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Homotopy Limits and Homotopy Colimits of Chain Complexes

Kensuke Arakawa

TL;DR

This work gives explicit, concrete models for homotopy colimits and limits of chain complexes using bar and cobar constructions, extending the Bousfield--Kan framework to (weakly) framed model categories via fat realizations. It shows that, under suitable structural hypotheses (e.g., simplicially admissible model structures on Ch(A) and AB4_κ-type conditions), the BK formula computes hocolimits and the cobar formula computes homotopy limits, with dual results. A central technical achievement is relating geometric realizations of simplicial chain complexes to totalizations of double complexes, and proving a suite of lemmas ensuring preservation of quasi-isomorphisms under these constructions. The paper also delineates practical criteria for when Ch(A) carries simplicially admissible model structures and provides concrete examples (injective/projective, Hovey–Gillespie, Cisinski–Déglise) to guide applications, situating the BK framework firmly within chain-homological contexts and its homotopical calculus. Overall, the results unify and extend methods for computing hocolimits and holimits in chain complexes across a broad class of model-categorical settings.

Abstract

We give a formula for homotopy limits and homotopy colimits of chain complexes using the cobar and bar constructions, also known as the Bousfield--Kan formula. Along the way, we show that the Bousfield--Kan formula computes homotopy colimits in any framed model categories.

Homotopy Limits and Homotopy Colimits of Chain Complexes

TL;DR

This work gives explicit, concrete models for homotopy colimits and limits of chain complexes using bar and cobar constructions, extending the Bousfield--Kan framework to (weakly) framed model categories via fat realizations. It shows that, under suitable structural hypotheses (e.g., simplicially admissible model structures on Ch(A) and AB4_κ-type conditions), the BK formula computes hocolimits and the cobar formula computes homotopy limits, with dual results. A central technical achievement is relating geometric realizations of simplicial chain complexes to totalizations of double complexes, and proving a suite of lemmas ensuring preservation of quasi-isomorphisms under these constructions. The paper also delineates practical criteria for when Ch(A) carries simplicially admissible model structures and provides concrete examples (injective/projective, Hovey–Gillespie, Cisinski–Déglise) to guide applications, situating the BK framework firmly within chain-homological contexts and its homotopical calculus. Overall, the results unify and extend methods for computing hocolimits and holimits in chain complexes across a broad class of model-categorical settings.

Abstract

We give a formula for homotopy limits and homotopy colimits of chain complexes using the cobar and bar constructions, also known as the Bousfield--Kan formula. Along the way, we show that the Bousfield--Kan formula computes homotopy colimits in any framed model categories.
Paper Structure (9 sections, 27 theorems, 68 equations)

This paper contains 9 sections, 27 theorems, 68 equations.

Table of Contents

  1. Bousfield--Kan Formula in (Weakly) Framed Model Categories
  2. Homotopy Colimits and Derived Functors
  3. Bousfield--Kan Formula in Framed Model Categories
  4. Geometric Realizations of Simplicial Chain Complexes
  5. Realizations and Totalizations of Double Complexes
  6. Two More Lemmas
  7. Main Results
  8. Proofs of the Main Theorems
  9. Simplicially Admissible Model Structures on Ch(A)

Key Result

Theorem 1

Let $\mathcal{A}$ be bicomplete abelian category, and suppose $\sf{Ch}\left(\mathcal{A}\right)$ is equipped with a model structure. Under a mild assumption on the model structure, the homotopy colimit of a small, pointwise cofibrant diagram $F:\mathcal{I}\to\sf{Ch}\left(\mathcal{A}\right)$ is modele

Theorems & Definitions (71)

  • Theorem 1: Theorem \ref{['thm:main1_precise']}
  • Theorem 2: Theorem \ref{['thm:main2_precise']}
  • Definition 1.1
  • Example 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • ...and 61 more