Forgetful functor

In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some or all of the object's structure or properties is 'forgotten' in the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature in some way: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure; this is in fact the most common case.

For example, there are several forgetful functors from the category of commutative rings. A (unital) ring, described in the language of universal algebra, is an ordered tuple ("R",+,*,"a","m",0,1) satisfying certain axioms, where "+" and "*" are binary functions on the set "R", "a" and "m" are unary operations corresponding to additive and multiplicative inverse, and 0 and 1 are nullary operations giving the identities of the two binary operations. Deleting the 1 gives a forgetful functor to the category of rings without unit; it simply "forgets" the unit. Deleting "*", "m", and 1 yields a functor to the category of abelian groups which assigns to each ring "R" the underlying additive abelian group of "R". To each morphism of rings is assigned the same function considered merely as a morphism of addition between the underlying groups. Deleting all the operations gives the functor to the underlying set "R".

It is beneficial to distinguish between forgetful functors which "forget structure" versus those which "forget properties". For example, in the above example of commutative rings, in addition to those functors which delete some of the operations, there are functors which forget some of the axioms. There is a functor from the category CRing to Ring which forgets the axiom of commutativity, but keeps all the operations. Occasionally the object may include extra sets not defined strictly in terms of the underlying set (in this case, it is a matter of taste which part to consider the underlying set, though this is rarely ambiguous in practice). For these objects, there are forgetful functors which forget the extra sets which are more general.

Most common objects studied in mathematics are constructed as underlying sets along with extra sets of structure on those sets (operations on the underlying set, privileged subsets of the underlying set, etc) which may satisfy some axioms. For these objects, a commonly considered forgetful functor is as follows.Let mathcal{C} be any category based on sets, e.g. groups - sets of elements - or topological spaces - sets of 'points'. As usual, write operatorname{Ob}(mathcal{C}) for the objects of mathcal{C} and write operatorname{Fl}(mathcal{C}) for the morphisms of the same. Consider the rule: :A in operatorname{Ob}(mathcal{C}),quad Amapsto |A|= the underlying set of A, :u in operatorname{Fl}(mathcal{C}),quad umapsto |u|= the morphism, u, as a map of sets. The functor |;;| is then the forgetful functor from mathcal{C} to mathbf{Set}, the category of sets.

Forgetful functors are almost always faithful. Concrete categories have forgetful functors to the category of sets -- indeed they may be "defined" as those categories which admit a faithful functor to that category.

Forgetful functors which only forget axioms are always fully faithful; every morphism which respects the structure between objects which satisfy the axioms automatically also respects the axioms. Forgetful functors which forget structures need not be full; there are morphisms which don't respect the structure. These functors are still faithful though; distinct morphisms which do respect the structure are still distinct when the structure is forgotten. Functors which forget the extra sets need not be faithful; distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set.

In the language of formal logic, a functor of the first kind is one which removes axioms. One of the second kind are those which remove predicates. The third kind are those which remove types.

An example of the first kind of the forgetful functor Ab → Grp. One of the second kind is the forgetful functor Ab → Set. A functor of the third kind is the functor Mod → Ab, where Mod is the fibred category of all modules over arbitrary ring. To see this, just choose a ring homomorphism between the underlying rings which does not change the ring action. Under the forgetful functor, this morphism yields the identity. Note that an object in Mod is a tuple which includes a ring and an abelian group, so it is a matter of taste which to forget.

Left Adjoint: Free

Forgetful functors tend to have left adjoints which are 'free' constructions. For example:
* free module: the forgetful functor from mathbf{Mod}(R) (the category of R-module) to mathbf{Set} has left adjoint operatorname{Free}_R, with Xmapsto operatorname{Free}_R(X), the free R-module with basis X.
* free group
* free lattice
* tensor algebra

For a more extensive list, see (Mac Lane 1997).

As this a fundamental example of adjoints, we spell it out:adjointness means that given a set "X" and an object (say, an "R"-module) "M", maps "of sets" X o M correspond to maps of modules operatorname{Free}_R(X) o M: every map of sets yields a map of modules, and every map of modules comes from a map of sets.

In the case of vector spaces, this is summarized as:"A map between vector spaces is determined by where it sends a basis, and a basis can be mapped to anything."

Symbolically::operatorname{Hom}_{mathbf{Mod}_R}(operatorname{Free}_R(X),M) = operatorname{Hom}_{mathbf{Set(X,operatorname{Forget}(M))

The counit of the free-forget adjunction is the "inclusion of a basis": X o operatorname{Free}_R(X).

Fld, the category of fields, furnishes an example of a forgetful functor with no adjoint. There is no field satisfying a free universal property for a given set.

References

* Mac Lane, Saunders. "Categories for the Working Mathematician", Graduate Texts in Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York, 1997. ISBN 0-387-98403-8


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Functor — For functors as a synonym of function objects in computer programming to pass function pointers along with its state, see function object. For the use of the functor morphism presented here in functional programming see also the fmap function of… …   Wikipedia

  • Representable functor — In mathematics, especially in category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i …   Wikipedia

  • Derived functor — In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Contents 1 Motivation 2 Construction… …   Wikipedia

  • Diagonal functor — In category theory, for any object a in any category where the product exists, there exists the diagonal morphism satisfying for , where πk …   Wikipedia

  • Adjoint functors — Adjunction redirects here. For the construction in field theory, see Adjunction (field theory). For the construction in topology, see Adjunction space. In mathematics, adjoint functors are pairs of functors which stand in a particular… …   Wikipedia

  • Limit (category theory) — In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint… …   Wikipedia

  • Concrete category — In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as… …   Wikipedia

  • Category of rings — In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (preserving the identity). Like many categories in mathematics, the category of rings is… …   Wikipedia

  • Free object — In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a clean… …   Wikipedia

  • Comma category — In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.