Examples
- In the category of Abelian groups and group homomorphisms, an injective object is a divisible group.
- In the category of modules and module homomorphisms, R-Mod, an injective object is an injective module. R-Mod has injective hulls (as a consequence, R-Mod has enough injectives).
- In the category of metric spaces and nonexpansive mappings, Met, an injective object is an injective metric space, and the injective hull of a metric space is its tight span.
- In the category of T0 spaces and continuous mappings, an injective object is always a Scott topology on a continuous lattice therefore it is always sober and locally compact.
- In the category of simplicial sets, the injective objects with respect to the class of anodyne extensions are Kan complexes.
- In the category of partially ordered sets and monotonic functions between posets, the complete lattices form the injective objects for order-embeddings, and the Dedekind–MacNeille completion of a partially ordered set is its injective hull.
- One also talks about injective objects in more general categories, for instance in functor categories or in categories of sheaves of OX modules over some ringed space (X,OX).
Read more about this topic: Injective Object
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