Quotienting a Category by an Object: Explained

[fallgesetz]

What does it mean to quotient a category by an object of the category?

In particular, the problem in front of me specifies the category I ( a full subcategory of Top which is compact, contains coproducts, and the one point space), an object X of I, and asks me to do stuff with I/X.

 

[Hurkyl]

This isn't a quotient, but a "slice category", a particular kind of "comma category".

Given any category E and object A, the category E/A is defined by:
. The objects of E/A are arrows of C of the form B --> A
. The arrows of E/A are commutative triangles of C with a distinguished vertex A (This arrow "points" in the same direction as the edge opposite A)

Or a more algebraic description
. Objects are arrows f of E such that codom(f) = A
. Hom_E/A(f,g) is the class of arrows h of E such that gh=f