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Functor category

In category theory, the functor category [ \mathcal{C}, \mathcal{D} ], also known as the category of functors from \mathcal{C} to \mathcal{D}, is defined such that its objects are all functors F: \mathcal{C} \to \mathcal{D} and its morphisms are natural transformations between these functors.^[1] This construction allows functors themselves to be treated as objects within a larger categorical framework, enabling the study of relationships among structure-preserving maps between categories.^[2] A functor F: \mathcal{C} \to \mathcal{D} is a mapping that assigns to each object A in \mathcal{C} an object FA in \mathcal{D}, and to each morphism f: A \to B in \mathcal{C} a morphism Ff: FA \to FB in \mathcal{D}, while preserving the composition of morphisms (F(g \circ f) = Fg \circ Ff) and identity morphisms (F(\mathrm{id}_A) = \mathrm{id}_{FA}).^[3] Functors thus serve as the "morphisms" in the category of categories, Cat, preserving the structural axioms of categories and facilitating comparisons across different mathematical domains, such as algebra and topology.^[4] The morphisms in the functor category, natural transformations, provide a way to compare functors pointwise while respecting their action on morphisms. Specifically, for functors F, G: \mathcal{C} \to \mathcal{D}, a natural transformation \alpha: F \to G consists of a family of morphisms \alpha_A: FA \to GA for each object A in \mathcal{C}, such that for every morphism f: A \to B in \mathcal{C}, the diagram

\begin{CD} FA @>{\alpha_A}>> GA \\ @V{Ff}VV @VV{Gf}V \\ FB @>>{\alpha_B}> GB \end{CD}

commutes (i.e., Gf \circ \alpha_A = \alpha_B \circ Ff).^[1] Natural transformations compose vertically (via componentwise composition) to form another natural transformation, and each functor has an identity natural transformation, ensuring that [ \mathcal{C}, \mathcal{D} ] satisfies the axioms of a category.^[5] This composition is associative, underscoring the functor category's role in higher-dimensional category theory, where it acts as a hom-category in the 2-category Cat.^[6] Functor categories are fundamental for constructing limits and colimits in enriched settings, representing actions (e.g., [\mathbb{M}, \mathbf{Set}] for monoid actions on sets), and generalizing to accessible categories in advanced applications like model theory and homotopy theory.^[5] Their study originated in the foundational work of Samuel Eilenberg and Saunders Mac Lane in 1945, where functors and natural transformations were introduced to unify diverse mathematical structures.^[4]

Definition and Construction

Formal Definition

In category theory, given two categories \mathcal{C} and \mathcal{D}, the functor category [\mathcal{C}, \mathcal{D}] (also denoted \mathrm{Fun}(\mathcal{C}, \mathcal{D})) is defined such that its objects are all functors F: \mathcal{C} \to \mathcal{D}.^[7]^[8]^[9] The morphisms in [\mathcal{C}, \mathcal{D}] from a functor F to a functor G: \mathcal{C} \to \mathcal{D} are the natural transformations \eta: F \Rightarrow G. Such a natural transformation consists of a family of morphisms \eta_c: F(c) \to G(c) in \mathcal{D}, one for each object c in \mathcal{C}, satisfying the naturality condition: for every morphism f: c \to c' in \mathcal{C}, the following diagram commutes in \mathcal{D},

\begin{CD} F(c) @>{\eta_c}>> G(c) \\ @V{F(f)}VV @VV{G(f)}V \\ F(c') @>>{\eta_{c'}}> G(c') \end{CD}

or equivalently, \eta_{c'} \circ F(f) = G(f) \circ \eta_c.^[7]^[8]^[9] This structure forms a category because the vertical composition of natural transformations is associative and the identity natural transformations serve as units. Specifically, for natural transformations \eta: F \Rightarrow G and \kappa: G \Rightarrow H, their composite \kappa \circ \eta: F \Rightarrow H is defined componentwise by (\kappa \circ \eta)_c = \kappa_c \circ \eta_c for each c \in \mathcal{C}, and this composition is associative since it inherits associativity from the morphisms in \mathcal{D}. The identity natural transformation on F has components \mathrm{id}_{F(c)} for each c, satisfying the unit laws for composition.^[7]^[8]^[9] Common notations for the functor category include [\mathcal{C}, \mathcal{D}] and \mathcal{D}^\mathcal{C}; when \mathcal{D} = \mathbf{Set}, it is often denoted ^\mathcal{C}\mathbf{Set}.^[7]^[8]^[9]

