Applying duality, this means that a morphism in some category " C " is a monomorphism if and only if the reverse morphism in the opposite category " C " op is an epimorphism.
42.
This means that a ( homo ) morphism f : A \ to B is a monomorphism if, for any pair of morphisms from any other object to, then f \ circ g = f \ circ h implies.
43.
In the category of vector spaces over a field " K ", every monomorphism and every epimorphism splits; this follows from the fact that linear maps can be uniquely defined by specifying their values on a basis.
44.
In particular, Mac Lane attempted to settle an ambiguity in usage for the terms epimorphism and monomorphism by introducing the terms " epic " and " monic, " but the distinction is not in common use.
45.
By dividing the two conditions on the induced map in the definition, one can define "'left semi-abelian categories "'by requiring that \ overline { f } is a monomorphism for each morphism f.
46.
Such a factorization of an arbitrary morphism into an epimorphism followed by a monomorphism can be carried out in all abelian categories and also in all the concrete categories mentioned above in the Examples section ( though not in all concrete categories ).
47.
As an example, the inclusion of the rational numbers into the real numbers is a monomorphism and an epimorphism, but it is clearly not an isomorphism; this example shows that "'Met "'is not a balanced category.
48.
Monomorphism is the ancestral state, and " X . micans ", which shows both mating strategies, have males partially covered in light hairs, showing the beginnings of sexual dimorphism that grows more prominent in species that require lek polygyny.
49.
Every morphism in a concrete category whose underlying function is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense.
50.
Every morphism in a concrete category whose underlying function is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense.
