Suppose I have a category C. And there are two objects X and Y with no morphisms between them. I've checked up "quotient category" on wikipedia, but there I can only make isomorphic objects with morphisms between them.


Is there a categorial notion available that I can use in this case? 



Collapsing objects in a category
Short answer: yes. There is a special class of 2-categorical limits called iso-inserters that does the trick. The paper to check is M. Kelly's "Elementary observations on 2-categorical limits", Bull. Austr. Math. Soc. 39 (1989), 301-317.


Instead of explaining what these are let me go about explaining how you would make two objects $a$ and $b$ isomorphic. First pass to the underlying graph of the category and insert two edges $a\to b$ and $b\to a$ then take the free category of this new graph. You have a graph morphism from the original category into this new graph. Quotient the category to force the graph morphism to be a functor. Now quotient the category again to make the edges you inserted to be mutually inverse. There is only a small snag to this construction: the category you end up may not be locally small, because inserting isomorphisms may create a proper class of new morphisms. This is where things like "calculus of fractions" come in.


Hope it helps, regards, G. Rodrigues