Objects and Morphisms

In the functor category \mathcal{D}^{\mathcal{C}}, the objects are covariant functors F: \mathcal{C} \to \mathcal{D}. Each such functor maps objects of \mathcal{C} to objects of \mathcal{D} and morphisms of \mathcal{C} to morphisms of \mathcal{D}, while preserving composition and identities: for all morphisms f: c \to c' and g: c' \to c'' in \mathcal{C}, F(g \circ f) = F(g) \circ F(f), and for each object c in \mathcal{C}, F(\mathrm{id}_c) = \mathrm{id}_{F(c)}.^[10] The morphisms between two objects F, G: \mathcal{C} \to \mathcal{D} in this category are natural transformations \eta: F \Rightarrow G. Such a natural transformation consists of components \eta_c: F(c) \to G(c) in \mathcal{D} for each object c in \mathcal{C}, satisfying the naturality condition that for every morphism f: c \to c' in \mathcal{C}, the following diagram commutes:

\begin{CD} F(c) @>{\eta_c}>> G(c) \\ @V{F(f)}VV @VV{G(f)}V \\ F(c') @>>{\eta_{c'}}> G(c') \end{CD}

This equality is expressed as \eta_{c'} \circ F(f) = G(f) \circ \eta_c.^[11]^[12] Natural transformations compose vertically: given \eta: F \Rightarrow G and \theta: G \Rightarrow H, their vertical composite is the natural transformation \theta \cdot \eta: F \Rightarrow H defined by components (\theta \cdot \eta)_c = \theta_c \circ \eta_c for each object c in \mathcal{C}. Horizontal composition of natural transformations is addressed in connections to broader categorical structures.^[11]^[10] For each functor F: \mathcal{C} \to \mathcal{D}, the identity natural transformation \mathrm{id}_F: F \Rightarrow F has components (\mathrm{id}_F)_c = \mathrm{id}_{F(c)} for each c in $\mathcal{C}); which acts as the unit for vertical composition of natural transformations.^[12]^[10]

Composition and Identities

In the functor category [ \mathcal{C}, \mathcal{D} ], the composition of morphisms is given by the vertical composition of natural transformations.^[8] For natural transformations \theta: G \Rightarrow H and \eta: F \Rightarrow G between functors F, G, H: \mathcal{C} \to \mathcal{D}, their composite \theta \cdot \eta: F \Rightarrow H is defined pointwise by

(\theta \cdot \eta)_c = \theta_c \circ \eta_c

for each object c \in \mathcal{C}, where \circ denotes the composition of morphisms in \mathcal{D}.^[8] This operation is natural because the defining commutative diagrams for \theta and \eta ensure that \theta \cdot \eta satisfies the naturality condition with respect to morphisms in \mathcal{C}.^[8] The vertical composition in [ \mathcal{C}, \mathcal{D} ] is associative, inheriting this property from the associativity of morphism composition in the codomain category \mathcal{D}.^[8] Specifically, for natural transformations \alpha: E \Rightarrow F, \eta: F \Rightarrow G, and \theta: G \Rightarrow H, the equality (\theta \cdot \eta) \cdot \alpha = \theta \cdot (\eta \cdot \alpha) holds pointwise, as

\bigl( (\theta \cdot \eta) \cdot \alpha \bigr)_c = \theta_c \circ \eta_c \circ \alpha_c = \bigl( \theta \cdot (\eta \cdot \alpha) \bigr)_c

