nLab object of monomorphisms



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.


Given a cartesian monoidal category CC and two objects ACA \in C and BCB \in C, there is a functor

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

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


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

See also

Created on January 10, 2023 at 20:56:52. See the history of this page for a list of all contributions to it.