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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.2012-12-02
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    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.2012-12-02
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    Also, this may be of interest to the OP: http://en.wikipedia.org/wiki/Groupoid.2012-12-02

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