TITLE: what are the product and coproduct in the category of topological groups
QUESTION [27 upvotes]: I know the limits in the categories of groups, abelian groups and topological spaces and was wondering about the same thing.

REPLY [9 votes]: @Martin: @Agusti Just come across your answer, Martin,  which rightly pointed out the use of Freyd representability to get a general result. In fact this was used in essence in the paper 
R. Brown and J.P.L. Hardy,  ``Topological groupoids I: universal
constructions'', Math. Nachr. 71 (1976) 273-286.
to give colimits of topological groupoids, and so in particular topological groups. 
As you say, it is difficult to get hold of specific properties of this topology; the construction is defined by its universal property, and that is all. 
However, in the case of free topological groups, there are some quite specific exploration of the properties, with papers by Markov, Graev, and S.A. Morris, P. Nickolas.  
Here are two papers which should be relevant to the discussion: 
Nickolas, Peter, 
Coproducts of abelian topological groups. 
Topology Appl. 120 (2002), no. 3, 403–426. 
Nickolas, Peter, A Kurosh subgroup theorem for topological groups. Proc. London Math. Soc. (3) 42 (1981), no. 3, 461–477.
Another way of dealing with this problem is to work in a convenient category of spaces, usually the category of compactly generated spaces, in which the product of identification maps is an identification map.