for each c \in \mathcal{C}.^[8] The identity morphism for a functor F: \mathcal{C} \to \mathcal{D} in [ \mathcal{C}, \mathcal{D} ] is the natural transformation \mathrm{id}_F: F \Rightarrow F defined by (\mathrm{id}_F)_c = \mathrm{id}_{F(c)} for each object c \in \mathcal{C}, where \mathrm{id}_{F(c)} is the identity morphism on F(c) in \mathcal{D}.^[8] This identity acts as the left and right unit for vertical composition: for any natural transformation \eta: F \Rightarrow G, \mathrm{id}_G \cdot \eta = \eta, and for \eta: H \Rightarrow F, \eta \cdot \mathrm{id}_F = \eta, holding pointwise due to the identities in \mathcal{D}.^[8] The construction of the functor category is itself functorial. The assignment [ \mathcal{C}, - ]: \mathbf{Cat} \to \mathbf{Cat} is a covariant functor, sending a functor K: \mathcal{D} \to \mathcal{E} to the functor K_*: [ \mathcal{C}, \mathcal{D} ] \to [ \mathcal{C}, \mathcal{E} ] defined by post-composition K_* (S) = K \circ S on objects and \mathrm{Nat}(K_* \theta, K_* \phi) = \mathrm{Nat}(\theta, \phi) on morphisms, preserving composition and identities.^[8] Dually, the assignment [ - , \mathcal{D} ]: \mathbf{Cat}^{\mathrm{op}} \to \mathbf{Cat} is contravariant, sending a functor L: \mathcal{C} \to \mathcal{B} to the functor L^*: [ \mathcal{B}, \mathcal{D} ] \to [ \mathcal{C}, \mathcal{D} ] defined by pre-composition L^* (T) = T \circ L on objects and \mathrm{Nat}(L^* \phi, L^* \theta) = \mathrm{Nat}(\phi, \theta) on morphisms.^[8] Together, these yield a bifunctor \mathbf{Cat}^{\mathrm{op}} \times \mathbf{Cat} \to \mathbf{Cat}.^[8]

Properties

Basic Categorical Properties

The functor category [\mathcal{C}, \mathcal{D}], also denoted \mathcal{D}^\mathcal{C}, is itself a category whose objects are functors from \mathcal{C} to \mathcal{D} and whose morphisms are natural transformations between such functors.^[8]^[7] The composition of natural transformations in this category is defined componentwise: for natural transformations \gamma: F \to G and \delta: G \to H, the composite \delta \circ \gamma has components (\delta \circ \gamma)_c = \delta_c \circ \gamma_c for each object c in \mathcal{C}.^[8] This composition is associative and admits identity natural transformations $1_F with components (1_F)_c = \mathrm{id}_{F(c)}, ensuring the axioms of a category hold coherently, just as in any category.^[8]^[7] The functor category [\mathcal{C}, \mathcal{D}] is locally small whenever \mathcal{C} is small and \mathcal{D} is locally small, meaning that the hom-sets \mathrm{Hom}_{[\mathcal{C},\mathcal{D}]}(F, G) = \mathrm{Nat}(F, G) are small sets for any functors F, G: \mathcal{C} \to \mathcal{D}.^[7]^[13] In general, if \mathcal{C} is not small, these hom-sets may fail to be small sets, though they remain classes.^[8]^[13] In the context of enriched category theory, if \mathcal{D} is enriched over a monoidal category \mathcal{V}, then the functor category [\mathcal{C}, \mathcal{D}] inherits this enrichment over \mathcal{V}, with hom-objects given by ends: [\mathcal{C}, \mathcal{D}](F, G) = \int_{c \in \mathcal{C}} \mathcal{D}(F c, G c).^[8] This structure preserves the enriched composition and identities from \mathcal{D}, generalizing the ordinary case where \mathcal{V} = \mathbf{Set}.^[8]

Limits, Colimits, and Adjoints

In the functor category [ \mathcal{C}, \mathcal{D} ], limits of a diagram consisting of functors \{ F_i : \mathcal{C} \to \mathcal{D} \}_{i \in I} are defined via the universal property in terms of cones of natural transformations. Specifically, a limit is a functor L : \mathcal{C} \to \mathcal{D} equipped with a family of natural transformations \pi_i : L \Rightarrow F_i for each i \in I, such that for any other functor K : \mathcal{C} \to \mathcal{D} with natural transformations \sigma_i : K \Rightarrow F_i, there exists a unique natural transformation \mu : K \Rightarrow L satisfying \pi_i \circ \mu = \sigma_i for all i. This universal property is satisfied pointwise: for each object c \in \mathcal{C}, the family \{ F_i(c) \}_{i \in I} in \mathcal{D} admits a limit cone with vertex L(c), and the components of the \pi_i are the corresponding projections in \mathcal{D}.^[14]^[8] If \mathcal{D} is complete (i.e., has all small limits), then so is [ \mathcal{C}, \mathcal{D} ], assuming \mathcal{C} is small, and the limits are computed pointwise. That is, for each c \in \mathcal{C},

