forgetful functor

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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, in the area of
category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...
, a forgetful
functor In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

(also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an
algebraic structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of a given
signature 's signature is the most prominent on the United States Declaration of Independence The United States Declaration of Independence is the pronouncement adopted by the Second Continental Congress meeting in Philadelphia Philadelphia, c ...
, 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 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. A ( unital) ring, described in the language of
universal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular Group (mathematics), groups as t ...
, is an ordered tuple (''R'', +, ×, ''a'', 0, 1) satisfying certain axioms, where "+" and "×" are binary functions on the set ''R'', ''a'' 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 group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s, which assigns to each ring ''R'' the underlying additive abelian group of ''R''. To each
morphism In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of rings is assigned the same
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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 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 $\mathcal$ be any category based on sets, e.g.
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
s—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. More specifically, a topological space is a ...
s—sets of 'points'. As usual, write $\operatorname\left(\mathcal\right)$ for the objects of $\mathcal$ and write $\operatorname\left(\mathcal\right)$ for the morphisms of the same. Consider the rule: :For all $A$ in $\operatorname\left(\mathcal\right), A\mapsto , A, =$ the underlying set of $A,$ :For all $u$ in $\operatorname\left(\mathcal\right), u\mapsto , u, =$ the morphism, $u$, as a map of sets. The functor $, \cdot,$ is then the forgetful functor from $\mathcal$ to Set, the
category of setsIn the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...
. 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 categoryFibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which ''inverse images'' (or ''pull-backs'') of ob ...
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 adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of ...
s, which are '
free Free may refer to: Concept * Freedom, having the ability to act or change without constraint * Emancipate, to procure political rights, as for a disenfranchised group * Free will, control exercised by rational agents over their actions and decis ...
' constructions. For example: *
free module In mathematics, a free module is a module (mathematics), module that has a Basis (linear algebra), basis – that is, a generating set of a module, generating set consisting of linearly independent elements. Every vector space is a free module, but, ...

: the forgetful functor from $\mathbf\left(R\right)$ (the category of $R$-
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) to $\mathbf$ has left adjoint $\operatorname_R$, with $X\mapsto \operatorname_R\left(X\right)$, the free $R$-module with
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other ba ...
$X$. *
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*
free latticeIn mathematics, in the area of order theory, a free lattice is the free object corresponding to a Lattice (order), lattice. As free objects, they have the universal property. Formal definition Any set (mathematics), set ''X'' may be used to generate ...
*
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', i ...
*
free categoryIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, adjoint to the forgetful functor from categories to quivers *
universal enveloping algebra In mathematics, a universal enveloping algebra is the most general (unital algebra, unital, associative algebra, associative) algebra that contains all representation of a Lie algebra, representations of a Lie algebra. Universal enveloping algebras ...
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'' $X \to , M,$ correspond to maps of modules $\operatorname_R\left(X\right) \to M$: 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: :$\operatorname_\left(\operatorname_R\left(X\right),M\right) = \operatorname_\left(X,\operatorname\left(M\right)\right).$ The unit of the free–forgetful adjunction is the "inclusion of a basis": $X \to \operatorname_R\left(X\right)$. 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.