## School of Mathematics, (2008)

### BibTeX

@MISC{Ehlers08schoolof,

author = {P. J. Ehlers and T. Porter and Dean Street and Wales U. K},

title = {School of Mathematics,},

year = {2008}

}

### OpenURL

### Abstract

We introduce a notion of join for (augmented) simplicial sets generalising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial

### Citations

933 |
Algebraic Topology
- Spanier
- 1966
(Show Context)
Citation Context ...rally from the ordinal sum on the base simplicial category ∆. 1991 Math. Subj. Class.: 18G30 1 Introduction The theory of joins of (geometric) simplicial complexes as given by Brown, [2], or Spanier, =-=[13]-=-, reveals the join operation to be a basic geometric construction. It is used in the development of several areas of geometric topology (cf. Hudson, [8]) whilst also being applied to the basic propert... |

303 |
P.: Simplicial Objects in Algebraic Topology
- May
- 1967
(Show Context)
Citation Context ...ics of ordinals. In the literature on simplicial set theory it seems rarely to be 1mentioned, yet it is sometimes there but hidden, for instance, in the Wconstruction for simplicial groups (see May, =-=[11]-=-) or simplicial groupoids (see Dwyer and Kan, [6]). In this context it occurs through the use of the Artin-Mazur codiagonal, [1], which assigns to a bisimplicial set or group, a much smaller model of ... |

274 |
Categories for the Working Mathematician. Graduate Texts in Mathematics
- Lane
- 1998
(Show Context)
Citation Context ...s is essentially the following: 4(i) take one copy of △ n for each non-degenerate n-simplex of X; and then (ii) glue them all together using the face and degeneracy maps of the simplicial set X (see =-=[10]-=-). Explicitly we have: Let X be a simplicial set. Define RX by: RX = ⊔n∈N ⊔x∈Xn △ n x Define an equivalence relation on RX as generated by the following relation: writing (p,x) for (p0, · · · ,pm) ∈ △... |

187 |
Complexe cotangent et déformations
- Illusie
- 1971
(Show Context)
Citation Context ...henever passage from bisimplicial objects to simplicial objects is needed; however, it may not always be the most efficient tool to use.) This codiagonal is linked with the total DEC functor (Illusie =-=[9]-=-, Duskin [5], Porter, [12], Bullejos et al [3]), which can be given explicitly in terms of the ordinal sum. In this brief note, it is shown that the ordinal sum leads naturally to a “join” operation o... |

129 |
Calculus of fractions and homotopy theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35
- Gabriel, Zisman
- 1967
(Show Context)
Citation Context ...ially says that the n-sphere in the category of topological spaces is the join of n + 1 copies of the 0-sphere. There are several simplicial models for the n-sphere. For instance, Gabriel and Zisman, =-=[7]-=-, p.26, define the simplicial circle, Ω, to be the coequaliser of the pair of morphisms △[0] δ0 δ1 � △[1], and the suspension of a pointed simplicial set X to be Ω ∧ X. This gives an n-sphere as being... |

53 |
P.,Piecewise Linear Topology
- Hudson
- 1969
(Show Context)
Citation Context ... complexes as given by Brown, [2], or Spanier, [13], reveals the join operation to be a basic geometric construction. It is used in the development of several areas of geometric topology (cf. Hudson, =-=[8]-=-) whilst also being applied to the basic properties of polyhedra relating to homology. The theories of geometric and abstract simplicial complexes run in a largely parallel way and when describing the... |

25 |
n-types of simplicial groups and crossed n-cubes
- Porter
- 1993
(Show Context)
Citation Context ...mplicial objects to simplicial objects is needed; however, it may not always be the most efficient tool to use.) This codiagonal is linked with the total DEC functor (Illusie [9], Duskin [5], Porter, =-=[12]-=-, Bullejos et al [3]), which can be given explicitly in terms of the ordinal sum. In this brief note, it is shown that the ordinal sum leads naturally to a “join” operation on augmented simplicial set... |

20 |
Homotopy Theory of Simplicial Groupoids
- Dwyer, Kan
- 1984
(Show Context)
Citation Context ...et theory it seems rarely to be 1mentioned, yet it is sometimes there but hidden, for instance, in the Wconstruction for simplicial groups (see May, [11]) or simplicial groupoids (see Dwyer and Kan, =-=[6]-=-). In this context it occurs through the use of the Artin-Mazur codiagonal, [1], which assigns to a bisimplicial set or group, a much smaller model of the homotopy type than does the diagonal. (The di... |

12 |
On the van Kampen theorem
- Artin, Mazur
- 1966
(Show Context)
Citation Context ...en, for instance, in the Wconstruction for simplicial groups (see May, [11]) or simplicial groupoids (see Dwyer and Kan, [6]). In this context it occurs through the use of the Artin-Mazur codiagonal, =-=[1]-=-, which assigns to a bisimplicial set or group, a much smaller model of the homotopy type than does the diagonal. (The diagonal is intuitively easier to use and tends to be “wheeled out” whenever pass... |

9 |
On cat”-groups and homotopy types
- Bullejos, Cegarra, et al.
- 1993
(Show Context)
Citation Context ...implicial objects is needed; however, it may not always be the most efficient tool to use.) This codiagonal is linked with the total DEC functor (Illusie [9], Duskin [5], Porter, [12], Bullejos et al =-=[3]-=-), which can be given explicitly in terms of the ordinal sum. In this brief note, it is shown that the ordinal sum leads naturally to a “join” operation on augmented simplicial sets, and the relation ... |

2 |
On the interpretation of Cohomology Theories definable through
- Duskin
- 1975
(Show Context)
Citation Context ...age from bisimplicial objects to simplicial objects is needed; however, it may not always be the most efficient tool to use.) This codiagonal is linked with the total DEC functor (Illusie [9], Duskin =-=[5]-=-, Porter, [12], Bullejos et al [3]), which can be given explicitly in terms of the ordinal sum. In this brief note, it is shown that the ordinal sum leads naturally to a “join” operation on augmented ... |