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Forgetful functor

In category theory, a forgetful functor is a functor U: \mathcal{C} \to \mathcal{X} that maps objects and morphisms from a category \mathcal{C} equipped with additional structure to a simpler category \mathcal{X}, typically the category of sets \mathbf{Set}, by assigning to each object its underlying set and to each morphism its underlying function while discarding the extra structure. For example, the forgetful functor U: \mathbf{Grp} \to \mathbf{Set} sends a group to its underlying set and a group homomorphism to the corresponding set function, ignoring the group operation and inverse. Similarly, U: \mathbf{Ring} \to \mathbf{Set} forgets the ring addition and multiplication, mapping rings to their underlying sets. Forgetful functors exhibit several key properties that make them central to algebraic . They are typically faithful, meaning they injectively embed the hom-sets of \mathcal{C} into those of \mathcal{X}, but not full, as not every in \mathcal{X} (such as arbitrary set functions) arises from one in \mathcal{C}. Many forgetful functors, such as U: \mathbf{Grp} \to \mathbf{Set}, admit a left known as the functor (e.g., the construction F: \mathbf{Set} \to \mathbf{Grp}), forming a free-forgetful adjunction that encodes the universal properties of free algebraic structures. As right adjoints, these functors preserve limits; for instance, U: \mathbf{Grp} \to \mathbf{Set} creates all small limits in \mathbf{Grp} from those in \mathbf{Set}, ensuring that products and equalizers of groups are computed via their underlying sets. Beyond , forgetful functors appear in topological and order-theoretic contexts, such as U: \mathbf{Top} \to \mathbf{Set}, which maps topological spaces to their underlying sets and continuous functions to set functions, forgetting the . This functor preserves both limits (e.g., products as Cartesian products) and colimits (e.g., coproducts as disjoint unions) and has both a left adjoint ( topology) and a right adjoint (indiscrete ). In broader applications, forgetful functors facilitate the study of and monadicity theorems, where a forgetful functor with a left is monadic if the category of algebras for the induced monad is equivalent to the original category, providing a framework for understanding varieties of universal algebras.

Fundamentals

Definition

In category theory, a category consists of a collection of objects and morphisms between them, composed associatively with identity morphisms for each object. A functor F: \mathcal{C} \to \mathcal{D} between categories \mathcal{C} and \mathcal{D} assigns to each object in \mathcal{C} an object in \mathcal{D}, and to each morphism in \mathcal{C} a morphism in \mathcal{D}, preserving composition and identities. A forgetful functor is a functor that forgets or drops some or all of the additional structure or properties of its input. Typically, it is a functor U: \mathcal{C} \to \mathbf{Set} (also called an underlying set functor), where \mathcal{C} is a category whose objects are structured sets (such as groups or rings) and \mathbf{Set} is the category of sets and functions. It maps each structured object in \mathcal{C} to its underlying set of elements, thereby discarding the additional structure like operations or relations, while mapping each morphism in \mathcal{C} to its underlying function between those sets, without regard for structure preservation. More generally, the target can be a category with less structure than \mathcal{C}. On objects, U acts by retaining only the elements of the structured set, ignoring any algebraic or relational properties that define membership in \mathcal{C}. On morphisms, if f: X \to Y is a structure-preserving map in \mathcal{C}, then U(f): U(X) \to U(Y) is simply the induced by f, which forgets whether f respects the structure of X and Y. Forgetful functors neither preserve nor reflect the forgotten structure, focusing solely on the bare set-theoretic aspects. Such functors are typically faithful but not full. means U is injective on hom-sets: if two morphisms in \mathcal{C} differ, their underlying functions differ, as the morphisms are determined by their action on elements. However, U is not full, since the hom-sets in \mathbf{Set} include all possible functions between underlying sets, but only a proper lift to structure-preserving morphisms in \mathcal{C}. For instance, the forgetful functor from the category of groups to \mathbf{Set} exhibits this behavior.

