TITLE: Categorical Construction of Quotient Topology?
QUESTION [12 upvotes]: The product topology is the categorical product, and the disjoint union topology is the categorical coproduct. But the arrows in the characteristic diagrams for the subspace and quotient topologies point the same way as in the diagrams for the product and disjoint union topologies, respectively (but there are different conditions on the "constructor" arrow). This leads me to wonder:

Are there categorical constructions that generalize the subspace and quotient topologies?

REPLY [3 votes]: Nobody gave this reference so I give it : http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html, "The joy of cats", especially chapter 21 p350. The notions of initial and final topologies are generalized. The point is that the underlying set functor from general topological spaces to sets is topological.