be the forgetful functor which assigns to each algebra its underlying vector space.

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

* The forgetful functor U: Grp → Set creates ( and preserves ) all small limits and filtered colimits ; however, U does not preserve coproducts.

* The free functor F: Set → Grp ( which assigns to every set S the free group over S ) is left adjoint to forgetful functor U and is, therefore, cocontinuous.

* The forgetful functor Top → Set lifts limits and colimits uniquely but creates neither.

The forgetful functor Met < sub > c </ sub > → Set lifts finite limits but does not lift them uniquely.

A forgetful functor between categories of algebraic structures " forgets " a part of a structure.

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 ".

( X, S ) and ( X, T ) are distinct objects in the category Top of topological spaces and continuous maps, but mapped to the same set X by the forgetful functor Top → Set.

Conversely, starting from this equivalence we can recover U as the composite Rel → Sup → Set of the forgetful functor for Sup with this embedding of Rel in Sup.

The forgetful functor which arises in this way is the contravariant powerset functor Set < sup > op </ sup > → Set.

# For technical reasons, the category Ban < sub > 1 </ sub > of Banach spaces and linear contractions is often equipped not with the " obvious " forgetful functor but the functor U < sub > 1 </ sub >: Ban < sub > 1 </ sub > → Set which maps a Banach space to its ( closed ) unit ball.

This way the free functor that builds the free object A from the set X becomes left adjoint to the forgetful functor.

The most general setting for a free object is in category theory, where one defines a functor, the free functor, that is the left adjoint to the forgetful functor.

This category has a functor,, the forgetful functor, which maps objects and functions in C to Set, the category of sets.

The forgetful functor is very simple: it just ignores all of the operations.

Other types of forgetfulness also give rise to objects quite like free objects, in that they are left adjoint to a forgetful functor, not necessarily to sets.

The forgetful functor U has both a left adjoint

In fact, the forgetful functor U: Top → Set uniquely lifts both limits and colimits and preserves them as well.

Unlike many algebraic categories, the forgetful functor U: Top → Set does not create or reflect limits since there will typically be non-universal cones in Top covering universal cones in Set.

The forgetful functor U: Grp → Set have a left adjoint given by the composite KF: Set → Mon → Grp where F is the free functor.

The category Cat has a forgetful functor U into the quiver category Quiv:

The monad axioms can be seen at work in a simple example: let be the forgetful functor from the category Grp of groups to the category Set of sets.

There are two forgetful functors from Grp:

An example of the first kind is the forgetful functor Ab → Grp.

The " forgetful " functor Met → Set assigns to each metric space the underlying set of its points, and assigns to each metric map the underlying set-theoretic function.

We have a forgetful functor Ord → Set which assigns to each preordered set the underlying set, and to each monotonic function the underlying function.

This is the adjoint functor – specifically the left adjoint – to the forgetful functor Vect < sub > C </ sup > → Vect < sub > R </ sup > from forgetting the complex structure.

One of the second kind is the forgetful functor Ab → Set.

There is a natural forgetful 2-functor i: Scin ( E ) → Fib ( E ) that simply forgets the splitting.

Indeed, there are two canonical ways to construct an equivalent split category for a given fibred category F over E. More precisely, the forgetful 2-functor i: Scin ( E ) → Fib ( E ) admits a right 2-adjoint S and a left 2-adjoint L ( Theorems 2. 4. 2 and 2. 4. 4 of Giraud 1971 ), and S ( F ) and L ( F ) are the two associated split categories.

This can be composed with forgetful functors from Ab to yield other forgetful functors, most importantly one to Set.

A category with a faithful functor to Set is ( by definition ) a concrete category ; in general, that forgetful functor is not full.

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

He is a Craig's wife who agonizes about tobacco ash on the living room rug and he is a forgetful genius who goes boating with the town baker when dignitaries from the local university have come to call.

This situation is typical of algebraic forgetful functors.

" One problem is that the clerks who compiled this document " were but human ; they were frequently forgetful or confused.

However, despite being increasingly forgetful, Maggie is lucid enough to manipulate Arthur when she wants to, usually for something devilish that will seriously inconvenience him.

In his writings of the early 1970s, he rejects what he regards as theological underpinnings of both Karl Marx and Sigmund Freud: " In Freud, it is judaical, critical sombre ( forgetful of the political ); in Marx it is catholic.

One might become forgetful or feel as if the information is on the tip of the tongue.

He is forgetful and clumsy, mis-speaking often with spoonerisms, and his spells tend to back-fire.

He is brave and honorable, but also somewhat forgetful and clumsy.

There is a forgetful functor from K-Vect to Ab, the category of abelian groups, which takes each vector space to its additive group.

That is, there is a forgetful functor from Cat to Quiv.