Examples

A canonical example of a forgetful functor arises in the category of groups, denoted \mathbf{Grp}, where the functor U: \mathbf{Grp} \to \mathbf{Set} maps each group G to its underlying set U(G) and each f: G \to H to the same viewed as a set map U(f), thereby forgetting the group multiplication and inverse operations. In the category of rings, \mathbf{Ring}, the forgetful functor U: \mathbf{Ring} \to \mathbf{Set} similarly sends a R to its underlying set U(R) and a to its action as a , discarding the addition, multiplication, and unity structure. A variant in ring theory involves the forgetful functor U: \mathbf{Ring} \to \mathbf{Ab}, where \mathbf{Ab} is the category of abelian groups; here, U(R) retains the additive abelian group structure of the ring but forgets the multiplication. For topological spaces, the category \mathbf{Top} admits the forgetful functor U: \mathbf{Top} \to \mathbf{Set}, which maps a topological space X to its underlying set U(X) and a continuous map to the corresponding set function, ignoring the topology defined by open sets. In the category of partially ordered sets, \mathbf{Pos}, the forgetful functor U: \mathbf{Pos} \to \mathbf{Set} assigns to each poset (P, \leq) its underlying set U(P) = P and to each order-preserving map its set-theoretic counterpart, forgetting the partial order relation.

Adjunctions

Left Adjoints

In algebraic categories arising from universal algebra, such as varieties of algebras defined by operations and equations, the forgetful functor U: \mathcal{C} \to \Set that maps an algebra to its underlying set always admits a left adjoint F: \Set \to \mathcal{C}, known as the free functor. This adjunction is characterized by a natural isomorphism of hom-sets \Hom_{\mathcal{C}}(F(X), C) \cong \Hom_{\Set}(X, U(C)) for any set X and algebra C in \mathcal{C}, where the bijection sends a group homomorphism F(X) \to C to its restriction on the generators X \subseteq U(F(X)). The free functor F constructs free objects in \mathcal{C}, which satisfy a universal property induced by the adjunction: for any algebra C in \mathcal{C} and function f: X \to U(C), there exists a unique homomorphism \tilde{f}: F(X) \to C extending f, making F(X) the "freest" algebra generated by X with no imposed relations beyond those of the variety. For instance, in the category of groups, F(X) is the free group on the set X, consisting of all reduced words formed from elements of X and their inverses, with the group operation defined by concatenation and reduction. Key properties of this adjunction include the fact that F preserves colimits, reflecting the "free" generation of algebraic structure from colimits in sets, such as coproducts becoming free products in groups. This colimit preservation ensures that free algebras behave well under categorical constructions like direct sums or coproducts. Representative examples include the free ring on a set X, which is the ring of non-commutative polynomials in elements of X with integer coefficients, satisfying the adjunction via unique ring homomorphisms extending set maps. Similarly, in the category of monoids, the free monoid on X consists of finite words over X under concatenation, and this aligns with the underlying set of the free category generated by X as objects and no non-identity morphisms, preserving the adjunction. This free-forgetful adjunction forms a cornerstone of universal algebra, central to Garrett Birkhoff's development of the subject in the 1930s through his axiomatization of varieties and emphasis on free generations.

