Full and faithful functors

In category theory, a full and faithful functor is a morphism between categories that induces bijections between the hom-sets of the source and target categories, thereby preserving and reflecting the relational structure encoded by morphisms.^[1] Specifically, given categories \mathcal{C} and \mathcal{D}, a functor F: \mathcal{C} \to \mathcal{D} is faithful if, for all objects A, B \in \mathcal{C}, the induced map F: \mathrm{Hom}_{\mathcal{C}}(A, B) \to \mathrm{Hom}_{\mathcal{D}}(F(A), F(B)) is injective, ensuring distinct morphisms in \mathcal{C} remain distinct in \mathcal{D}.^[2] It is full if the same map is surjective, meaning every morphism in \mathcal{D} between images of objects from \mathcal{C} arises from a unique morphism in \mathcal{C}.^[3] Together, these properties make F an isomorphism on hom-sets, akin to an embedding that captures the "internal logic" of \mathcal{C} within \mathcal{D}.^[1] Full and faithful functors play a central role in structural comparisons across categories, often serving as embeddings of subcategories or realizing equivalences up to isomorphism.^[1] For instance, the inclusion functor of a full subcategory inherits these properties, embedding the subcategory without loss or addition of morphisms.^[2] A classic example is the forgetful functor from the category of groups \mathbf{Grp} to sets \mathbf{Set}, which is faithful—preserving distinct group homomorphisms as distinct set functions—but not full, as not every set function between underlying sets arises from a group homomorphism.^[3] In contrast, the Yoneda embedding Y: \mathcal{C} \to [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}], which maps objects to representable functors, is full and faithful, establishing a foundational isomorphism via the Yoneda lemma that represents any functor as a colimit of representables.^[1] These functors exhibit strong reflective properties, such as reflecting isomorphisms (if F(f) is an isomorphism, then f is) and reflecting limits and colimits.^[1] In the context of adjunctions, a right adjoint G \dashv F is full and faithful precisely when the counit \epsilon: FG \to 1 consists of isomorphisms, implying F reflects the reflective subcategory structure.^[1] Such functors are indispensable for Kan extensions, where a full and faithful K: \mathcal{M} \to \mathcal{C} ensures the right Kan extension along K yields a natural isomorphism.^[1] Overall, full and faithful functors provide a precise tool for dissecting categorical equivalences and universal constructions, underpinning much of modern algebraic and topological abstraction.^[3]

Definitions

Faithful functors

A functor F: \mathcal{C} \to \mathcal{D} between categories \mathcal{C} and \mathcal{D} is faithful if, for every pair of objects A, B in \mathcal{C}, the map it induces on hom-sets, \mathrm{Hom}_{\mathcal{C}}(A, B) \to \mathrm{Hom}_{\mathcal{D}}(F(A), F(B)), defined by f \mapsto F(f), is injective.^[1] This condition ensures that distinct morphisms in \mathcal{C} remain distinct after applying F, preserving the separation of arrows between objects without collapsing any.^[1] Formally, F is faithful if \forall A, B \in \mathrm{Ob}(\mathcal{C}), \, F(f_1) = F(f_2) implies f_1 = f_2 for all morphisms f_1, f_2: A \to B in \mathcal{C}.^[4] This injectivity captures the idea of an embedding-like behavior on the morphism level, where F acts as a one-to-one correspondence into the hom-sets of \mathcal{D}.^[1] The notion of a faithful functor was introduced by Samuel Eilenberg and Saunders Mac Lane in the 1940s as part of the foundational development of category theory, specifically appearing in their 1945 paper where they defined a "faithful representation" of a category as one preserving distinct mappings.^[4] This concept arose in the context of algebraic topology and homology, aiming to formalize structure-preserving maps that maintain morphism distinctions.^[4] To verify that a given functor F is faithful, one examines each pair of objects A, B in \mathcal{C} and confirms that the function f \mapsto F(f) sends no two distinct morphisms to the same element in \mathrm{Hom}_{\mathcal{D}}(F(A), F(B)), typically by direct computation or structural argument in concrete examples.^[1]

