Editing Concrete category (section)

== Definition ==

A '''concrete category''' is a pair (''C'',''U'') such that

*''C'' is a category, and 
*''U'' : ''C'' → '''Set''' (the category of sets and functions) is a [[faithful functor]].

The functor ''U'' is to be thought of as a [[forgetful functor]], which assigns to every object of ''C'' its "underlying set", and to every morphism in ''C'' its "underlying function".

It is customary to call the morphisms in a concrete category ''homomorphisms'' (e.g., group homomorphisms, ring homomorphisms, etc.) Because of the faithfulness of the functor ''U'', the homomorphisms of a concrete category may be formally identified with their underlying functions (i.e., their images under ''U''); the homomorphisms then regain the usual interpretation as "structure-preserving" functions.

A category ''C'' is '''concretizable''' if there exists a concrete category (''C'',''U'');
i.e., if there exists a faithful functor ''U'':&nbsp;''C'' → '''Set'''.  All small categories are concretizable: define ''U'' so that its object part maps each object ''b'' of ''C'' to the set of all morphisms of ''C'' whose [[codomain]] is ''b'' (i.e. all morphisms of the form ''f'': ''a'' → ''b'' for any object ''a'' of ''C''), and its morphism part maps each morphism ''g'': ''b'' → ''c'' of ''C'' to the function ''U''(''g''): ''U''(''b'') → ''U''(''c'') which maps each member ''f'': ''a'' → ''b'' of ''U''(''b'') to the composition ''gf'': ''a'' → ''c'', a member of ''U''(''c'').  (Item 6 under [[concrete category#Further examples|Further examples]] expresses the same ''U'' in less elementary language via presheaves.)  The [[concrete category#Counter-examples|Counter-examples]] section exhibits two large categories that are not concretizable.