Functor category ($[\mathcal{C},\mathcal{D}]$)

The functor category has functors $\mathcal{C}\to \mathcal{D}$ as objects and natural transformations as morphisms.

A natural transformation between functors to the Category of sets is a monomorphism in the functor category exactly when every component is an injective function. Componentwise cancellation proves sufficiency. For necessity, the Yoneda lemma converts two elements with equal component images into two transformations from a covariant representable functor; monicity cancels these transformations, forcing the elements to coincide. The argument remains valid for a locally small category in an ambient universe where the functor category is formed.

For small diagrams with complete codomain, choose each objectwise categorical limit. For $u:d\to d^{\prime }$, its projections uniquely define the arrow $L(u)$ through $p_j(d^{\prime })L(u)=H_j(u)p_j(d)$. The universal property proves identity and composition laws. The same projection calculation makes every categorical cone factorization a natural transformation. Hence these are genuine categorical limits in the functor category, and the objectwise forgetful functor is a limit-creating functor.

The constant diagram functor sends $C$ to the functor on $J$ with constant object value $C$ and identity arrow values; a morphism $f$ becomes the natural transformation with every component $f$. When all $J$-shaped categorical limits exist, it is a left adjoint to the limit functor.

For a topological space $S$, let $\Delta A$ be the constant presheaf of sets on its open-set category, with identity restriction maps, including the value at the empty open set. A natural transformation $P\to \Delta A$ is determined by its map $P(\varnothing )\to A$; its component at $U$ composes with $P(U)\to P(\varnothing )$. A transformation $\Delta A\to P$ is determined by $A\to P(S)$; its component at $U$ restricts sections from $S$. These two bijections prove the displayed adjunctions. This concerns presheaves, so no sheafification is involved.

For a nonempty topological space $S$, the constant diagram functor on its open-set presheaf category lies in this five-term adjunction chain. The leftmost functor sends $A$ to the categorical presheaf equal to $A$ at the empty open set and empty elsewhere. The rightmost sends $A$ to the presheaf equal to $A$ at $S$ and a singleton elsewhere. A natural transformation from the former is determined by its empty-open component; a transformation into the latter is determined by its whole-space component. The middle adjunctions follow from restriction maps to the empty open set and from the whole space. Neither endpoint extends: the leftmost functor fails to preserve the terminal object, and the rightmost fails to preserve the initial object.

A presheaf on a category $\mathcal{C}$ is a functor $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$. A morphism $f:C\to D$ induces the restriction map $X(f):X(D)\to X(C)$, reversing direction. The indexing category need not be small. For the category of open subsets of a topological space, this specializes to a presheaf of sets on a topological space.

The presheaf category on a small category $\mathcal{C}$ is the functor category $[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$. Its limits and colimits are computed pointwise.

Each object is a presheaf on a category, and its morphisms are natural transformations.

For a categorical presheaf $P$ on a small category, its category of elements has objects $(c,x)$ with $x\in P(c)$ and arrows $u:(c,x)\to (d,y)$ satisfying $P(u)y=x$. The canonical natural transformation $H_c\to P$ sends $v:a\to c$ to $P(v)x$. These maps exhibit $P$ as the displayed colimit. At $a$, a representative $(c,x,v)$ maps to $P(v)x$, with inverse $z\mapsto (a,z,1_a)$. The arrow $v:(a,P(v)x)\to (c,x)$ identifies every representative with its canonical inverse image, proving the bijection and the colimit universal property.

For a small category $\mathcal{C}$, its presheaf category is the free enlargement under small colimits. Each categorical presheaf $P$ is the colimit of the diagram sending $(C,x)\in \int P$ to the representable presheaf $yC$. The map at $(C,x)$ sends $f:A\to C$ to $P(f)x$. It is surjective at each $A$ because $z\in P(A)$ is the image of $1_A$ at $(A,z)$. If two representatives have the same image $z$, they are both identified with the representative $1_A$ at $(A,z)$ by arrows in the category of elements, proving injectivity. Thus $P\cong {\mathrm{colim}}_{(C,x)\in \int P}yC$. For a cocomplete target, a functor on $\mathcal{C}$ extends by these colimits, uniquely up to natural isomorphism among colimit-preserving extensions.

