# Colimits of topological groupoids

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.

• 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. Oct 16, 2010 at 0:14
• math.stackexchange.com/questions/5095/… (I'm pretty sure the same idea works for groups replaced by groupoids). Oct 17, 2010 at 16:20
• @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). Oct 17, 2010 at 17:33
• link to HArdy-Brown article: pages.bangor.ac.uk/~mas010/pdffiles/brown-hardy1.pdf (from R. Brown web page) Oct 17, 2011 at 10:03

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!