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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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    $\begingroup$ I think Freyd-Scedrov Categories, Allegories qualifies, even though I wouldn't recommend it as an introduction to category theory. $\endgroup$
    – t.b.
    Jul 30, 2011 at 14:29
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    $\begingroup$ Objects can be identified with their identity arrows, so I don't know if this approach is really all that different. $\endgroup$
    – Qiaochu Yuan
    Jul 30, 2011 at 14:31
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    $\begingroup$ Here is a arrows-only definition of a category, if you haven't already seen it. As Qiaochu has remarked, it's just clever trickery, albeit one that leads to inequivalent generalisations. $\endgroup$
    – Zhen Lin
    Jul 30, 2011 at 15:21
  • $\begingroup$ @Zhen: Could you expand on your comment? Are you just thinking of the "internalization" and $n$-ification mentioned on the nlab or do you have something more interesting in mind? $\endgroup$
    – t.b.
    Jul 30, 2011 at 15:47
  • $\begingroup$ @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... $\endgroup$
    – Zhen Lin
    Jul 30, 2011 at 16:41

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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.

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