A Grothendieck topology assigns covering sieves to each object. The maximal sieve covers; pulling back a covering sieve gives a covering sieve; and a sieve locally covering along all arrows of a covering sieve itself covers. It defines the sheaves on a site. Families of arrows generate sieves and can be used to generate the topology.

The dense sieves form the double-negation Grothendieck topology. Its sheaf topos is a Boolean topos: the union of a subobject with its pseudocomplement is dense and thus becomes the whole object after sheafification. This need not be the atomic topology, which requires every nonempty sieve to be dense.

Consider a site indexed by natural-number objects with arrows only downwards, the cancellation condition $fh=gk\Rightarrow f=g$ for equally typed $f,g$, and the separation condition that arrows $f:m\to n$, $g:m\to n+1$ admit $h,k:m+1\to m$ with $fh=fk$ but $gh\ne gk$. Its dense-sieve sheaf topos is a Boolean topos and a two-valued topos. The representables are separated. No nonempty partial map $y_n\to y_{n+1}$ exists, by the separation condition and naturality. The associated sheaves $A_n$ consequently admit no maps $y_n\to A_{n+1}$. Each has nonzero support and maps epimorphically to one, but their product has an empty value at every stage $m$, because its $A_{m+1}$ factor does.

On the opposite of the category of finitely presented unital commutative rings, a basic covering family is $A\to A[f_i^{-1}]$, where finitely many $f_i$ generate the unit ideal. Include the empty cover of the zero ring. Localization commutes with base change, so these families define a coverage on a category. Their sheaf topos classifies the coherent theory of local rings. This coverage is subcanonical: representable functors obey the usual localization gluing equations.

A coverage specifies covering families on a small category, stable under refinement along arrows: for a covering family into $C$ and $f:B\to C$, there is a covering family into $B$ whose composites with $f$ factor through the original family. Passing to generated sieves and imposing maximality and local transitivity gives its associated Grothendieck topology, with the same sheaves on a site. One can equivalently specify the saturated covering sieves directly. Rigidity refers to these covering sieves, so it is unchanged by replacing a family presentation by its generated topology.

A coverage on a category is rigid when each object is covered by the sieve generated by arrows from J-irreducible objects of a site. Restriction to their full subcategory identifies the sheaves on a site with its presheaf category. A presheaf on that subcategory extends by the right Kan extension $C\mapsto \mathrm{Nat}(yC{|}_{\mathcal{D}},F)$: every covering sieve restricts to the full representable on $\mathcal{D}$, which proves the sheaf condition directly.

An object of a site is J-irreducible when its only covering sieve on a category is the maximal sieve. In particular the empty sieve cannot cover it. Every arrow from a J-irreducible object belongs to any covering sieve on its codomain: the pulled-back sieve must contain the identity. This makes the induced topology on the full subcategory of J-irreducible objects trivial.

A site is a category equipped with a Grothendieck topology, specifying which sieves are covering. The sheaves on a site satisfy unique amalgamation of matching families on those sieves. A small site presents a Grothendieck topos.

The atomic topology declares exactly the nonempty sieves covering. It exists precisely when arrows with a common codomain admit a common refinement. On nonempty finite sets and surjections this follows from the fiber product. Its J-closed sieves are only the empty and maximal sieves.

Use a small skeleton of nonempty finite sets and surjections, with every nonempty sieve covering. Fiber-product projections are surjections, so the atomic coverage exists. Surjections are effective quotients of their kernel pairs, making the site subcanonical. Its sheaves admit the primitive decomposition of an atomic finite-surjection sheaf.

An element $x\in F(n)$ is primitive if it does not descend along any surjection $n\to n-1$. All elements at cardinality one are primitive. Repeated descent reaches a primitive ancestor in finitely many steps, and the primitive-element kernel rigidity lemma makes the ancestor unique up to bijection.

