I am trying to better understand the topic of direct and inverse limits, sadly only in a topological context ( and not categorical). Namely, I know what is the limit of a direct spectrum {Y_i,f_{ij}} where fs are connection maps: the free union of spaces equipped with the equivalence relation of having a common successor.
Intuitively it seems to me we are putting together little spaces in a consistent way, to build a "homogeneous" object which does not shows pathological "discrete like" behaviour.
Hence, I thought of a generic infinite dimensional vector space as a good example, but I have not found such a construction. Any help, even with other examples of construction of simple well known spaces would be very useful