How do we build colimits in the category Top of topological spaces? This is easy: we take the quotient of a big disjoint sum. How do we build colimits in the category Dcpo of dcpos? This is a much more complicated question. All colimits indeed exist in Dcpo, and this has been shown by various authors over time, but this is complex. I will explain why. Then I will explain what the definition of quotients should be in Dcpo, and how one can build them. Funnily, this is related to other questions, such as the existence of d-completions, for example. Read the full post.