A monomorphism in a derivator is the generalization to the context of a derivator of the notion of monomorphism in ordinary category theory. Viewing a derivator as the “shadow” of an (∞,1)-category, the notion of monomorphism therein coincides with the notion of monomorphism in an (∞,1)-category.
Let denote the category
Let be a prederivator and a morphism in . By one of the axioms of a derivator, there exists an object representing , which is unique up to non-unique isomorphism. We say that is a monomorphism in if is a pullback square.
It is well-known and easy to verify that a morphism in a 1-category is a monomorphism if and only if the square
is a pullback. Therefore, in representable prederivators this definition reduces to the usual notion of monomorphism.
In the homotopy derivator of an -category, one can check that this reduces to the usual notion of monomorphism in an (∞,1)-category.