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Cofibrance and Completion

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Andrei Radulescu-Banu

Algebraic Topology · Submitted 8 Dec 2006

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This allows us to construct a Bousfield-Kan completion functor with respect to a triple, and for pointed cofibrantly-generated model categories a Bousfield-Kan spectral sequence that computes the relative homotopy groups of the Bousfield-Kan completion of an object.

how it is stated · asserted 1.00 read the record →

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arXiv IDmath/0612203 (version 1)
Pith IDS332PDYA, Pith's permanent record of this paper and its review
CitationAndrei Radulescu-Banu. "Cofibrance and Completion." arXiv:math/0612203 (2006). https://pith.science/paper/S332PDYA
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,607 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.65 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.24 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

For a cofibrantly generated Quillen model category, we show that the cofibrant replacement functor constructed using the small object argument admits a cotriple structure. If all acyclic cofibrations are monomorphisms, the fibrant replacement functor constructed using the small object argument admits a triple structure. For a triple in the base category, the associated cosimplicial resolution is not necessarily homotopy invariant. However using a mix of the triple with the cofibrant replacement cotriple we construct a 'homotopically correct' version of the cosimplicial resolution of the triple. This allows us to construct a Bousfield-Kan completion functor with respect to a triple, and for pointed cofibrantly-generated model categories a Bousfield-Kan spectral sequence that computes the relative homotopy groups of the Bousfield-Kan completion of an object. This is the text of my PhD thesis, worked under the supervision of Prof. Haynes Miller, submitted on Feb. 1999 at MIT.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 For a cofibrantly generated Quillen model category, we show that the cofibrant replacement functor constructed using the small object argument admits a cotriple structure. result 0.79 0.93 0.22 asserted 1.00
s1 If all acyclic cofibrations are monomorphisms, the fibrant replacement functor constructed using the small object argument admits a triple structure. result 0.93 0.80 0.11 asserted 1.00
s2 For a triple in the base category, the associated cosimplicial resolution is not necessarily homotopy invariant. background 0.38 0.47 0.05 asserted 0.80
s3 However using a mix of the triple with the cofibrant replacement cotriple we construct a 'homotopically correct' version of the cosimplicial resolution of the triple. contribution 0.65 0.92 0.17 asserted 0.99
s4 This allows us to construct a Bousfield-Kan completion functor with respect to a triple, and for pointed cofibrantly-generated model categories a Bousfield-Kan spectral sequence that computes the relative homotopy groups of the Bousfield-Kan completion of an object. contribution 0.65 0.90 0.61 asserted 1.00
s5 This is the text of my PhD thesis, worked under the supervision of Prof. background 0.91 0.44 0.01 asserted 0.87
s6 Haynes Miller, submitted on Feb. 1999 at MIT. background 0.98 0.13 0.01 asserted 0.65

What kind of paper the classifier reports this is: theoretical result 0.89 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

For a cofibrantly generated Quillen model category, we show that the cofibrant replacement functor constructed using the small object argument admits a cotriple structure. If all acyclic cofibrations are monomorphisms, the fibrant replacement functor constructed using the small object argument admits a triple structure. For a triple in the base category, the associated cosimplicial resolution is not necessarily homotopy invariant. However using a mix of the triple with the cofibrant replacement cotriple we construct a 'homotopically correct' version of the cosimplicial resolution of the triple. This allows us to construct a Bousfield-Kan completion functor with respect to a triple, and for pointed cofibrantly-generated model categories a Bousfield-Kan spectral sequence that computes the relative homotopy groups of the Bousfield-Kan completion of an object. This is the text of my PhD thesis, worked under the supervision of Prof. Haynes Miller, submitted on Feb. 1999 at MIT.

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