"Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological space, how many times does it show up as an object in Top? Once? Countably many times? Uncountably many times? Is there a semantic crisis if we don't identify all homeomorphic topological spaces to the same object?



Confusion over a point in basic category theory
Well, such categories are actually proper classes. For instance, spaces that are homeomorphic to $\mathbb{R}$ will be given on ANY set of the same cardinality. So how many there are, depends on how many such sets there are...but there are quite a lot of sets. In fact, more than any individual set worth. (Though on the details there, I'm not sure about what happens for fixed cardinality)


We CAN identify them, and we get a [skeleton](http://ncatlab.org/nlab/show/skeleton). However, we usually don't do this, because there's no natural way to pick one object in each class, and one of the important catch phrases for category theory is "never make a choice." (Naturally, this is a vast oversimplification, but it's a good rule of thumb)


