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Can someone give me some examples of mathematical structures where the associated morphisms are forced to be injective (e.g. fields)?


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There is no notion of injective morphism in a category. Do you mean monomorphism? Or do you refer to concrete categories and define injectivity via the forgetful functor? – Martin Brandenburg Apr 14 '13 at 8:44
Yes, concrete categories. – Henry Apr 14 '13 at 8:57

Schur's lemma is a more broad result in this direction. The rule of thumb is that you know for a given category (of course, when applicable) what the kernel is going to look like, so if your object doesn't have nontrivial subobjects that look like that (like a field, or simple module) nonzero morphisms are going to be injective.

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In any algebraic setting where you have a category $C$ with a good notion of kernel, and where a morphism is mono (or becomes injective after applying the forgetful functor to $Set$ in the concrete case) if, and only if, its kernel is $0$ you can produce a category where all non-zero morphisms are monos as follows. The category $D$ in the full subcategory of $C$ spanned by the simple objects, where an object $X$ in $C$ is simple if it has no non-trivial normal subobjects (a normal subobject is a subobject which is a kernel of some morphism).

In particular, the category of simple groups and group homomorphisms is an example, as well as many other similar situations.

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Trivial homomorphisms are still not injective. But you also cannot really discard them ... – Martin Brandenburg Apr 14 '13 at 9:50

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