Partition the elements of a sheaf by their primitive-ancestor equivalence classes under bijection. Each class gives a sheaf subfunctor, because its membership is preserved and reflected along covering surjections. A representative primitive element names an epic representable map onto its class component. Thus every sheaf is the coproduct of these components, each an atom in a topos.

If primitive elements have equal restrictions along $\alpha :P\to m$ and $\beta :P\to n$, then these surjections have equal kernels. To rule out $\alpha (a)=\alpha (b)$ with unequal $\beta$ images, identify those two images by $q:n\to n-1$. The set of pairs equal under $\alpha$ and $q\beta$ maps surjectively both to $P$ by projection and to the kernel pair of $q$ by applying $\beta$. Injective restrictions force the kernel-pair matching condition on the primitive element, contradicting descent along $q$.

For any sheaf $F$ and surjection $q:n\to k$, $F(k)\to F(n)\rightrightarrows F(n{\times }_{k}n)$ is an equalizer. In particular every $F(q)$ is injective. These follow from the sheaf condition on the sieve generated by the covering arrow $q$ and control primitive elements.

For each pair $f:V\to U$, $g:W\to U$, require $fh=gk$ for some arrows $h:T\to V$, $k:T\to W$. This is necessary and sufficient for stability of nonempty sieves under pullback; the other topology axioms then follow. Nonempty finite sets with arbitrary functions fail it at distinct singleton-to-two-point maps.

A Grothendieck topology is subcanonical when every representable functor is a sheaf on a site. It is also called standard in some topos-theory terminology. A nontrivial covering quotient can fail this condition by identifying distinct sections of a representable.

A categorical presheaf is a sheaf for $J$ if every matching family on a covering sieve has a unique amalgamation. Equivalently restriction $\mathrm{Nat}(yU,F)\to \mathrm{Nat}(S,F)$ is bijective for each covering sieve $S\hookrightarrow yU$. Sheaves on a small site form a Grothendieck topos.

A sieve on $U$ is a subfunctor of the representable functor $yU$, equivalently a family of arrows into $U$ closed under precomposition. For $f:V\to U$, the pullback sieve consists of arrows $g$ into $V$ for which $fg$ belongs to the original sieve.

A sieve $R$ on $A$ is dense when every arrow $f:B\to A$ has a further composite $fh$ in $R$. Equivalently, every pullback sieve $f^{\ast }R$ is nonempty. This condition is stable under pullback and transitive refinement and defines the double-negation topology.

A sieve is J-closed when membership is local with respect to the Grothendieck topology $J$. Pullback preserves J-closedness. Their presheaf is the subobject classifier of the sheaf topos: a section of a sheaf is sent to the sieve of arrows on which it belongs to the chosen subsheaf.

Every presheaf category is a topos. Finite limits are pointwise; the exponential isand the subobject classifier assigns to $C$ the set of sieves on $C$.

If every slice category $\mathcal{C}/A$ is essentially small, the displayed exponential object and the set of sieves on a category at each $A$ are small even when $\mathcal{C}$ is large. Together with pointwise finite limits, they give the elementary-topos constructions in a universe admitting the large diagrams. This does not force the entire functor category to be locally small in the original universe: a large discrete category has terminal slices, but maps between its constant-one and constant-two presheaves are arbitrary class-indexed binary families.

For categorical presheaves $P,Q$ on a small category, restrictions in the displayed exponential object act by precomposition with $H_u\times 1_P$. Evaluation sends $(\alpha ,x)$ at $c$ to ${\alpha }_c(1_c,x)$. The currying of $t:R\times P\to Q$ sends $z\in R(c)$ to the natural transformation whose value on $(v:a\to c,x\in P(a))$ is $t_a(R(v)z,x)$. Evaluation and currying are inverse and natural, proving the Cartesian closed category structure.

The arrow category of $\mathcal{C}$ has morphisms $A_0\to A_1$ of $\mathcal{C}$ as objects and commutative squares as morphisms. It is the functor category $[2,\mathcal{C}]$, where $2$ is the category with one nonidentity arrow.

