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category theory

Contents

Idea

In a concrete category, an injective hull of an object $A$ is an extension $A \stackrel{m}{\longrightarrow} B$ of $A$ such that $B$ is injective and $m$ 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.

Examples

• In Vect every object $A$ has an injective hull, $A \stackrel{id_A}{\longrightarrow} A$. In other words, every vector space is already an injective object.
• In Pos every object has an injective hull, its MacNeille completion.
• In Ab every object has an injective hull. The embedding $\mathbb{Z} \hookrightarrow \mathbb{Q}$ is an example.
• In the category of fields and algebraic field extensions, every object has an injective hull, its algebraic closure.
• In the category of metric spaces and short maps, the injective hull is a standard construction also known as the tight span? (see Wikipedia).

Generalization

Given a class $\mathcal{E}$ of objects in a category, an $\mathcal{E}$-hull (or $\mathcal{E}$-envelope) of an object $A$ is a map $h\colon A\longrightarrow E$ such that the following two conditions hold:

1. Any map $k\colon A\longrightarrow E'$ to an object in $\mathcal{E}$ factors through $h$ via some map $f: E\longrightarrow E'$.

2. Whenever a map $f\colon E\longrightarrow E$ satisfies $f\circ h = h$ then it must be an automorphism.

On the other hand, given a class $\mathcal{H}$ of morphisms in a category, an $\mathcal{H}$-injective hull of an object $A$ is a map $h:A\to E$ in $\mathcal{H}$ such that:

1. $E$ is a $\mathcal{H}$-injective object and

2. $h$ is $\mathcal{H}$-essential, i.e. if $k\circ h \in \mathcal{H}$ then $k\in\mathcal{H}$.

References

Discussion in homological algebra:

Discussion in general concrete categories: