## Normalisation for the fundamental crossed complex of (2006)

### BibTeX

@MISC{Brown06normalisationfor,

author = {Ronald Brown and Rafael Sivera},

title = {Normalisation for the fundamental crossed complex of},

year = {2006}

}

### OpenURL

### Abstract

a simplicial set

### Citations

149 |
Elements of homotopy theory
- Whitehead
- 1978
(Show Context)
Citation Context ...ex of (A,∂). This is the normalisation theorem. In homotopy, rather than homology, theory, there is another and more complicated basic formula, known as the Homotopy Addition Lemma (HAL) (or theorem) =-=[22, 29]-=-. It has roughly the same import as (1), namely it gives ‘the boundary of a simplex’, but it takes account also of: ∗ Brown was supported by a Leverhulme Emeritus Fellowship, 2002-2004, which also sup... |

123 | On the algebra of cubes
- Brown, Higgins
- 1981
(Show Context)
Citation Context ...motopy classification theorem in the non simply connected case, and includes many classical results. It is a considerable work to set up all these results, which require moving to a cubical category, =-=[11]-=-, for the proofs. It is planned that the book in preparation, [16], which will give a full account in one place of these main properties, will make these results more accessible and so make them more ... |

75 | Colimit theorems for relative homotopy groups
- BROWN, HIGGINS
- 1981
(Show Context)
Citation Context ...omplexes were found in a sequence of papers by Brown and Higgins. These facts are: (i) The functor Π : FTop → Crs from the category of filtered spaces to crossed complexes preserves certain colimits, =-=[12]-=-; (ii) the category Crs is monoidal closed, [13], with an exponential law of the form Crs(A ⊗ B,C) ∼ = Crs(A,CRS(B,C)). (exponential law) (iii) The category Crs has a unit interval object written {0} ... |

67 | On the 3-type of a complex - Lane, Whitehead - 1950 |

50 | Groupoids and crossed objects in algebraic topology - Brown - 1999 |

44 | Tensor product and homotopies for ω-groupoids and crossed complexes - Brown, Higgins - 1987 |

44 | The classifying space of a crossed complex
- Brown, Higgins
- 1991
(Show Context)
Citation Context ...ndamental crossed complex Π(X∗) of the filtered space (see below). Blakers associated to a reduced crossed complex C a simplicial set which nowadays we would call the nerve NC of the crossed complex, =-=[15]-=-; the definition uses the Homotopy Addition Lemma in an essential way, although a detailed and elementary proof of that was available only in 1953, [22]. Blakers’ concept was taken up in J.H.C. Whiteh... |

37 | On adding relations to homotopy groups - Whitehead - 1941 |

30 |
Simple homotopy types
- Whitehead
- 1950
(Show Context)
Citation Context ...ing the relative Hurewicz theorem, and is also a consequence of the Generalised van Kampen Theorem of [12]. We now give a proposition and a counterexample due in the crossed module case to Whitehead, =-=[33]-=-, which illustrate some of the difficulties of working with free crossed modules. Theorem 4.1 Let C be the free crossed complex on R∗, and suppose S∗ ⊆ R∗ generates a subcrossed complex B of C. Let F ... |

22 |
Identities among relations, in Low dimensional topology, Ed. R.Brown and
- Brown, Huebschmann
- 1982
(Show Context)
Citation Context ... morphism π1B → π1C is injective. Proof We use the functor ∆ : Crs → Chn to chain complexes with a groupoid of operators which was introduced in [14], generalising work of Whitehead in CHII. See also =-=[17]-=- for the low dimensional and reduced cases. First of all, we know that a subgroupoid of a free groupoid is free. Also in dimensions > 2 Cn is the free π1C-module on the basis Rn. So injectivity, under... |

20 | Crossed complexes and chain complexes with operators
- Brown, Higgins
- 1990
(Show Context)
Citation Context ... the induced morphism F → C is injective if the induced morphism π1B → π1C is injective. Proof We use the functor ∆ : Crs → Chn to chain complexes with a groupoid of operators which was introduced in =-=[14]-=-, generalising work of Whitehead in CHII. See also [17] for the low dimensional and reduced cases. First of all, we know that a subgroupoid of a free groupoid is free. Also in dimensions > 2 Cn is the... |