L(c) = \lim_{i \in I} F_i(c),

where the limit on the right is taken in \mathcal{D}, and the natural transformations \pi_i have components \pi_i(c) : L(c) \to F_i(c) given by the limit projections in \mathcal{D}. This pointwise construction ensures that the evaluation functors \mathrm{ev}_c : [ \mathcal{C}, \mathcal{D} ] \to \mathcal{D}, which send a functor F to F(c), preserve these limits.^[14]^[8]^[6] Colimits in [ \mathcal{C}, \mathcal{D} ] are defined dually: a colimit of \{ F_i \} is a functor \mathrm{colim}_i F_i : \mathcal{C} \to \mathcal{D} with natural transformations \iota_i : F_i \Rightarrow \mathrm{colim}_i F_i such that for any K with \tau_i : F_i \Rightarrow K, there is a unique \nu : \mathrm{colim}_i F_i \Rightarrow K with \nu \circ \iota_i = \tau_i for all i. This property holds pointwise, and if \mathcal{D} is cocomplete and \mathcal{C} is small, then [ \mathcal{C}, \mathcal{D} ] is cocomplete with colimits computed pointwise:

(\mathrm{colim}_{i \in I} F_i)(c) = \mathrm{colim}_{i \in I} F_i(c)

for each c \in \mathcal{C}, with the \iota_i components being the colimit inclusions in \mathcal{D}. The evaluation functors \mathrm{ev}_c then create these colimits, reflecting the structure back to \mathcal{D}. For instance, in the category of sets, coproducts in the functor category correspond to disjoint unions taken at each point.^[14]^[8]^[6] Adjunctions between categories \mathcal{C} and \mathcal{D} lift to adjunctions between their functor categories. Suppose F : \mathcal{C} \to \mathcal{D} is left adjoint to G : \mathcal{D} \to \mathcal{C} (i.e., F \dashv G). For any category \mathcal{E}, the precomposition functors induce an adjunction F^* \dashv G^* : [ \mathcal{C}, \mathcal{E} ] \leftrightarrows [ \mathcal{D}, \mathcal{E} ], where G^*(H) = H \circ G for H : \mathcal{D} \to \mathcal{E} and F^*(K) = K \circ F for K : \mathcal{C} \to \mathcal{E}. The unit and counit of this adjunction are derived pointwise from those of F \dashv G: the unit \eta^* : \mathrm{id}_{[ \mathcal{C}, \mathcal{E} ]} \Rightarrow G^* F^* has components \eta^*_K = G(\eta_K) \circ \eta (natural in K), and the counit \epsilon^* : F^* G^* \Rightarrow \mathrm{id}_{[ \mathcal{D}, \mathcal{E} ]} has components \epsilon^*_H = F(\epsilon_H) \circ \epsilon_{GH} (natural in H). This construction preserves the hom-set bijections naturally, ensuring that natural transformations between functors correspond via the original adjunction. An example arises in representation theory, where forgetful functors from group representations to sets induce adjoints via induction and coinduction in the respective functor categories.^[14]^[8]

