# Objects whose morphisms are all injective

Apart from fields, what other mathematical objects have "naturally" as morphisms maps which are all injective?

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This is an uninteresting example, but you could just artificially define a category with one object whose only morphism is the identity. Similarly, every group defines a category with one object, together with $\vert G \vert$ isomorphisms. –  jmracek Dec 2 '12 at 7:20
I think you may want to broaden the scope of your question by replacing "injectivity" by "monomorphisms", which is a generalization of injectivity. Also @jmracek I do not think your example about groups is at all uninteresting. –  Rankeya Dec 2 '12 at 7:27
Also, this may be of interest to the OP: en.wikipedia.org/wiki/Groupoid. –  Rankeya Dec 2 '12 at 7:28