# Looking for an “arrows-only” intro to category theory

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!

• I think Freyd-Scedrov Categories, Allegories qualifies, even though I wouldn't recommend it as an introduction to category theory. – t.b. Jul 30 '11 at 14:29
• Objects can be identified with their identity arrows, so I don't know if this approach is really all that different. – Qiaochu Yuan Jul 30 '11 at 14:31
• 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. – Zhen Lin Jul 30 '11 at 15:21
• @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? – t.b. Jul 30 '11 at 15:47
• @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... – Zhen Lin Jul 30 '11 at 16:41

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).