I am wondering if for any category $C$ (at least a small category), we can find a graph $G$ (at least a small graph), such that $C$ is the free category generated by the graph $G$.

I think this result comes in handy, if we want to construct a category, given any other such category $J$, by appending a family of objects and arrows to $J$.

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    $\begingroup$ Every category is the quotient category of a free category. This fits to the usual pattern, first take the free object and then mod out the appropriate relations. $\endgroup$ – Martin Brandenburg Jun 14 '14 at 13:24

No, obviously not. Take, for instance, any non-trivial group considered as a one-object category. More generally, one observes that the only isomorphisms in a category freely generated by a graph are the identities.

  • $\begingroup$ Any non-trivial non-free group, no ? $\endgroup$ – Pece Jun 15 '14 at 5:56
  • $\begingroup$ Any non-trivial non-free monoid would do; but the only free group that is also a free monoid is the trivial one. $\endgroup$ – Zhen Lin Jun 15 '14 at 6:01
  • $\begingroup$ Of course, I forgot the quotient making $gg^{-1}$ the identity (even in a free group). Sorry for the useless comment. ;) $\endgroup$ – Pece Jun 15 '14 at 6:04

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