nLab object of monomorphisms

Idea

In the same way that internal homs in a cartesian monoidal category behave as an internal version of hom-sets between two objects, the object of monomorphism should behave as an internal version of the sub-hom-set of monomorphisms between two objects.

These generalize injection sets in the category of sets Set.

Definition

Given a cartesian monoidal category $C$ and two objects $A \in C$ and $B \in C$, there is a functor

$$X \mapsto \mathrm{Mono}_{C/X}(A \times X, B \times X)$$

which takes an object $X \in C$ to a monomorphism between objects $A \times X$ and $B \times X$ in the slice category $C/X$. A representing object of the above functor, if it exists, is the object of monomorphisms between $A$ and $B$ and is denoted as $[A, B]_\mathrm{mono}$.

Properties

If the category is a cartesian closed category, then there is a monomorphism $i:\mathrm{Mono}_C([A, B]_\mathrm{mono}, [A, B])$ between the object of monomorphism between $A$ and $B$ and the internal hom from $A$ to $B$.

See also

• injection set
• embedding type
• exponential object, internal hom, object of isomorphisms