Right Adjoints

In category theory, a forgetful functor U: \mathcal{C} \to \Set admits a right adjoint R: \Set \to \mathcal{C} precisely when, for every set X, there exists a \mathcal{C}-object R(X) called the cofree \mathcal{C}-object on X, equipped with a natural transformation \epsilon: U R \to \mathrm{id}_{\Set} (the counit) such that every set map f: U(c) \to X for c \in \mathcal{C} extends uniquely to a \mathcal{C}-morphism \hat{f}: c \to R(X) satisfying U(\hat{f}); \epsilon_X = f. This adjunction satisfies the isomorphism \Hom_{\mathcal{C}}(c, R(X)) \cong \Hom_{\Set}(U(c), X) naturally in c and X. Such right adjoints exist less frequently than left adjoints to forgetful functors, often requiring \mathcal{C} to be a topological concrete category over \Set where minimal or indiscrete structures suffice to make the correspondence hold without additional constraints. In contrast to left adjoints, which freely generate structure and are common in algebraic settings, right adjoints to forgetful functors typically produce objects with maximal or indiscrete structure, preserving limits as all right adjoints do. A canonical example occurs in the category \Top of topological spaces, where the forgetful functor U: \Top \to \Set has right adjoint R(X) assigning to each set X the indiscrete (or trivial) topological space on X, whose only open sets are \emptyset and X. In this case, every function f: |Y| \to X from the underlying set of a topological space Y extends uniquely to a continuous map \hat{f}: Y \to R(X), since the preimage under any function of an open set in the indiscrete topology is either empty or the whole space, both of which are open. This adjunction highlights how the minimal topology on R(X) imposes no continuity restrictions, ensuring the bijection \Hom_{\Top}(Y, R(X)) \cong \Hom_{\Set}(U(Y), X). For the category \Pos of posets and order-preserving maps, however, the forgetful functor U: \Pos \to \Set does not admit a right adjoint. An attempt to construct R(X) as an indiscrete poset fails because the universal relation x \leq y for all x, y \in X violates antisymmetry (a requirement for posets) when |X| > 1, preventing every set function from extending to an order-preserving map without collapsing elements. This absence underscores that right adjoints require compatibility with the specific axioms of \mathcal{C}, such as reflexivity and transitivity in orders, which indiscrete constructions cannot always satisfy. In categories like \Pos, completeness and cocompleteness alone are insufficient; the solution set condition in the adjoint functor theorem fails for infinite X. Another instance arises in the category \Cat of small categories, where the forgetful functor U: \Cat \to \Set to the set of objects has a right adjoint R(X) sending X to the (or indiscrete) category on X, with objects X and exactly one morphism between any pair of objects (including identities). Here, functors from a category C to R(X) correspond bijectively to functions from the objects of C to X, as the unique morphisms in R(X) impose no functoriality constraints beyond object mapping. This construction parallels the indiscrete case in \Top, emphasizing how right adjoints often yield objects with "universal arrows" that minimize structural impositions.

Applications

In Universal Algebra

In universal algebra, forgetful functors play a central role in the study of varieties, which are classes of algebras defined by a fixed signature of operations and a set of equational identities. For a variety V of algebras over a signature \Sigma, the forgetful functor U: V \to \mathbf{Set} maps each algebra to its underlying set while preserving the structure of homomorphisms. This functor admits a left adjoint F_V: \mathbf{Set} \to V, known as the free algebra functor, which constructs the free algebra F_V(X) generated by a set X. The adjunction F_V \dashv U is characterized by the natural bijection \hom_V(F_V(X), A) \cong \hom_{\mathbf{Set}}(X, U(A)) for any algebra A \in V, with the unit of the adjunction providing the canonical inclusion of generators X \hookrightarrow U(F_V(X)). A fundamental theorem in states that every V admits such a free-forgetful adjunction, ensuring the existence of free objects parametrized by arbitrary sets. This result enables the classification of algebras in V up to via homomorphisms from free algebras, as every algebra is a of a free one by a . For instance, in the variety of abelian groups (equivalently, \mathbb{Z}-modules), the free functor F_V sends a set X to the \bigoplus_{x \in X} \mathbb{Z}, where each basis element corresponds to a from X. This construction exemplifies how the adjunction captures the universal property of free resolutions in algebraic structures. The free-forgetful adjunction underpins key developments in equational and systems, where terms in the serve as normal forms for evaluating identities within the . It also facilitates Birkhoff's , which characterizes varieties as precisely those equationally defined classes closed under the formation of homomorphic images (H), subalgebras (S), and direct products (P)—the HSP . This closure property ensures that varieties are stable under these operations, allowing systematic study of algebraic structures through their free objects and congruences. However, not all categories of algebraic structures qualify as varieties in this sense; for example, the category of fields over a fixed characteristic lacks free algebras on arbitrary sets, as field extensions cannot be freely generated without imposing additional non-equational axioms like the axiom of choice or transcendence conditions. This limitation highlights that the free-forgetful adjunction is tied specifically to equational theories, excluding non-varietal structures.