Size Considerations

In category theory, the size of the functor category [ \mathcal{C}, \mathcal{D} ], also denoted \mathcal{D}^\mathcal{C}, is determined by the cardinalities of the collections of objects and morphisms in \mathcal{C} and \mathcal{D}. A category is small if both its class of objects and its class of morphisms form sets, while it is locally small if the hom-sets between any pair of objects are sets. If \mathcal{C} is small and \mathcal{D} is locally small, then [ \mathcal{C}, \mathcal{D} ] is locally small, as the sets of natural transformations between functors are constructed as equalizers of products indexed by the small set of morphisms in \mathcal{C}, yielding sets via the local smallness of \mathcal{D}.^[8] This ensures that the hom-sets in the functor category remain manageable within set-theoretic bounds. When \mathcal{C} is large—meaning its objects or morphisms form a proper class—the functor category [ \mathcal{C}, \mathcal{D} ] typically has a proper class of objects, even if \mathcal{D} is locally small, because the functors are specified by class-indexed assignments. In such cases, the hom-sets \mathrm{Nat}(F, G) between functors F, G: \mathcal{C} \to \mathcal{D} may also be proper classes, as they involve components indexed over the class of morphisms in \mathcal{C}. To rigorously handle these large functor categories without foundational paradoxes, techniques such as accessible categories or Grothendieck universes are employed; a Grothendieck universe \mathcal{U} is a transitive set closed under pairing, power sets, and unions of small families, allowing one to define \mathcal{U}-small categories whose objects and morphisms lie in \mathcal{U}, and ensuring that if \mathcal{C} is \mathcal{U}-small and \mathcal{D} is a \mathcal{U}-category (locally \mathcal{U}-small), then [ \mathcal{C}, \mathcal{D} ] is also a \mathcal{U}-category. A notable instance arises with presheaf categories [ \mathcal{C}, \mathbf{Set} ]. When \mathcal{C} is small, this forms an elementary topos, possessing all finite limits and colimits, subobject classifiers, and other topos-theoretic structure, as the category of sets provides the necessary completeness and the smallness of \mathcal{C} ensures the existence of required Kan extensions and representables. However, for large \mathcal{C}, such as \mathcal{C} = \mathbf{Set}, [ \mathcal{C}, \mathbf{Set} ] becomes a proper class category, necessitating careful set-theoretic foundations like those in ZFC with global choice or universe axioms to define its structure and verify properties like 2-categorical composition. This distinction highlights the foundational role of smallness: small functor categories admit set-based treatments akin to ordinary categories, while large ones demand extensions of set theory to avoid inconsistencies in handling class-sized collections.

Examples

Elementary Examples

One of the simplest examples of a functor category arises when considering the terminal category \mathbf{1}, which consists of a single object * equipped with only its identity morphism \mathrm{id}_* : * \to *. The functor category [\mathbf{1}, \mathbf{D}] has as objects all functors F : \mathbf{1} \to \mathbf{D}, each of which selects a single object F(*) \in \mathrm{Ob}(\mathbf{D}) since \mathbf{1} has only one object, with the unique morphism \mathrm{id}_* mapping to the identity on that object. Morphisms in [\mathbf{1}, \mathbf{D}] are natural transformations between such functors, which reduce to ordinary morphisms in \mathbf{D} between the corresponding objects F(*) \to G(*). Thus, [\mathbf{1}, \mathbf{D}] is isomorphic to \mathbf{D} itself, where the isomorphism sends each object of \mathbf{D} to the constant functor on that object and each morphism in \mathbf{D} to the corresponding constant natural transformation.^[15] Dually, the functor category [\mathbf{C}, \mathbf{1}] provides another elementary case. Here, there is a unique functor F : \mathbf{C} \to \mathbf{1}, which sends every object of \mathbf{C} to * and every morphism in \mathbf{C} to \mathrm{id}_*. Consequently, [\mathbf{C}, \mathbf{1}] has a single object (this unique functor) and a single morphism (the identity natural transformation on it). Therefore, [\mathbf{C}, \mathbf{1}] is isomorphic to \mathbf{1} itself. A slightly more involved but still introductory example is the presheaf category [\mathbf{C}^\mathrm{op}, \mathbf{Set}], where \mathbf{Set} denotes the category of sets. Objects here are contravariant functors F : \mathbf{C} \to \mathbf{Set}, assigning to each object c \in \mathrm{Ob}(\mathbf{C}) a set F(c) and to each morphism f : c \to c' in \mathbf{C} a function F(f) : F(c') \to F(c) that preserves identities and composition. Morphisms in [\mathbf{C}^\mathrm{op}, \mathbf{Set}] are natural transformations, which are families of functions \eta_c : F(c) \to G(c) for G : \mathbf{C} \to \mathbf{Set} that commute with the actions on morphisms, i.e., \eta_c \circ F(f) = G(f) \circ \eta_{c'} for all f : c \to c'. This category captures assignments of sets to objects of \mathbf{C} equipped with compatible actions under morphisms, serving as a foundational setting for many constructions in category theory.^[15] To illustrate concretely, consider a small finite category \mathbf{C} with two objects A, B and morphisms including identities \mathrm{id}_A, \mathrm{id}_B plus a single non-identity morphism g : A \to B, and take \mathbf{D} = \mathbf{Set}. A functor F : \mathbf{C} \to \mathbf{Set} is then specified by sets F(A), F(B) and a function F(g) : F(A) \to F(B), with F(\mathrm{id}_A) = \mathrm{id}_{F(A)} and F(\mathrm{id}_B) = \mathrm{id}_{F(B)} automatically. For instance, one such functor might assign F(A) = \{1, 2\}, F(B) = \{a, b, c\}, and F(g) mapping $1 \mapsto a, $2 \mapsto b; another could assign singleton sets everywhere with F(g) the unique function between them. Natural transformations between two such functors F, H are functions \eta_A : F(A) \to H(A) and \eta_B : F(B) \to H(B) satisfying \eta_B \circ F(g) = H(g) \circ \eta_A, ensuring compatibility with the structure of \mathbf{C}. This explicit enumeration highlights how functors amount to families of sets indexed by objects, linked by maps induced by morphisms.^[11]

