Forgetful functors are almost always faithful.

Forgetful functors that only forget axioms are always fully faithful ; every morphism that respects the structure between objects that satisfy the axioms automatically also respects the axioms.

Forgetful functors that forget structures need not be full ; some morphisms don't respect the structure.

Forgetful functors tend to have left adjoints, which are ' free ' constructions.

* Forgetful functor –

It is where Christian meets Apollyon in the place known as " Forgetful Green.

Forgetful.

Forgetful of national calamity and of personal wrong, he looked to Prussia as affording the best example of an organized system of national education ; and he was persuaded that " to carry back the education of Prussia into France afforded a nobler ( if a bloodless ) triumph than the trophies of Austerlitz and Jena.

A Forgetful Nation: On Immigration and Cultural Identity in the United States, Duke UP, 2005.

Richard Hunt, who, in Jon Stone's words, joined the Muppets as a " wild-eyed 18-year-old and grew into a master puppeteer and inspired teacher ", created Gladys the Cow, Forgetful Jones, Don Music, and the construction worker Sully.

# Mr. Forgetful ... Hey, Waiter!

# Mr. Forgetful the World's Best Actor ( January 20, 1997 )

Another staple throughout the 1950s and early-1960s was a short visit to the North Pole with Santa Claus and Forgetful the Elf.

He has a surprisingly " sweet tooth " for one who has not yet started teething, as demonstrated once when Forgetful Ness lost him, resulting in his crawling into the back door of Mrs McToffee's sweet shop to steal sweeties, and on another occasion when he had the measles and Angus and Elspeth bought him a whole jar of sticky sweets to cheer him up.

; Forgetful Ness: A tie-wearing Nessie who, when he does call the children, sometimes even forgets that it was he who called.

Eventually, Warlock, an independent record label, issued a debut single, " Ode to a Forgetful Mind ", in 1989, but it went unnoticed.

The label of Charley Straight's recording of Forgetful Blues for Paramount, made in 1923.

* The functor category D < sup > C </ sup > has as objects the functors from C to D and as morphisms the natural transformations of such functors.

The Yoneda lemma is one of the most famous basic results of category theory ; it describes representable functors in functor categories.

* Adjoint functors: A functor can be left ( or right ) adjoint to another functor that maps in the opposite direction.

The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors.

One can compose functors, i. e. if F is a functor from A to B and G is a functor from B to C then one can form the composite functor G ∘ F from A to C. Composition of functors is associative where defined.

Identity of composition of functors is identity functor.

The collection of all functors C → D form the objects of a category: the functor category.

More generally, if C is an arbitrary abelian category that has enough injectives, then so does C < sup > I </ sup >, and the right derived functors of the inverse limit functor can thus be defined.

In the case where C satisfies Grothendieck's axiom ( AB4 *), Jan-Erik Roos generalized the functor lim < sup > 1 </ sup > on Ab < sup > I </ sup > to series of functors lim < sup > n </ sup > such that

This means that T is left adjoint to the forgetful functor U ( see the section below on relation to adjoint functors ).

It also clarifies how the embedded category, of representable functors and their natural transformations, relates to the other objects in the larger functor category.

Mapping each object A in C to its associated hom-functor h < sup > A </ sup > = Hom ( A ,–) and each morphism f: B → A to the corresponding natural transformation Hom ( f ,–) determines a contravariant functor h < sup >–</ sup > from C to Set < sup > C </ sup >, the functor category of all ( covariant ) functors from C to Set.

The meaning of Yoneda's lemma in this setting is that the functor h < sup >–</ sup > is fully faithful, and therefore gives an embedding of C < sup > op </ sup > in the category of functors to Set.

An important property of adjoint functors is that every right adjoint functor is continuous and every left adjoint functor is cocontinuous.

The functors ( V, U ) are then a pair of adjoint functors, with V left-adjoint to U and U right-adjoint to V.

* The category of presheaves on a topological space X is a functor category: we turn the topological space into a category C having the open sets in X as objects and a single morphism from U to V if and only if U is contained in V. The category of presheaves of sets ( abelian groups, rings ) on X is then the same as the category of contravariant functors from C to Set ( or Ab or Ring ).

Both of these functors are, in fact, right inverses to U ( meaning that UD and UI are equal to the identity functor on Set ).

Instead, if f: T → S and g: U → T are morphisms in E, then there is an isomorphism of functors

There are two forgetful functors from Grp:

* For any category C and object A of C the Hom functor Hom ( A ,–): C → Set preserves all limits in C. In particular, Hom functors are continuous.

If the chain complex depends on the object X in a covariant manner ( meaning that any morphism X → Y induces a morphism from the chain complex of X to the chain complex of Y ), then the H < sub > n </ sub > are covariant functors from the category that X belongs to into the category of abelian groups ( or modules ).

* objects are functors Γ: Q → C,

Here FG: D → D and GF: C → C, denote the respective compositions of F and G, and I < sub > C </ sub >: C → C and I < sub > D </ sub >: D → D denote the identity functors on C and D, assigning each object and morphism to itself.

The following statements are equivalent for functors F: C → D and G: D → C:

The functor category D < sup > C </ sup > has all the formal properties of an exponential object ; in particular the functors from E × C → D stand in a natural one-to-one correspondence with the functors from E to D < sup > C </ sup >.

But it turns out that ( if A is " nice " enough ) there is one canonical way of doing so, given by the right derived functors of F. For every i ≥ 1, there is a functor R < sup > i </ sup > F: A → B, and the above sequence continues like so: 0 → F ( A ) → F ( B ) → F ( C ) → R < sup > 1 </ sup > F ( A ) → R < sup > 1 </ sup > F ( B ) → R < sup > 1 </ sup > F ( C ) → R < sup > 2 </ sup > F ( A ) → R < sup > 2 </ sup > F ( B ) → ....

The right derived functors of the covariant left-exact functor F: A → B are then defined as follows.