In Ordered Structures

In the category of partially ordered sets and functions, the forgetful functor U: \mathbf{Pos} \to \mathbf{Set} maps each poset to its underlying set, discarding the partial relation \leq. This functor admits a left adjoint, known as the discrete poset functor D: \mathbf{Set} \to \mathbf{Pos}, which assigns to each set X the poset DX equipped with the discrete where x \leq y if and only if x = y. The adjunction D \dashv U arises from the universal property that every function f: X \to PY (where PY is the underlying set of a poset P) extends uniquely to a map \tilde{f}: DX \to P given by \tilde{f}(x) = f(x), since the discrete imposes no additional constraints beyond equality. However, U does not have a right adjoint, as it fails to preserve certain colimits; for instance, the coequalizer in \mathbf{Pos} of the inclusion \mathbb{Q} \hookrightarrow \mathbb{R} and the constant map sending all rationals to $0collapses to a singleton due to the density of\mathbb{Q}in\mathbb{R}, whereas the corresponding coequalizer in \mathbf{Set}is the quotient\mathbb{R}/\mathbb{Q}$ of cardinality continuum. The forgetful functor U: \mathbf{Pos} \to \mathbf{Set} is faithful, meaning it injects hom-sets: distinct monotone maps between posets induce distinct functions on underlying sets, as monotone maps are precisely the order-preserving functions. In the subcategory of complete lattices and join-preserving maps, or more generally in the category Lat of lattices (posets equipped with binary meet and join operations satisfying the lattice axioms) and lattice homomorphisms, the forgetful functor U: \mathbf{Lat} \to \mathbf{Set} similarly discards the algebraic operations. This functor has a left adjoint, the free lattice functor F: \mathbf{Set} \to \mathbf{Lat}, which constructs on a set X the free lattice FX generated by X as the Lindenbaum-Tarski algebra of terms built from elements of X using meet (\wedge) and join (\vee), modulo the equational axioms of lattices (commutativity, associativity, , and distributivity where applicable). The elements of FX can be represented in canonical form using Whitman’s condition for finite free lattices, ensuring the adjunction F \dashv U via the universal property that lattice homomorphisms from FX to a lattice L correspond bijectively to functions from X to the underlying set of L. In , forgetful functors from categories of ordered structures like complete partial orders (cpos) to \mathbf{Set} play a key role in modeling and of programming languages. For example, the category CPO of cpos (posets with all directed suprema) and Scott-continuous functions (monotone maps preserving directed sups) has a forgetful functor to \mathbf{Set}, and the Scott on a cpo—generated by the complements of principal down-sets and upward-closed sets—equips the poset with a topological structure where Scott-continuous functions coincide with continuous maps in this . This allows ordered domains into sets while preserving computable approximations via finite elements, facilitating the of recursive definitions and fixed points without directly relying on the full order for .

