Homology and standard constructions

Barr, Michael; Beck, Jon

doi:10.1007/bfb0083087

Cited by 146 publications

(170 citation statements)

References 27 publications

Supporting

Mentioning

167

Contrasting

Unclassified

Order By: Relevance

“…In Sections 2 to 6 we consider the coefficients corresponding to the split extension (3), and study the corresponding (co)homology theory. After recalling some background in Section 1, in Section 2 we introduce the (co)homology.…”

Section: Ccg N (T G µ) → H N G → H N B(t G µ) → · · · (1)mentioning

confidence: 99%

See 1 more Smart Citation

(Co)homology of crossed modules with coefficients in a $\pi_1$-module

Paoli

2003

Homology, Homotopy and Applications

View full text Add to dashboard Cite

show abstract

Section: Ccg N (T G µ) → H N G → H N B(t G µ) → · · · (1)mentioning

confidence: 99%

“…The following are elementary properties of the (co)homology which can be deduced from well known general facts about cotriple (co)homology in an algebraic category [3]. In what follows a projective crossed module means a projective object in the category of crossed modules.…”

Section: Definition 3 Let φ = (T G µ) Be a Crossed Module Definementioning

confidence: 99%

(Co)homology of crossed modules with coefficients in a $\pi_1$-module

Paoli

2003

Homology, Homotopy and Applications

View full text Add to dashboard Cite

show abstract

“…It is by now well known that (/' and hence U are tripleable in each of the cases A, S, L, and G of [3] (which are, respectively, unitary ZC-algebras, supplemented unitary ZC-algebras, Lie algebras over K, and groups), and that the H"(X, M) defined by Barr and Rinehart agree with the resulting cotriple cohomology groups (see e.g. [2]). Thus assumption (3) above is true.…”

Section: H Van Osdolmentioning

confidence: 98%

Relative Barr-Rinehart and cotriple cohomology groups are isomorphic

Osdol¹

1984

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The theorem, stated in the title of this article, is proved.Several people have asked about the relationship between the relative cohomology groups defined by Barr and Rinehart [3], and those of [6]. The point of this note is to show that these groups (in the cases considered by Barr and Rinehart) coincide.In a sense, this result adds perspective on the creative tension which exists between derived functors and Ext groups. Barr has examples (see, e.g. [1]) which show that, in the theory of commutative rings, the functor which classifies extensions is definitely not a derived functor. By the results of Beck [4] and Duskin [5] (see also [6]) any algebraic theory defines a cohomology theory, the "cotriple-derived functors", which classifies extensions. The paper of Barr and Rinehart can thus be viewed as a statement that for certain theories these intrinsically defined groups can equally well be computed by the more classical approach of "ordinary" derived functors on a module category. This note shows that the Barr-Rinehart relative groups Exl"R(S/S2, M) also appear naturally in the cotriple theory. This raises the following unanswered question: what conditions on an algebraic theory make it possible to compute the intrinsic long exact cohomology sequences via ordinary derived functors? Here we shall give conditions, but they are not really conditions on the theory.

show abstract

“…For an object M ∈ M, the triple resolution α : Various authors including Barr-Beck [2], Bousfield-Kan [18], and BenderskyThompson [7] have used triple resolutions to define right derived functors, completions, or homotopy spectral sequences, and we can now fit these constructions into our framework. Starting with a triple, we shall find a compatible class of injective models giving the following:…”

Section: Triples and Triple Resolutionsmentioning

confidence: 99%

Cosimplicial resolutions and homotopy spectral sequences in model categories

Bousfield

2003

Geom. Topol.

108

View full text Add to dashboard Cite

We develop a general theory of cosimplicial resolutions, homotopy spectral sequences, and completions for objects in model categories, extending work of Bousfield-Kan and Bendersky-Thompson for ordinary spaces. This is based on a generalized cosimplicial version of the Dwyer-Kan-Stover theory of resolution model categories, and we are able to construct our homotopy spectral sequences and completions using very flexible weak resolutions in the spirit of relative homological algebra. We deduce that our completion functors have triple structures and preserve certain fiber squares up to homotopy. We also deduce that the Bendersky-Thompson completions over connective ring spectra are equivalent to Bousfield-Kan completions over solid rings. The present work allows us to show, in a subsequent paper, that the homotopy spectral sequences over arbitrary ring spectra have well-behaved composition pairings.

show abstract

Homology and standard constructions

Cited by 146 publications

References 27 publications

(Co)homology of crossed modules with coefficients in a $\pi_1$-module

(Co)homology of crossed modules with coefficients in a $\pi_1$-module

Relative Barr-Rinehart and cotriple cohomology groups are isomorphic

Cosimplicial resolutions and homotopy spectral sequences in model categories

Contact Info

Product

Resources

About