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Regarding the morphisms-only-definition of a category (which is equivalent to the usual one dealing with objects and morphisms), I would like to ask: Which examples of categories in practice appear naturally as a class of morphisms with a partial composition rule, without an explicit or evident class of objects? Are there examples which really convince us that the morphisms-only-definition is natural? This may be a vague question, but in principle I would like to know if there is any pedagogical value in the morphisms-only-definition of a category (apart from its usefulness for e.g. mathematical logic).

I think that there many groupoids which appear only via their morphisms. In fact, the original conception of a groupoid was that of a 'group' with a partially defined group multiplication (curiously, it seems that categories are usually not seen as 'monoids' with a partially defined multiplication). Therefore I would like to favor examples which are no groupoids.

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    $\begingroup$ The first example that comes to mind is monoids: they're defined as a class of "morphisms" with a total composition rule, the unique object doesn't appear naturally IMO. $\endgroup$ – Najib Idrissi Apr 6 '15 at 9:41
  • $\begingroup$ @NajibIdrissi: Funny example! $\endgroup$ – Martin Brandenburg Apr 6 '15 at 9:45
  • $\begingroup$ Yes, and i was thinking at groups, but the OP has already discounted this. I do not think thereis anything better than "monoids with a partial composition" After all this what categories really are. $\endgroup$ – magma Apr 6 '15 at 17:29
  • $\begingroup$ $n$-Cobordisms come to mind as a category named after their morphisms, but I don't think they quite meet the criterion of being most naturally defined without mentioning objects at all: partitioning the boundary components into domain and codomain is extra structure on the manifold. $\endgroup$ – tcamps Apr 7 '15 at 2:17

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