It is also the comma category $({\mathrm{id}}_{\mathcal{C}}\downarrow {\mathrm{id}}_{\mathcal{C}})$.

Write a presheaf on $0\to 1$ as an arrow $r:B\to A$. Evaluation $\Gamma (r)=B$ has the displayed chain, where $\Pi (r)=A$, $\Delta (S)=(S\stackrel{1}{\to }S)$, $\nabla (S)=(S\to 1)$ and $\Lambda (S)=(\varnothing \to S)$. The Hom-set bijections follow by solving the commuting-square equation. The functor $\nabla$ has no right adjoint because it fails to preserve the initial object: $\nabla (\varnothing )=(\varnothing \to 1)$ is not $(\varnothing \to \varnothing )$.

The category of injective functions is the full subcategory of $[2,\mathbf{Set}]$ whose objects are injective functions. It is cartesian closed: products are computed pointwise, and exponentiating an injection by any arrow again gives an injection.

A covariant functor is representable when it is naturally isomorphic to $\mathcal{C}(A,-)$ for some object $A$; a contravariant functor is representable when it is naturally isomorphic to $\mathcal{C}(-,A)$.

Suppose a small category has equalizers. If a categorical presheaf $P$ is a retract $P\stackrel{i}{\to }{H}_B\stackrel{p}{\to }P$, with $pi=1_P$, then $i$ equalizes $ip$ and $1_{H_B}$. The Yoneda lemma writes $ip=H_b$ for an endomorphism $b:B\to B$. An equalizer $A\to B$ of $b$ and $1_B$ gives the same presheaf equalizer, since the Yoneda embedding preserves categorical limits. Uniqueness of equalizers gives $P\cong H_A$.

On a complete category that is locally small, a set-valued functor is representable exactly when it preserves small categorical limits and its category of elements has a weakly initial set. Categorical limit preservation makes that comma category complete. The initial-object lemma for complete categories with a weakly initial set then supplies a universal element, hence a representation of a functor.

If $\mathcal{C}$ has small coproducts in a category and $U\cong \mathcal{C}(R,-)$, then $F(S)={\coprod }_{s\in S}R$ is a left adjoint to $U$. Maps from this coproduct to $X$ are families of maps $R\to X$, naturally equivalent to functions $S\to UX$. Conversely, a set-valued right adjoint $U$ is represented by $F(1)$, because maps from a singleton evaluate to elements.

A representable functor $\mathcal{C}(R,-)$ preserves every small categorical limit that exists. A map from $R$ into a limit is exactly a compatible family of maps from $R$ to the diagram objects, and such a family is exactly an element of the corresponding limit of sets. The comparison bijection is the one induced by the limit projections.

A categorical presheaf is representable if its elements projection has a colimit and the presheaf preserves the opposite-indexed categorical limit of that projection. Compatibility of all distinguished elements then supplies a universal element at the colimit object. The preservation requirement must cover this diagram even when its indexing category is large.

Let $\mathcal{C}$ be the ordered category of all ordinals with an added greatest object $\infty$. The categorical presheaf with $X(\alpha )=\{\ast \}$ at ordinals and $X(\infty )=\varnothing$ preserves every small categorical limit: a small supremum is an ordinal unless its diagram contains $\infty$. Its category of elements consists of all ordinals, and their projection has the large colimit $\infty$. Nevertheless $X$ is not representable: an ordinal representer has bounded support, while the representer $\infty$ has a nonempty value at $\infty$. Thus preservation must include the possibly large opposite elements diagram. The dual functor $X^{\mathrm{op}}:\mathcal{C}\to {\mathbf{Set}}^{\mathrm{op}}$ also disproves the comma-projection criterion for a right adjoint under small-colimit preservation alone.

Every functor $F:\mathcal{C}\to \mathbf{Set}$ on a small category is the colimit of the covariant representable functors $\mathcal{C}(A,-)$ indexed by the opposite of its category of elements. The cocone indexed by $(A,x)$ sends $f:A\to B$ to $F(f)(x)$. By the Yoneda lemma, a compatible cocone into $G$ is exactly a natural transformation $F\to G$. The opposite indexing direction is essential because precomposition reverses the representing-object arrow.