Full functors

A functor F: \mathcal{C} \to \mathcal{D} between categories is called full if, for every pair of objects A, B in \mathcal{C}, the induced map on hom-sets F: \hom_{\mathcal{C}}(A, B) \to \hom_{\mathcal{D}}(F(A), F(B)) is surjective.^[1] This means that every morphism g: F(A) \to F(B) in \mathcal{D} arises as the image under F of some morphism f: A \to B in \mathcal{C}, formally expressed as: for all A, B \in \mathrm{Ob}(\mathcal{C}) and all g: F(A) \to F(B) in \mathcal{D}, there exists f: A \to B in \mathcal{C} such that F(f) = g.^[1] Full functors preserve the entirety of the morphism structure between their images, ensuring no arrows are "missed" in the target category.^[1] They arise naturally in the context of full embeddings and full subcategories, where a subcategory \mathcal{S} \subseteq \mathcal{D} is full if it contains all morphisms of \mathcal{D} between the objects of \mathcal{S}; the inclusion functor of such a subcategory is then full by definition.^[1] An example of a non-full functor is the forgetful functor from the category of partially ordered sets (Poset), where objects are posets and morphisms are order-preserving maps, to the category of sets (Set), where morphisms are all functions; this functor is not full because not every function between underlying sets preserves the order relation.^[5] In contrast to faithful functors, which are injective on hom-sets, fullness emphasizes this surjective aspect on morphisms.^[1]

Full and faithful functors

A functor F: \mathcal{C} \to \mathcal{D} between categories is full and faithful if it is both full and faithful, meaning that for every pair of objects A, B in \mathcal{C}, the induced map F: \mathcal{C}(A, B) \to \mathcal{D}(F A, F B) on hom-sets is a bijection.^[6] This bijectivity ensures that F provides an exact correspondence between morphisms in \mathcal{C} and those between their images in \mathcal{D}, preserving the relational structure without collapse or omission.^[6] Formally, the condition is that each hom-set map is an isomorphism of sets: \mathcal{C}(A, B) \cong \mathcal{D}(F A, F B).^[6] Such functors, often termed fully faithful, integrate the surjectivity of full functors with the injectivity of faithful ones to yield a complete embedding of the morphism structure.^[6] A key property is that full and faithful functors reflect isomorphisms: if f: A \to B in \mathcal{C} is such that F f is an isomorphism in \mathcal{D}, then f itself is an isomorphism in \mathcal{C}.^[6] This reflection follows directly from the bijectivity on hom-sets, as the existence of an inverse for F f implies, via the inverse map on hom-sets, an inverse for f.^[6] Unlike mere inclusions of subcategories, which may fail to capture all morphisms between included objects (i.e., not full), a full and faithful functor guarantees no missing or extraneous morphisms in the image, establishing a precise isomorphic correspondence on hom-sets.^[6]

Properties

Preservation and reflection

Full and faithful functors preserve limits and colimits that exist in the source category. Specifically, if F: \mathcal{C} \to \mathcal{D} is full and faithful and a diagram in \mathcal{C} admits a limit, then the image diagram in \mathcal{D} admits a limit given by applying F to the limiting cone in \mathcal{C}.^[7] The same holds for colimits, as the bijectivity on hom-sets ensures that the universal property transfers directly to the codomain.^[8] In addition to preservation, full and faithful functors reflect a wide range of categorical properties. They reflect isomorphisms: if F(f) is an isomorphism in \mathcal{D}, then f is an isomorphism in \mathcal{C}, since the injectivity and surjectivity on hom-sets imply that f admits an inverse whenever its image does.^[9] More generally, they reflect limits and colimits; for instance, if the image of a diagram under F has a limit in \mathcal{D}, then the original diagram has a limit in \mathcal{C}.^[10] This reflection property follows from the fact that cones over the original diagram correspond bijectively to cones over the image diagram via F, preserving the universal mapping property.^[8] A key theorem states that if F: \mathcal{C} \to \mathcal{D} is full and faithful, then F reflects limits: a diagram in \mathcal{C} has a limit if and only if its image under F has a limit in \mathcal{D}.^[7] Formally, for a cone \eta: X \to \Delta A_i in \mathcal{C} over a diagram (A_i)_{i \in I}, this cone is limiting if and only if the image cone F(\eta): F(X) \to \Delta F(A_i) is limiting in \mathcal{D}. The dual statement holds for colimits.^[10]

Relation to equivalences and adjoints

A functor F: \mathcal{C} \to \mathcal{D} between categories is an equivalence of categories if and only if it is full, faithful, and essentially surjective on objects. This characterization highlights the central role of full and faithful functors in establishing equivalences, as the additional condition of essential surjectivity ensures that every object in \mathcal{D} is isomorphic to one in the image of F. Essential surjectivity means that for every object d in \mathcal{D}, there exists an object c in \mathcal{C} such that d \cong F(c). In the context of adjoint functors, full and faithful functors arise prominently in reflective and coreflective subcategories. Specifically, if \mathcal{B} is a reflective subcategory of \mathcal{A}, then the inclusion functor i: \mathcal{B} \to \mathcal{A} (the right adjoint to the reflection functor) is full and faithful. Dually, in a coreflective subcategory, the inclusion functor (now the left adjoint to the coreflection) is full and faithful. These situations underscore how full and faithful functors capture the "inclusion" of subcategories while preserving all structure and morphisms between their objects. Full and faithful functors also play a key role in Kan extensions, where they ensure that pointwise Kan extensions are absolute. That is, if K: \mathcal{A} \to \mathcal{B} is full and faithful, then the pointwise left (or right) Kan extension of any functor along K exists (under suitable completeness or cocompleteness assumptions on the codomain) and is preserved by arbitrary functors out of the codomain, making it absolute. This absoluteness property facilitates computations and extensions in categorical constructions, as the Kan extension behaves independently of the ambient category.

