Topological groupoids: I. universal constructions

Brown, Ronald; Hardy, J. P. L.

doi:10.1002/mana.19760710123

Cited by 28 publications

(12 citation statements)

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“…Examples 4.11. Proposition 4.10 applies when K = V , because the discrete complex is flag; then colim tmg G K is isomorphic to the free product of m copies of G, whose classifying space is the m-fold wedge m j=1 BG j (by [6], for example). On the other hand, when K is the non-flag complex ∂(m), Example 4.3 confirms that B colim tmg G K is BG m , whereas colim + (BG) K is the fat wedge subspace.…”

Section: Colimitsmentioning

confidence: 99%

Colimits, Stanley-Reisner Algebras, and Loop Spaces

Panov

Ray

Vogt

2003

Categorical Decomposition Techniques in Algebraic Topology

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We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: right-angled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); Stanley-Reisner algebras and coalgebras; Davis and Januszkiewicz's spaces DJ(K) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend well-known results on Artin and Coxeter groups by confirming that the relevant circulation group is homotopy equivalent to the space of loops ΩDJ(K). We define homotopy colimits for diagrams of topological monoids and topological groups, and show they commute with the formation of classifying spaces in a suitably generalised sense. We deduce that the homotopy colimit of the appropriate diagram of topological groups is a model for ΩDJ(K) for an arbitrary complex K, and that the natural projection onto the original colimit is a homotopy equivalence when K is flag. In this case, the two models are compatible.

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Section: Colimitsmentioning

confidence: 99%

Colimits, Stanley-Reisner Algebras, and Loop Spaces

Panov

Ray

Vogt

2003

Categorical Decomposition Techniques in Algebraic Topology

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“…It was proven in [2] that if G is a topological groupoid then G/N can be endowed with a unique topology such that it makes G/N into a topological groupoid, the projection π : G → G/N is continuous and for every continuous morphism of topological groupoids (φ, f ) :…”

Section: Definition 213 a Topological Groupoid Is Principal Ifmentioning

confidence: 99%