18 | Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems
- Brown
(Show Context)
Citation Context ...both categories are also required for some results. For example the natural transformation of Theorem 1.1 is easy to see in the cubical category. For a survey on crossed complexes and their uses, see =-=[9]-=-. (6)sdraft RB May 31, 2006 8 2 Generating complexes Let C be a crossed complex, and let R∗ be a family of subsets Rn ⊆ Cn for all n � 0. We have to explain what is meant by the subcrossed complex 〈R∗... |

16 | Theory and applications of crossed complexes
- Tonks
- 1993
(Show Context)
Citation Context ...mmediate that the projection induces an isomorphism of fundamental groupoids and of the homologies of the universal covers at all base points. ✷ For further work on crossed complexes, see for example =-=[1, 3, 4, 27, 28]-=-. The first three works refer to crossed complexes as crossed chain complexes. It can be argued that the category Crs gives a linear approximation to homotopy theory: that is, crossed complexes incorp... |

13 |
Graphs of groups: word computations and free crossed resolutions
- Moore
- 2001
(Show Context)
Citation Context ...mmediate that the projection induces an isomorphism of fundamental groupoids and of the homologies of the universal covers at all base points. ✷ For further work on crossed complexes, see for example =-=[1, 3, 4, 27, 28]-=-. The first three works refer to crossed complexes as crossed chain complexes. It can be argued that the category Crs gives a linear approximation to homotopy theory: that is, crossed complexes incorp... |

12 | On relative homotopy group of the product filtration, the James construction and a formula of Hopf
- Baues, Brown
- 1993
(Show Context)
Citation Context ...[15]. (v) There is for filtered spaces X∗,Y∗ a natural transformation η : ΠX∗ ⊗ ΠY∗ → Π(X∗ ⊗ Y∗), which is an isomorphism if X∗,Y∗ are the skeletal filtrations of CW-complexes, [15], and more widely, =-=[2]-=-. The first result is a kind of Generalised van Kampen Theorem, GvKT, and has consequences which include the relative Hurewicz Theorem, and nonabelian results in dimension 2, [20], not seemingly obtai... |

12 |
A model structure for the homotopy theory of crossed complexes. Cahiers Topologie Geom. Differentielle Categoriques 30
- Brown, Golasinski
- 1989
(Show Context)
Citation Context ... in the first instance we have R0 = F0, and F1 is the free groupoid on the graph (R1,R0,s,t). We assume this concept as known; it is fully treated in [7]. For free crossed complexes, we refer also to =-=[10, 15]-=- for more details. Secondly, R2 comes with a function w : R2 → F1 given by the restriction of δ2. We require that the inclusion R2 → F2 makes F2 the free crossed F1-module on R2. By this stage, the fu... |

12 | Salleh, Free crossed resolutions of groups and presentations of modules of identities among relations
- Brown, Razak
(Show Context)
Citation Context ...much more complicated than the chain complexes which are the setting for the boundary formula (1). Yet formulae of these type occur frequently in mathematics, for example in the cohomology of groups, =-=[19]-=-, in differential geometry, [24], and in the cohomology of stacks, [6]. The formal structure required for this HAL is known as a crossed complex of groupoids, and they form the objects of a category w... |

10 |
Algebraic Homotopy, volume 15 of Cambridge
- Baues
- 1989
(Show Context)
Citation Context ...mmediate that the projection induces an isomorphism of fundamental groupoids and of the homologies of the universal covers at all base points. ✷ For further work on crossed complexes, see for example =-=[1, 3, 4, 27, 28]-=-. The first three works refer to crossed complexes as crossed chain complexes. It can be argued that the category Crs gives a linear approximation to homotopy theory: that is, crossed complexes incorp... |

10 |
On the tensor algebra of a nonabelian group and applications
- Baues
(Show Context)
Citation Context |

10 | On the twisted cobar construction
- Baues, Tonks
- 1997
(Show Context)
Citation Context |

10 | Algebraic Topology - Brown - 1968 |

10 | Computation and homotopical applications of induced crossed modules
- Brown, Wensley
- 1959
(Show Context)
Citation Context ...15], and more widely, [2]. The first result is a kind of Generalised van Kampen Theorem, GvKT, and has consequences which include the relative Hurewicz Theorem, and nonabelian results in dimension 2, =-=[20]-=-, not seemingly obtainable by other means. The fourth result is a homotopy classification theorem in the non simply connected case, and includes many classical results. It is a considerable work to se... |