Examples

Concrete categories

In concrete categories, full and faithful functors often arise as inclusions of subcategories or as forgetful functors that preserve structure without adding extraneous morphisms. A prominent example is the inclusion functor i: \mathbf{FinSet} \to \mathbf{Set}, where \mathbf{FinSet} is the category of finite sets and functions between them. This functor is full and faithful because, for any finite sets A and B, the hom-sets satisfy \mathbf{FinSet}(A, B) \cong \mathbf{Set}(A, B); every function between finite sets in \mathbf{Set} is automatically a morphism in \mathbf{FinSet}, inducing a bijection on hom-sets. In contrast, the forgetful functor U: \mathbf{Grp} \to \mathbf{Set}, which sends a group to its underlying set and a group homomorphism to its underlying function, is faithful but not full. Faithfulness holds because distinct group homomorphisms between groups G and H differ as functions on the underlying sets UG and UH, so the induced map \mathbf{Grp}(G, H) \to \mathbf{Set}(UG, UH) is injective. However, it is not full, as there exist functions in \mathbf{Set}(UG, UH) that do not preserve the group operations and thus are not group homomorphisms; for instance, given nontrivial groups G and H, the set of all functions \mathbf{Set}(UG, UH) properly contains the set of group homomorphisms \mathbf{Grp}(G, H). Specifically, \mathbf{Grp}(G, H) bijects with the subset of \mathbf{Set}(UG, UH) consisting of functions that preserve the group multiplication and identity. Similarly, the underlying set functor U: \mathbf{Top} \to \mathbf{Set}, which maps a topological space to its set of points and a continuous map to its underlying function, is faithful but not full. It is faithful since distinct continuous maps between spaces X and Y differ as set functions, yielding an injective map \mathbf{Top}(X, Y) \to \mathbf{Set}(UX, UY). Fullness fails because continuous maps form a proper subset of all set functions; for example, between the real line with the standard topology and itself, the discontinuous step function is in \mathbf{Set}(\mathbb{R}, \mathbb{R}) but not in \mathbf{Top}(\mathbb{R}, \mathbb{R}). Thus, \mathbf{Top}(X, Y) bijects with the subset of \mathbf{Set}(UX, UY) comprising continuous functions.

Abstract categories

In abstract categories, the inclusion functor of a full subcategory provides a paradigmatic example of a full and faithful functor. By definition, such an inclusion preserves all objects of the subcategory and includes every morphism between those objects exactly as it appears in the ambient category, ensuring bijectivity on hom-sets.^[1] This embedding realizes the subcategory as structurally indistinguishable from its image within the larger category, reflecting limits, colimits, and isomorphisms while preserving the categorical structure without alteration.^[1] The nerve functor N: \mathbf{Cat} \to \mathbf{sSet}, which assigns to each small category its nerve—a simplicial set whose n-simplices are the functors from the opposite of the standard n-simplex category to the given category—exemplifies a full and faithful embedding in the context of simplicial sets. This functor induces a bijection between natural transformations of functors between categories and simplicial maps between their nerves, thereby embedding the category of small categories fully and faithfully into the category of simplicial sets.^[11] Consequently, categories are realized as special simplicial sets (those with a single non-degenerate n-simplex for each n), preserving the compositional structure of composition and identities through the simplicial face and degeneracy maps.^[12] A canonical full and faithful functor in any locally small category \mathcal{C} is the Yoneda embedding y: \mathcal{C} \to [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}], which sends each object c \in \mathcal{C} to its representable functor \mathcal{C}(c, -): \mathcal{C} \to \mathbf{Set} and each morphism f: c \to c' to the induced natural transformation \mathcal{C}(c, -) \to \mathcal{C}(c', -). The Yoneda lemma establishes that this functor is full and faithful by providing a natural bijection \mathrm{Nat}(y(c), F) \cong F(c) for any presheaf F, ensuring bijectivity on hom-sets between representables and reflecting the universal property of hom-functors.^[1] This embedding realizes objects of \mathcal{C} as their "structure of morphisms," embedding the category densely into the presheaf category without loss of structural information. In the category of posets \mathbf{Poset}, where objects are partially ordered sets and morphisms are order-preserving maps, the inclusion of a subposet via the identity map on elements yields a full and faithful functor. Since hom-sets in poset categories contain at most one morphism (corresponding to the order relation), the inclusion is faithful by preserving the unique possible arrows injectively and full by ensuring that any order relation between images in the ambient poset lifts bijectively to the subposet when both elements reside there.^[1] More generally, an order embedding—an injective order-preserving map that reflects the order (i.e., f(x) \leq f(y) implies x \leq y)—induces a full and faithful functor between the associated category structures, embedding the subposet rigidly while preserving the relational structure.

