Ok. I'm not insisting since I'm not fond of extensible structures anyway. But it is certainly not odd to make morphisms the underlying set of a category. The fact that an "arrows only" definition of a category is possible shows that it's that latter set which is more important. Also, it is common to say that a monoid (resp. group) is a category (resp. groupoid) with one object. (This is actually what led me to this proposal, see the linked post with the question by David Starner.)
Indeed, one often says things like "the category of groups" instead of "the category of groups and group morphisms" or "the category of morphisms", but this is an abuse of language. It's ok in this case since the morphisms are obvious, but it is not always so. As you noted, considering posets as thin categories seems to contradict this. The same goes with metric spaces as (\R, \leq)-enriched categories. But actually, these are examples of categorification, so it's expected that there is a shift by 1 (from n-morphisms to (n+1)-morphisms). Benoît -- You received this message because you are subscribed to the Google Groups "Metamath" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/metamath/84285818-c976-4912-a4bd-2c9af466e79e%40googlegroups.com.
