In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, more specifically in the area of
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's
structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
or
properties mapping to the output. For an
algebraic structure
In mathematics, an algebraic structure or algebraic system consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplicatio ...
of a given
signature
A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...
, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
is simply to take the
underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case.
Overview
As an 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
satisfying certain axioms, where
and
are binary functions on the set
,
is a unary operation corresponding to additive 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
and 1 yields a functor to the category of
abelian groups, which assigns to each ring
the underlying additive abelian group of
. 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
.
It is beneficial to distinguish between forgetful functors that "forget structure" versus those that "forget properties". For example, in the above example of commutative rings, in addition to those functors that delete some of the operations, there are functors that forget some of the axioms. There is a functor from the category CRing to Ring that 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, which part to consider the underlying set is a matter of taste, though this is rarely ambiguous in practice). For these objects, there are forgetful functors that forget the extra sets that 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
be any category based on
sets, e.g.
groups—sets of elements—or
topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s—sets of 'points'. As usual, write
for the
objects of
and write
for the morphisms of the same. Consider the rule:
:For all
in
the underlying set of
:For all
in
the morphism,
, as a map of sets.
The functor
is then the forgetful functor from
to 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 that admit a faithful functor to that category.
Forgetful functors that only forget axioms are always
fully faithful, since 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. These functors are still faithful however because distinct morphisms that do respect the structure are still distinct when the structure is forgotten. Functors that forget the extra sets need not be faithful, since 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 removes axioms, a functor of the second kind removes predicates, and a functor of the third kind remove types. An example of the first kind is 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 rings. To see this, just choose a ring homomorphism between the underlying rings that 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 which to forget is a matter of taste.
Left adjoints of forgetful functors
Forgetful functors tend to have
left adjoints, which are '
free' constructions. For example:
*
free module: the forgetful functor from
(the category of
-
modules) to
has left adjoint
, with
, the free
-module with
basis .
*
free group
*
free lattice
*
tensor algebra
*
free category, adjoint to the forgetful functor from categories to
quivers
*
universal enveloping algebra
For a more extensive list, see (Mac Lane 1997).
As this is 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''
correspond to maps of modules
: 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:
:
The
unit of the free–forgetful adjunction is the "inclusion of a basis":
.
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.
See also
*
Adjoint functors
*
Functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s
*
Projection (set theory)
References
*
Mac Lane, Saunders. ''Categories for the Working Mathematician'', Graduate Texts in Mathematics 5, Springer-Verlag, Berlin, Heidelberg, New York, 1997.
External links
*
{{DEFAULTSORT:Forgetful Functor
Functors