Generalizations

To enriched categories

In the context of V-enriched category theory, where V is a monoidal category and categories are enriched over V, the notions of full and faithful functors are generalized using the structure of V. A V-functor F: \mathcal{C} \to \mathcal{D} between V-categories is V-faithful if, for all objects A, B \in \mathcal{C}, the induced map on hom-objects F_{A,B}: \mathcal{C}(A,B) \to \mathcal{D}(FA, FB) is a monomorphism in V; it is V-full if each such map is an epimorphism in V; and it is V-fully faithful (or V-full and V-faithful) if each map is an isomorphism in V.^[13] This extends the ordinary case over V = Set, where monomorphisms and epimorphisms correspond to injections and surjections on hom-sets.^[13] A concrete instance arises when V = Ab, the category of abelian groups, in the theory of additive categories. Here, V-faithful functors induce injective group homomorphisms on hom-groups, while V-full functors induce surjective group homomorphisms, reflecting the additive structure where composition is mediated by abelian group operations.^[13] V-fully faithful functors retain key properties from the unenriched setting, notably in reflecting enriched limits and weighted colimits. Specifically, V-fully faithful functors reflect enriched limits and weighted colimits: if F maps a V-cone over a diagram in \mathcal{C} to a limit cone in \mathcal{D} (or a weighted colimit cocone), then the original V-cone in \mathcal{C} is a limit (or weighted colimit), with the mediating morphisms forming an isomorphism in V.^[13] These concepts were developed in the 1970s as part of the foundational work on enriched category theory by G. M. Kelly and Ross Street, among others in the Australian school, providing a systematic framework that addresses gaps in earlier basic category theory texts.^[13]^[14]

To (∞,1)-categories

In the context of (\infty,1)-categories, a functor F: \mathcal{C} \to \mathcal{D} is fully faithful if, for every pair of objects A, B \in \mathcal{C}, the induced map on mapping spaces \mathrm{Map}_{\mathcal{C}}(A, B) \to \mathrm{Map}_{\mathcal{D}}(FA, FB) is an equivalence of spaces (regarded as \infty-groupoids).^[15] This condition ensures that F embeds \mathcal{C} as a full sub-(\infty,1)-category of \mathcal{D}, preserving the full homotopy-theoretic structure of homotopical morphisms between objects.^[16] This notion recovers the classical definition of a full and faithful functor upon applying the homotopy category functor h: the zeroth homotopy groups \pi_0(\mathrm{Map}_{\mathcal{C}}(A, B)) and \pi_0(\mathrm{Map}_{\mathcal{D}}(FA, FB)) recover the hom-sets \mathcal{C}(A, B) and \mathcal{D}(FA, FB), and an equivalence of spaces induces a bijection on these \pi_0 groups.^[15] Thus, fully faithful (\infty,1)-functors generalize the ordinary case by accounting for higher homotopical data beyond mere set bijections. A key property is that fully faithful (\infty,1)-functors reflect (\infty,1)-limits and colimits: if a diagram in \mathcal{C} is sent by F to a limit (or colimit) diagram in \mathcal{D}, then the original diagram was already a limit (or colimit) in \mathcal{C}.^[10] This reflection arises because the universal property of limits and colimits in (\infty,1)-categories is detected via mapping spaces, which F preserves up to equivalence.^[10] In applications to stable homotopy theory, fully faithful (\infty,1)-functors facilitate embeddings such as the Yoneda embedding of the (\infty,1)-category of finite spectra S_{\mathrm{fin}}^\infty into the stable (\infty,1)-category of spectra S_\infty, preserving colimits and enabling the study of infinite spectra as Ind-completions of finite ones.^[17] This embedding underpins much of modern stable homotopy theory by allowing computations in the smaller category to inform the larger one.^[17]

Full and faithful functors

Definitions

Faithful functors

Full functors

Full and faithful functors

Properties

Preservation and reflection

Relation to equivalences and adjoints

Examples

Concrete categories

Abstract categories

Generalizations

To enriched categories

To (∞,1)-categories

References