A representation of $F:\mathcal{C}\to \mathbf{Set}$ is an object $A$ together with a universal element $a\in F(A)$ such thatis a bijection for every $B$, naturally in $B$. Equivalently, it is a specified natural isomorphism $\mathcal{C}(A,-)\cong F$.

Two representations of a functor have unique mutually inverse morphisms carrying their specified universal elements to one another. Composing those morphisms preserves each universal element, so injectivity of the representing bijection makes the composites identities. Arbitrary isomorphisms of the underlying objects need not preserve the specified elements.

If $H:{\mathcal{A}}^{\mathrm{op}}\times \mathcal{B}\to \mathbf{Set}$ has chosen representations ${\psi }_B:\mathcal{A}(-,GB)\cong H(-,B)$, the Yoneda lemma determines a unique functor $G$ on morphisms by $\mathcal{A}(-,Gb)={\psi }_{B^{\prime }}^{-1}H(-,b){\psi }_B$. This makes the representing natural isomorphisms natural in $B$.

The universal element in a representation of a functor is the image of $1_A$ under the representing natural isomorphism. Any two universal elements induce unique mutually inverse maps between their representing objects, so representations are unique up to unique compatible isomorphism.

The Yoneda lemma gives natural bijectionsand their contravariant analogues.

For a presheaf on a category, the categorical coend identifies $(fh,x)$ with $(h,X(f)x)$. The displayed isomorphism sends the class of $(h:U\to W,x\in X(W))$ to $X(h)x$, with inverse $y\mapsto [1_U,y]$. The coend relation makes these inverse and natural. Consequently every presheaf is the colimit of representables indexed by its category of elements.

The Yoneda embedding sends $A$ to $\mathcal{C}(-,A)$. The Yoneda lemma identifies natural transformations $yA\to yB$ with morphisms $A\to B$, so the embedding is full and faithful. It preserves every limit that exists in $\mathcal{C}$.

For $f:A\to B$, the Yoneda embedding sends $f$ to postcomposition $\mathcal{C}(-,A)\to \mathcal{C}(-,B)$. If this is epic in the presheaf category, its component at $B$ is surjective, so $1_B=fg$ for some $g:B\to A$. Conversely such a section makes the natural transformation split epic. Pointwise surjectivity follows directly by doubling the target presheaf along the image subpresheaf: the two inclusions agree on the image but differ on any element outside it.

For a small Cartesian closed category, the Yoneda embedding into its presheaf category preserves exponential objects. At $C$, the exponential in a presheaf category has value $\mathrm{Nat}(YC\times YA,YB)$. Since $YC\times YA\cong Y(C\times A)$, the Yoneda lemma identifies this with $\mathcal{C}(C\times A,B)\cong \mathcal{C}(C,B^A)$. Every identification is natural in all three objects.

The covariant version of the Yoneda embedding sends an object to its outgoing representable functor. Precomposition reverses arrows. The Yoneda lemma gives $\mathrm{Nat}(Y(A),Y(B))\cong \mathcal{C}(B,A)$, so it is full and faithful. It carries existing colimits in $\mathcal{C}$ to pointwise categorical limits.

A natural transformation in a set-valued functor category is an epimorphism exactly when every component is a surjective function. More generally, limits and colimits in a functor category are computed pointwise whenever they exist in the codomain.

For a natural transformation $a:X\to Y$ of set-valued functors, form the pushout by taking two copies of each $Y(C)$ and identifying the two copies of every element in the image of $a_C$. Naturality makes all induced maps well-defined. If $y\in Y(C)$ is outside that image, its two copies remain distinct. Thus the two maps $Y\rightrightarrows Y{⨿}_{X}Y$ agree after $a$ but differ at $y$, proving $a$ is not an epimorphism. Conversely, componentwise surjective functions allow cancellation at every object, proving that $a$ is an epimorphism. This argument does not assume a natural choice of preimages.