This entry is about the basic notion of “monic” in relation to morphisms in category theory. For the notion of “monic” in relation to polynomials in commutative algebra, see at monic polynomial. The notion of monomorphism is the generalization of the notion of injective map of sets from the category Set to arbitrary categories. The formally dual concept is that of epimorphism, which similarly generalizes (or strengthens) the concept of surjective function. Common jargon includes “is a mono” or “is monic” for “is a monomorphism”, and “is an epi” or “is epic” for “is an epimorphism”, and “is an iso” for “is an isomorphism”. A morphism in some category is called a monomorphism (sometimes abbrieviated to mono, or described as being monic), if for every object and every pair of parallel morphisms then Stated more abstractly, this says that is a monomorphism if for every the hom-functor takes it to an injective function between hom-sets Since injective functions are precisely the monomorphisms in Set (example below) this may be stated as saying that is a monomorphism if is a monomorphism for all objects . Finally, being a monomorphism in a category means equivalently that it is an epimorphism in the opposite category . (monomorphisms in preorders) In a preorder, all arrows are mono because they satisfy the required condition vacuously (any pair of parallel arrows is equal in a preorder). (monomorphisms in ) The monomorphisms in the category Set of sets and functions between them are precisely the injective functions. The monomorphisms in the category Cat of categories and functors between them are precisely the embeddings of categories, i.e. the injective-on-objects faithful functors. But beware that the converse fails: The following lists some examples of morphisms that are both monomorphisms and epimorphisms, but not necessarily isomorphisms. In the category of Hausdorff topological spaces, the inclusion of a dense subspace is an epimorphism. See this Prop. for...