Full and faithful functors

Faithful functors and here also to be discussed fully and fully faithful functors, which are closely related to it, are considered in the mathematical theory of category theory functors with special properties.

Definitions

Be a functor between two categories. Such a functor assigns definition of every object and morphism, where and objects are, an object or a morphism to, with certain compatibility conditions are satisfied.

For each pair of objects one has a picture:

One calls the faithful functor (or fully or fully faithful ) if the pictures injective for each pair of objects from (or surjective or bijective ) are. Instead of fully faithful we also find the name completely true.

Embeddings

Is a functor, so, the terms true, full and fully faithful only morphism between any two objects, they do not refer to the classes of all objects and all morphisms, in particular the faithfulness of the functor does not say that one of the pictures

Is injective, which is not the case in general. To understand the context of these terms and the use of the above definitions, here the following simple statement is proved:

• If the functor is faithful, so is injective if it is injective.

Is injective and are, it follows, therefore, by assumption, and thus. Therefore is injective. ( For this direction, the faithfulness of the functor is not needed. )

Conversely, suppose that is injective, and be with. Then there are objects of the category, so that and morphisms are respectively. We have to show. To apply the faithfulness of the functor, we have and show. Since functor, we obtain morphisms and. Since, follows and. Because, by assumption, is injective, we get and. Therefore, and the loyalty of supplies as desired.

This is called a functor is an embedding if it is injective. For a faithful functor embedding property is according to the above, is equivalent to the injectivity of.

If the functor is an embedding, the objects form the morphisms, a subcategory of which is denoted by. Since, in general, is not the case for arbitrary functors that are not embeddings, embeddings play an important role in category theory.

Full fidelity functors

If the functor is an embedding, and is a full functor, so is a full subcategory of. This motivates the name full functor in the above definitions. So is a fully faithful functor, so that is injective, so defines an embedding to a full subcategory.

Full fidelity functors are important for the category theory because of the following statement:

• Be a fully faithful functor and a morphism of the category. Then: is isomorphism is isomorphism.

The direction from left to right is very simple. Is namely isomorphism, then there is by definition a morphism and further. As functor is followed and the same, that is, is an isomorphism.

Full fidelity is required for the inversion. Indeed, if an isomorphism, then there is a morphism with and. There is full, there is a morphism with. Then follows and the same. Because of Loyalty and now follows, that is, is an isomorphism.