Notable Specific Cases

The presheaf category [\mathbf{C}^{\mathrm{op}}, \mathbf{Set}], often denoted \widehat{\mathbf{C}}, has as objects all contravariant functors from a small category \mathbf{C} to the category of sets, with morphisms given by natural transformations between such functors.^[16] This category serves as a "generalized universe" of sets indexed by \mathbf{C}, where objects can be thought of as generalized sets or spaces parametrized by the structure of \mathbf{C}, and it is cocomplete with all small colimits computed pointwise.^[17] Presheaf categories are fundamental in algebraic geometry and topology, providing a topos-theoretic framework for studying sheaves and schemes without assuming additional structure like a Grothendieck topology. When \mathbf{C} is a small category and \mathbf{Ab} is the category of abelian groups, the functor category [\mathbf{C}, \mathbf{Ab}] forms an abelian category, inheriting exactness properties from \mathbf{Ab} via pointwise kernels and cokernels.^[18] Objects in this category are covariant functors assigning to each object in \mathbf{C} an abelian group and to each morphism a group homomorphism, making it a key setting for homological algebra over categories, such as computing derived functors in a categorical context. This structure is particularly useful in representation theory and algebraic K-theory, where it models chain complexes or modules over category rings.^[18] For a quiver \mathbf{Q}, viewed as a small category with objects as vertices and morphisms as paths along directed edges, the category of representations [\mathbf{Q}, \mathbf{Vect}_k] over a field k consists of functors assigning to each vertex a finite-dimensional vector space and to each arrow a linear map between those spaces, with natural transformations as morphisms.^[19] This category captures the linear algebraic essence of quiver representations, which classify indecomposable modules over path algebras and connect to topics like invariant theory and Kac-Moody algebras through its Krull-Schmidt property and Auslander-Reiten theory.^[19] Representations here emphasize the combinatorial and finite-dimensional aspects, distinguishing them from more general quiver functors by focusing on vector space assignments. Simplicial sets arise as the presheaf category [\Delta^{\mathrm{op}}, \mathbf{Set}], where \Delta is the simplex category with objects finite ordinals = \{0, 1, \dots, n\} and morphisms generated by face and degeneracy maps.^[20] Objects are functors encoding sequences of sets X_n for n \geq 0, equipped with simplicial operators modeling higher-dimensional simplices and their gluings, forming a cartesian closed category that models topological spaces combinatorially.^[21] This category is central to homotopy theory, where Kan complexes within it represent \infty-groupoids, and it supports geometric realization functors to topological spaces, highlighting its role in bridging combinatorics and topology.^[20]

Relations to Other Concepts

Connection to 2-Categories

The 2-category \mathbf{[Cat](/page/Cat)} has small categories as 0-cells, functors between them as 1-cells, and natural transformations as 2-cells.^[22]^[23] In this structure, the hom-category \mathbf{Cat}(C, D) between two categories C and D is precisely the functor category [C, D], whose objects are functors C \to D and whose morphisms are natural transformations between such functors.^[22]^[23] This embedding positions functor categories as the basic building blocks for composition in \mathbf{Cat}. In \mathbf{Cat}, natural transformations admit a horizontal composition operation, which enables the 2-categorical structure. Specifically, for natural transformations \eta \colon F \Rightarrow F' with F, F' \colon C \to D and \theta \colon G \Rightarrow G' with G, G' \colon D \to E, the horizontal composite \eta * \theta \colon F ; G \Rightarrow F' ; G' (where ; denotes ordinary functor composition) is defined componentwise by

