There are various categories associated with topology. The most fundamental is Top  Category of Topological Spaces. There is a category of the subset structure of open sets. There are also categories of fibres and sheaves.
Top  Topological Spaces
Consider a category C as follows:
 Objects are spaces
 Morphisms are structure preserving morphisms between these spaces
For a noncategory theory view of this structure see page here.
Subset Structure of Open Sets as Category
Subsets give interesting structure to open sets. This allows us to define a category of topological spaces. We can also further elaborate this subset structure to get fibre bundles and sheaves.
In this diagram the open set U is a subset of V. UV We can represent this as an arrow from V to U V>U 

This has the properties we expect from a category, for instance, the identity map: U>U (identity map) 

and composition (U>V)*(V >W) = U>W (composition) 
For a noncategory theory view of this structure see page here.
Bn  Category of Bundles
A fibre bundle is a function f:(A>I) This is described on the following pages: 
We can make this a category where:
This triangle must commute. So the elements (germs) of a stalk in 'A' must map to the same stalk in 'B'. This is a comma category as discussed on this page. 
The Bn Category can lead on to the concept of a Topos.
For more information about the Bn category see this page.
Category of Presheaves
Objects: C^{op} > setcontraveriant functors X: C > set (written X: C^{op} > set to indicate contraveriance) 

Morphisms:are natural transformations N: X > Y 
For a noncategory theory view of this structure see page here.