8 | Crossed complexes, and free crossed resolutions. for amalgamated sums and HNN-extensions of groups
- Brown, Moore, et al.
(Show Context)
Citation Context ... category Crs of crossed complexes: our viewpoint, following that of CHII, is that Crs should be seen as a basic category for applications in algebraic topology, with better realisability properties, =-=[15, 18]-=-, than the more usual chain complexes with a group of operators. From the point of view of practical algebraic topology, important additional facts on crossed complexes were found in a sequence of pap... |

7 |
Abstract and Simple Homotopy Theory, World Scientific
- Kamps, Porter
- 1997
(Show Context)
Citation Context ...extending trivially in higher dimensions than 1. This crossed complex, which is isomorphic to ΠE 1 ∗, can be given the structure of a unit interval object {0} ⇉ I in the category Crs (see for example =-=[23]-=-). This allows us to define homotopies of morphisms B → C of crossed complexes as morphisms I ⊗ B → C, or, equivalently, as morphisms I → CRS(B,C). The detailed structure of this cylinder object I ⊗ C... |

5 |
On the classification of 2-gerbes and 2-stacks, Astérisque
- Breen
- 1994
(Show Context)
Citation Context ...r the boundary formula (1). Yet formulae of these type occur frequently in mathematics, for example in the cohomology of groups, [19], in differential geometry, [24], and in the cohomology of stacks, =-=[6]-=-. The formal structure required for this HAL is known as a crossed complex of groupoids, and they form the objects of a category which we write Crs. This category is complete and cocomplete. It contai... |

4 |
The homotopy addition theorem
- Hu
- 1953
(Show Context)
Citation Context ...ex of (A,∂). This is the normalisation theorem. In homotopy, rather than homology, theory, there is another and more complicated basic formula, known as the Homotopy Addition Lemma (HAL) (or theorem) =-=[22, 29]-=-. It has roughly the same import as (1), namely it gives ‘the boundary of a simplex’, but it takes account also of: ∗ Brown was supported by a Leverhulme Emeritus Fellowship, 2002-2004, which also sup... |

2 |
Some relations between homotopy and homology
- Blakers
- 1948
(Show Context)
Citation Context ...erature, but they follow from our conventions given in section 5 for crossed complexes and their ‘cylinder object’ I ⊗ C. Crossed complexes, but called group systems, were first defined by Blakers in =-=[5]-=-. This concept combined into a single structure the fundamental group π1(X,x) and the relative homotopy groups πn(Xn,Xn−1,x),n � 2 associated to a filtered space X∗, but only in the reduced case, i.e.... |

2 |
reprinted as a Theory and Applications of Categories Reprint
- Higgins, Categories, et al.
- 1971
(Show Context)
Citation Context ...s, or composition operators. The tensor product, and corresponding notions of an ‘algebra’, allow for more structure, as in [3, 4]. We assume work on groupoids, for example normal subgroupoids, as in =-=[7, 21]-=-. 1 Basic definitions for crossed complexes We use relative homotopy theory to construct a functor where FTop is the category of filtered spaces, whose objects Π : FTop → Crs (3) X∗ : X0 ⊆ X1 ⊆ · · · ... |

2 |
Combinatorics of curvature and the Bianci identity
- Kock
- 1996
(Show Context)
Citation Context ...hain complexes which are the setting for the boundary formula (1). Yet formulae of these type occur frequently in mathematics, for example in the cohomology of groups, [19], in differential geometry, =-=[24]-=-, and in the cohomology of stacks, [6]. The formal structure required for this HAL is known as a crossed complex of groupoids, and they form the objects of a category which we write Crs. This category... |

1 |
Topology and Groupoids, (2006) Booksurge LLC; (revised and retitled version of previous editions
- Brown
- 1968
(Show Context)
Citation Context ...s, or composition operators. The tensor product, and corresponding notions of an ‘algebra’, allow for more structure, as in [3, 4]. We assume work on groupoids, for example normal subgroupoids, as in =-=[7, 21]-=-. 1 Basic definitions for crossed complexes We use relative homotopy theory to construct a functor where FTop is the category of filtered spaces, whose objects Π : FTop → Crs (3) X∗ : X0 ⊆ X1 ⊆ · · · ... |