in print only in proposition 3.13 of

  • Tibor Beke, Sheafifiable homotopy model categories (arXiv, pdf)

(at least according to the remark below that).

The projective model structure is discussed after that in

  • Mark Hovey, Model category structures on chain complexes of sheaves, Trans. Amer. Math. Soc. 353, 6 (pdf)

Revised on December 29, 2009 08:07:49 by Toby Bartels (71.31.210.131)
Edit | Back in time (5 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: canonical model structure, model category, category with weak equivalences, homotopy category, model structure on simplicial sets, category of fibrant objects, model structure on topological spaces, monoidal model category, topology, path space object, small object argument, Dold-Kan correspondence, homotopy limit, model structure on simplicial presheaves, Higher Topos Theory, Thomason model structure, chain complex, derived category, Bousfield-Kan map, simplicial model category, enriched model category, model structure on strict omega-groupoids, fiber sequence, proper model category, Quillen equivalence, global model structure on simplicial presheaves, Quillen adjunction, 2009 July changes, model structure on sSet-enriched presheaves, model structure on chain complexes, combinatorial model category, cofibrantly generated model category, homotopy image, Reedy model structure, model structure on functors, (infinity,1)-categorical hom-space, model structure for Cartesian fibrations, derived noncommutative geometry, model structure on dg-algebras, Quillen bifunctor, model category theory - contents, model structure on cosimplicial rings, model structure on dendroidal sets, tractable model category, Theta-space, model structure on homotopical presheaves, cellular model category, model structure on an over category, Bousfield localization of model categories, model structure on dg-coalgebras, canonical model structure on Cat, cocycle, model structure for complete Segal spaces, model structure for quasi-categories, model structure on categories with weak equivalences, model structure on algebraic fibrant objects, model structure on sSet-categories, stable model category, Hochschild cohomology, model structure on cubical sets, model structure for left fibrations, model structure on operads, transferred model structure, model structure for weak complicial sets, homotopy Kan extension, Cisinski model structure, fat simplex, resolution, factorization lemma