(also nonabelian homological algebra)
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In a concrete category, an injective hull of an object is an extension of such that is injective and is an essential embedding. It is the dual concept to projective cover.
Beware that, in general, there is no way of making the assignment of the injective hull to an object into a functor such that there is a natural transformation from the identity functor to that functor.
Given a class of objects in a category, an -hull (or -envelope) of an object is a map such that the following two conditions hold:
Any map to an object in factors through via some map .
Whenever a map satisfies then it must be an automorphism.
On the other hand, given a class of morphisms in a category, an -injective hull of an object is a map in such that:
is a -injective object and
is -essential, i.e. if then .
projective object, projective presentation, projective cover, projective resolution
injective object, injective presentation, injective envelope, injective resolution
flat object, flat resolution
Discussion in homological algebra:
Discussion in general concrete categories:
See also:
On injective hulls of partially ordered monoids:
Last revised on June 29, 2024 at 16:06:43. See the history of this page for a list of all contributions to it.