(\eta * \theta)_c = \theta_{F'(c)} \cdot G(\eta_c)

for each object c \in C, with \cdot denoting vertical composition of morphisms in E. This is equivalent to G'(\eta_c) \cdot \theta_{F(c)} by the naturality of \theta.^[23] This operation satisfies associativity and unit laws strictly, as \mathbf{Cat} is a strict 2-category.^[23] A key aspect of this composition involves whiskering, where a functor acts on a natural transformation: for instance, the left whiskering G * \eta yields a natural transformation F ; G \Rightarrow F' ; G with components G(\eta_c) \colon G(F(c)) \to G(F'(c)), and similarly for right whiskering \eta * H \colon F ; H \Rightarrow F' ; H.^[22] Whiskering preserves naturality and underpins the horizontal composition, as (\eta * \theta)_c can equivalently be expressed via whiskered terms that equalize by the naturality squares of \eta and \theta.^[23] Functor categories also arise as slices within the higher 2-category \mathbf{2\text{-}Cat} of 2-categories, pseudofunctors, and modifications. For example, the category of functors from a 2-category B to \mathbf{[Cat](/page/Cat)} (indexing 2-categories) can be realized as a slice construction in \mathbf{2\text{-}Cat}, generalizing the role of [C, D] in \mathbf{Cat}.^[22] Such slices preserve the 2-categorical enrichment, allowing functor categories to model fibered structures in higher dimensions.^[24] While \mathbf{[Cat](/page/Cat)} is a strict 2-category—meaning composition and identities hold without isomorphism—weak 2-categories (bicategories) relax these to coherent isomorphisms. Nonetheless, if C and D are ordinary (1-)categories, the hom-category [C, D] remains a strict category, with composition of natural transformations being strictly associative and unital.^[22] This strictness holds regardless of the ambient 2-category's weakness, as [C, D] lacks inherent higher cells beyond natural transformations.^[22]

Role in Representable Functors

The functor category [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}] serves as the codomain for the Yoneda embedding, a canonical functor y: \mathcal{C} \to [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}] that maps each object c \in \mathcal{C} to the representable functor y(c) = \Hom_{\mathcal{C}}(-, c) and each morphism f: c \to c' to the induced natural transformation y(f): y(c) \to y(c'). This embedding is fully faithful, preserving and reflecting the hom-sets of \mathcal{C}, thereby embedding \mathcal{C} as a full subcategory of the functor category.^[25]^[26] The Yoneda lemma establishes a profound connection between objects of \mathcal{C} and functors into \mathbf{Set}, stating that for any locally small category \mathcal{C} and any functor F: \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}, the set of natural transformations satisfies

\Nat(y(c), F) \cong F(c)

naturally in c and F. This isomorphism implies that every functor F is determined up to natural isomorphism by its action on representable functors, underscoring the dense subcategory generated by the image of y within [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}].^[27]^[28] A functor F: \mathcal{C} \to \mathbf{Set} is representable if it is naturally isomorphic to \Hom_{\mathcal{C}}(-, x) for some object x \in \mathcal{C}, meaning F lies in the essential image of the Yoneda embedding (adjusted for covariance). Representability captures universal properties in category theory, such as limits or colimits, by reducing them to concrete objects in \mathcal{C}. More generally, for a category \mathcal{D} with sufficient structure (e.g., cocompleteness), \mathcal{C} admits a full embedding into [\mathcal{C}, \mathcal{D}] via a suitable nerve or realization functor, generalizing the Yoneda construction.^[28]^[27]^[29] The duality between covariant and contravariant functors is reflected in the identification of the category of contravariant functors from \mathcal{C} to \mathcal{D} with [\mathcal{C}^{\mathrm{op}}, \mathcal{D}], which interchanges the roles of domains and codomains in the functor category framework. This perspective highlights how representability in contravariant settings corresponds to corepresentability in covariant ones, facilitating proofs via adjointness and embedding theorems.^[27]^[28]

