Let $G$ and $H$ be two topological groupoids. Suppose that I have two morphisms $G \rightrightarrows H$ and I want to take the 2-coequalizer of these maps. I'd like an explicit description of (a particular model for) this weak colimit. I can do this very easily for groupoids in SET by constructing a groupoid with objects $G_0 \coprod H_0$ where I desribe the arrows in terms of generators and relations. However, then I don't know what topology to put on the arrows. I'd also be happy, if instead of this, someone knew an explicit description of a weak pushout diagram of topological groupoids.
$\begingroup$ Explicit descriptions of the topologies of pushouts (even coproducts) of topological groups are pretty complicated. I would imagine the same is true for groupoids. Brown and Hardy's paper "Topological Groupoids I: Universal constructions" may be a starting point. $\endgroup$– Jeremy BrazasOct 16, 2010 at 0:14
$\begingroup$ math.stackexchange.com/questions/5095/… (I'm pretty sure the same idea works for groups replaced by groupoids). $\endgroup$– Martin BrandenburgOct 17, 2010 at 16:20
$\begingroup$ @Martin, coproducts are trivially given by the coproduct of the objects and arrows- and yes, since there's no such thing as a "weakened" coproduct, this is a weak 2-colimit. However, I was more concerned with constructing weak coequalizers (or weak pushouts) precisely because it is these notions which DO NOT agree with their non-weak version (and are difficult to write down apparently). $\endgroup$– David CarchediOct 17, 2010 at 17:33
$\begingroup$ link to HArdy-Brown article: pages.bangor.ac.uk/~mas010/pdffiles/brown-hardy1.pdf (from R. Brown web page) $\endgroup$– Buschi SergioOct 17, 2011 at 10:03
General colimits of topological groupoids are shown to exist in the paper with Lew Hardy referred to by Jeremy (Math. Nachr. 71 (1976) 273-286.); essentiall,y existence is an application of the Freyd adjoint functor theorem. This also makes it quite difficult to describe the topology explicitly, but in practice we often want only the universal property.
Homotopy colimits are constructed as colimits: for example a homotopy pushout is a double mapping cylinder, where the topologically discrete groupoid $\mathcal I$, the groupoid version of the unit interval, with two objects $0,1$, takes the place of the usual unit interval. The general homotopy colimit is more complicated, and I think you would have to refer to specialised papers, or ask Tim Porter!