Generalizations

Beyond Sets

Forgetful functors generalize beyond codomains in the category of sets, allowing mappings U: \mathcal{C} \to \mathcal{D} where \mathcal{D} possesses less structure than \mathcal{C}, thereby partially forgetting properties while preserving intermediate algebraic features. For instance, the forgetful functor U: R\text{-Mod} \to \text{Ab} from the category of modules over a ring R to the category of abelian groups discards the R-action but retains the underlying abelian group structure, mapping an R-module M to its additive group (M, +) and R-linear maps to group homomorphisms. This construction exemplifies a "partial forgetting," contrasting with full forgetful functors to \text{Set} by maintaining nontrivial categorical structure in the codomain. Such functors often admit adjoints that recover the forgotten structure. For U: R\text{-Mod} \to \text{Ab}, the left adjoint sends an abelian group A to the induced R-module R \otimes_{\mathbb{Z}} A, while the right adjoint maps A to \text{Hom}_{\mathbb{Z}}(R, A), endowing it with a module structure via the ring action on the domain. Moreover, this U is exact, preserving short exact sequences, as kernels and cokernels in R\text{-Mod} coincide with those in \text{Ab} upon forgetting the scalars. A notable example arises in , where the forgetful functor from the category of s over a k (of characteristic zero) to the category of Lie algebras equips an A with the Lie [a, b] = ab - ba, effectively forgetting the full associative multiplication while retaining the derived structure; this functor has the universal enveloping algebra construction as its left . Historically, this adjunction underpins deformation theory, linking infinitesimal deformations (modeled by Lie algebras) to associative structures via . In enriched category theory, forgetful functors extend to settings over a monoidal category V, such as V = \text{[Ab](/page/AB)}, where one may forget the V-enrichment of a category, mapping to the underlying ordinary category (enriched over \text{Set}) and thus partially discarding hom-object structures in V. For example, the category of abelian groups admits an \text{[Ab](/page/AB)}-enrichment via hom-groups, and the forgetful functor to \text{Set}-enriched categories forgets this additive enrichment while preserving the discrete morphisms. These partial forgettings highlight how adjunctions in enriched contexts can reconstruct enrichments, analogous to free constructions in algebraic categories.

Reflexive Subcategories

A subcategory \mathcal{C} of a category \mathcal{D} is called reflective if the inclusion functor \Inc: \mathcal{C} \to \mathcal{D}, which forgets the additional structure present in \mathcal{C}, admits a left adjoint R: \mathcal{D} \to \mathcal{C}; this left adjoint R is termed the reflector. The adjunction R \dashv \Inc induces a unit natural transformation \eta: \Id_{\mathcal{D}} \to \Inc \circ R, with the components \eta_d: d \to \Inc(R(d)) being the reflection maps that are initial among all morphisms from d to objects in \mathcal{C}. This structure ensures that every object in \mathcal{D} has a canonical "reflection" into \mathcal{C}, universal with respect to the subcategory's properties. In the context of forgetful functors, the inclusion \Inc typically forgets the structure defining \mathcal{C}, such as algebraic operations or topological axioms, while the existence of the left adjoint R guarantees that any object in \mathcal{D} can be equipped with a "free" or minimal structure making it isomorphic to an object in \mathcal{C}. This reflection process parallels free constructions in , where the builds the necessary while preserving the underlying . A prominent example arises in topology: the full subcategory of sober topological spaces is reflective in the category \Top of all topological spaces, with the sobrification functor serving as the reflector that assigns to each space its sober hull via the Alexandrov topology on the set of irreducible closed sets. This reflection ensures that non-sober spaces, like the real line with the cofinite topology, are mapped to their sober counterparts, which satisfy the sobriety condition that every irreducible closed set is the closure of a unique point. Reflective subcategories inherit significant stability properties: they are closed under all limits that exist in the ambient \mathcal{D}, meaning that limits in \mathcal{C} are computed as in \mathcal{D} and land in \mathcal{C}. Moreover, the reflector R preserves these limits, reflecting the fact that the inclusion \Inc is monadic, inducing an idempotent on \mathcal{D} whose algebras form \mathcal{C}. These features make reflective subcategories particularly useful for studying completions and approximations within broader categorical frameworks. In theory, reflective subcategories play a central role in constructing localizations, where the reflector corresponds to inverting a of morphisms to yield a subcategory of "local" objects that behaves like a under suitable conditions. Such localizations preserve finite limits and are essential for defining sheaves and étale maps in geometric contexts, with the unit of the adjunction identifying the "weak equivalences" relative to the subcategory. A key theorem establishes that algebraic categories—varieties of universal algebras over the category —are reflective subcategories of , with the forgetful functor from the algebra category to having a left adjoint given by the free algebra functor that adjoins the necessary operations freely to sets. This reflectivity underscores the monadic nature of such categories, ensuring that algebraic structures can be freely generated while preserving the underlying sets.