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[PDF] maclane-categories.pdf - MIT Mathematics
... Mac Lane. Categories for the. Working Mathematician. Second Edition. Springer. Page 4. Saunders Mac Lane. Professor Emeritus. Department of Mathematics.
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Categories for the Working Mathematician - SpringerLink
This book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors.
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[PDF] Intro to Category Theory: Natural Transformations 1 ... - cs.wisc.edu
With category theory we first define categories, which are a way to formalize structure preserving transformations called morphisms.
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[PDF] Chapter 4 - Basic category theory - MIT OpenCourseWare
In this section we give the standard definition of categories and functors. These, together with natural transformations (Section 4.3), form the backbone of ...
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[PDF] NATURALITY
A natural isomorphism is a natural transformation. ϑ : F → G which is an isomorphism in the functor category Fun(C, D). Page 13. i. EXAMPLES OF NATURAL ...<|control11|><|separator|>
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[PDF] ON THE SIZE OF CATEGORIES
Dec 30, 1995 · Key words and phrases: small, locally small, small homsets, idempotent, presheaf category. c⃝ Peter Freyd and Ross Street 1995. Permission to ...
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[PDF] Locally Presentable and Accessible Categories
189 Locally presentable and accessible categories, J. ADAMEK & J. ROSICKY. 190 Polynomial invariants of finite groups, DJ. BENSON. 191 Finite geometry and ...
[15]
[PDF] Category Theory in Context Emily Riehl
Mar 1, 2014 · ... (co)limits in question. In such cases, a functor that creates (co) ... Mac Lane. Categories for the working mathematician, volume 5 of ...
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[PDF] Saunders Mac Lane - Categories for the Working Mathematician
This occurs at several levels. On the first level, categories provide a convenient con- ceptual language, based on the notions of category, functor, natural.
[17]
presheaf in nLab
Oct 9, 2021 · Given a small category C C of “primitive objects”, we can think of a functor F : C op → Set F:\: C^{op} \to Set as being a more complex object ...Idea · Definition · Remarks · Properties
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category of presheaves in nLab
Oct 6, 2025 · Proposition 2.9. Let P : C op → Set P:C^{op}\to Set be a presheaf. Then there is an equivalence of categories. PSh ( ∫ C P ) ≃ PSh ( C ) / P .<|control11|><|separator|>
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abelian category in nLab
Mar 15, 2025 · The notion of abelian category is an abstraction of basic properties of the category Ab of abelian groups, more generally of the category R R ...
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[PDF] Lectures on Representations of Quivers by William Crawley-Boevey
The theory of representations of quivers touches linear algebra, invariant theory, finite dimensional algebras, free ideal rings, Kac-Moody Lie algebras, and ...
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simplicial set in nLab
Jun 10, 2025 · A simplicial set is like a combinatorial space built up out of gluing abstract simplices to each other. Equivalently, it is an object equipped with a rule.Missing: sources | Show results with:sources
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[PDF] An elementary illustrated introduction to simplicial sets
A simplicial set is a contravariant functor X : ∆ → Set (equivalently, a covariant functor X : ∆op → Set). The reader should compare this with the categorical ...
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[PDF] a 2-categories companion - Department of Mathematics
An enriched category is a category in which the hom-functors take their values not in Set, but in some other category V .
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[PDF] The (strict) 2-category of categories - Yigal Kamel
This proves that horizontal composition is associative. Thus, categories, functors, and natural transformations define a strict 2-category.<|separator|>
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[PDF] arXiv:2211.12122v2 [math.CT] 23 May 2024
May 23, 2024 · Abstract. We show that 2-categories of the form B-Cat are closed under slicing, pro- vided that we allow B to range over bicategories ...
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[PDF] 1 Categories
In fact, this is a functor of categories. Yoneda's lemma basically asserts that this is a fully faithful embedding, i.e. every category can be thought of as ...
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An elementary proof of the naturality of the Yoneda embedding
Jul 21, 2023 · The Yoneda embedding is an example of a functor, and the main interest of this article - while classically easy to define as c ↦→ homC(−,c), the ...
[28]
[PDF] introduction to category theory and the yoneda lemma
Oct 9, 2022 · We can further say that if for each object X ∈ C,ηX is an isomorphism in the category D, then F and G are naturally isomorphic. Natural ...
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[PDF] an introduction to category theory and the yoneda lemma
Aug 28, 2020 · the natural transformation µ : A → B consists of just one function ... An important example of a functor category is the category of presheaves.
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EMBEDDING OF CATEGORIES 42 - American Mathematical Society
Abstract. In this paper we generalize the notion of exact functor to an arbitrary category and show that every small category has a full embedding into a ...