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I have often seen it remarked in passing that the "collection of objects" that appears in the standard definition of a category is, strictly speaking, superfluous, and that it is possible to give an equivalent definition of categories that dispenses with it altogether and uses morphisms only. But, after this nod to such a possibility, it is dropped.

Is there any good introduction to category theory that takes the arrows-only approach in earnest?

Thanks!

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    @Theo: Yes, I was referring to the comment in nLab which indicates that an internal one-sorted category may not give rise to an internal two-sorted category if the background category does not have split idempotents. But I don't personally know why one might consider such internal categories...2011-07-30

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I've already answered this question elsewhere.

But I just wanted to warn you. I remember when I first learned category theory from MacLane's "Category theory for working mathematicians" I was fascinated by this object-free definition of category, especially because it shows more easily why monoids are one object-categories.

By the way I've learn later that this is not the best way to presents categories mostly because this kind of definition made it a little artificial presenting many classical examples: structured sets and morphisms between them, but also preorder and poset, in my personal opinion, are more easly understood as categories through the classical definition.

Reading the link that Zhen Lin posted in a comment, it seems to me that this kind of definition it's also more complex to generalize to the $\infty$-dimensional version (but maybe that's just me).

Edit: I also think this post can be interesting for this discussion.