Quotients of Group Actions as Colimits

Our next goal is to interpret the quotients of group actions as colimits. Assume, that X is some mathematical object (e.g., topological space, vector space, abstract simplicial complex, graph), and assume that a group G acts on X. Let us first informally contemplate what a quotient X/G should be. 

Imagine that X itself consists of some kind of elements (points, vectors, simplices). Then the most natural thing to do would be to glue together two elements whenever one is mapped to the other by the group action. In other words, take orbits of elements as the new elements. 

A problem arises when one also tries in a consistent way to impose additional structure to make sure that the collection of orbits of X can be made into the mathematical object of the same nature (read belonging to the same category) as A. In many situations this will lead to additional identifications, which can be controlled with a varied degree of success. 

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