Adjoint functors ($L⊣R$)

An adjunction is a natural bijection $\mathcal{D}(LB,A)\cong \mathcal{C}(B,RA)$.

A functor admitting a further left adjoint must itself be a right adjoint and preserve all existing categorical limits. A functor admitting a further right adjoint must preserve all existing colimits. Failure to preserve a single terminal object, equalizer, initial object or coproduct therefore rules out the corresponding extension. In the adjoint chain for the objects of a category, connected components fail an equalizer and the indiscrete category functor fails a binary coproduct.

For small categories, the object functor $O$ has left adjoint the discrete category functor and right adjoint the indiscrete category functor. Connected components, defined by zigzags of morphisms, give a left adjoint ${\pi }_0$ to the discrete functor. There is no further left adjoint: ${\pi }_0$ fails the equalizer of two functors selecting distinct objects of the indiscrete two-object category. There is no further right adjoint: the indiscrete functor fails the coproduct of two singleton sets.

Discretization $D$ and indiscretization $J$ retain underlying sets. Every function $DX\to Y$ is continuous, and every function $X\to JY$ is continuous, giving the adjunction. Its induced monad and comonad are idempotent, since repeated discrete topology or indiscrete topology construction changes nothing.

A Boolean algebra homomorphism $B\to \mathcal{P}(A)$ is equivalent to a function assigning to every $a\in A$ an ultrafilter on $B$: test membership of $a$ in every image subset. This gives a left adjoint valued in the opposite category of sets. Its unit sends an algebra element $b$ to the set of ultrafilters containing $b$. On an infinite power-set algebra this map is not surjective, because the subset consisting of the principal ultrafilters cannot arise this way in the presence of a nonprincipal ultrafilter.

An adjunction $L⊣R$ is idempotent when its induced monad has invertible multiplication. This is equivalent to its induced comonad having invertible comultiplication. For an idempotent monad, the two maps ${\eta }_{TA}$ and $T{\eta }_A$ agree, since both invert the multiplication. If $a:TA\to A$ is an algebra, naturality then gives ${\eta }_{A}a=Ta{\eta }_{TA}=T(a{\eta }_A)=1$. Thus every algebra unit is invertible. Apply this to $R{\epsilon }_B$ to obtain the self-dual adjunction condition.

A theorem giving sufficient conditions for a functor to have an adjoint functor. The general adjoint functor theorem uses a complete category, locally small categories, preservation of categorical limits and a solution-set condition. The Special adjoint functor theorem obtains that condition from a well-powered source and a small cogenerating family. Precise hypotheses matter in both forms.

The left Kan extension of $Q:\mathcal{C}\to \mathcal{E}$ along $F:\mathcal{C}\to \mathcal{D}$ is universal among functors on $\mathcal{D}$ equipped with a transformation from $Q$ to their restriction along $F$. When it exists, it is left adjoint to precomposition. For small indexing categories and cocomplete $\mathcal{E}$, it is computed by comma-category colimits.

For small categories and a set-valued functor $Q$, each displayed colimit exists in the Category of sets. A representative $(c,u:Fc\to d,x\in Qc)$ maps under a natural transformation $\eta :Q\to GF$ to $G(u){\eta }_c(x)$. The colimit identifications make this well-defined, proving the adjunction between left Kan extension and precomposition.

For small categories $\mathcal{C},\mathcal{D}$, a functor $F:\mathcal{C}\to \mathcal{D}$ and $Q:\mathcal{C}\to \mathbf{Set}$, choose an infinite cardinal number $\kappa$ bounding the number of morphisms of $\mathcal{D}$ and the total number of elements of $Q$. Given a natural transformation $\eta :Q\to GF$, generate a subfunctor $G^{\prime }\subseteq G$ from all ${\eta }_c(x)$ by applying every outgoing $\mathcal{D}$-morphism. Each $G^{\prime }(d)$ has at most $\kappa$ elements. Label bounded values by subsets of $\kappa$; only a set of such functors and transformations exists. These factorizations give the solution-set condition for precomposition, whose left adjoint is the left Kan extension.

For a functor $U:\mathcal{C}\to \mathcal{D}$ between locally small categories, with $\mathcal{C}$ a complete category, $U$ has a left adjoint if and only if it preserves small categorical limits and satisfies the solution-set condition. The dual statement uses cocompleteness, small-colimit preservation and weakly terminal solution sets to characterize existence of a right adjoint.

A functor $F:\mathcal{X}\to \mathcal{A}$ preserving the possibly large colimits of the projections $U_A:(F\downarrow A)\to \mathcal{X}$ has a right adjoint exactly when all those colimits exist. The universal cocone makes its vertex a terminal object of $(F\downarrow A)$, hence a representation of $\mathcal{A}(F-,A)$.

For adjunctions $F_i⊣G_i$, taking mates gives a natural bijectionIf $\alpha :F_1\to F_2$, its right mate isTaking mates reverses vertical composition.

A Galois connection between posets consists of monotone maps $f:A\to B$ and $g:B\to A$ satisfying $f(a)\le b$ exactly when $a\le g(b)$; equivalently, $f$ is left adjoint to $g$ in the poset-enriched sense.

For a function $p:A\to B$, order each power set by inclusion. The inverse image map $p^{\ast }(V)=p^{-1}(V)$ has a left adjoint ${\exists }_p(U)=p(U)$ and a right adjoint ${\forall }_p(U)=\{b\in B:p^{-1}(\{b\})\subseteq U\}$. The two adjunctions follow from $p(U)\subseteq V⟺U\subseteq p^{-1}(V)$ and $p^{-1}(V)\subseteq U⟺V\subseteq {\forall }_p(U)$. Empty fibers satisfy the second condition vacuously, so points outside the image of $p$ belong to ${\forall }_p(U)$ for every $U$.

For $F⊣G$, the unit $\eta :1\to GF$ and counit $\epsilon :FG\to 1$ are the natural transformations corresponding to identity morphisms under the adjunction. They satisfy ${\epsilon }_{F}F\eta =1_F$ and $G\epsilon {\eta }_G=1_G$.

For $F⊣G$, the counit is the natural transformation $\epsilon :FG\Rightarrow 1$ corresponding to the identity of $GY$ under the adjunction at each $Y$. It participates in the triangle identities for an adjunction.

For $F⊣G$, the unit is the natural transformation $\eta :1\Rightarrow GF$ corresponding to the identity of $FX$ under the adjunction at each $X$. It participates in the triangle identities for an adjunction.

For $F⊣G$ with adjunction unit $\eta$ and adjunction counit $\epsilon$, the identities are ${\epsilon }_{FC}F{\eta }_C=1_{FC}$ and $G{\epsilon }_D{\eta }_{GD}=1_{GD}$. They make the transpose maps $f\mapsto Gf\eta$ and $g\mapsto \epsilon Fg$ inverse.

Let $F:\mathcal{C}\to \mathcal{D}$, $G:\mathcal{D}\to \mathcal{C}$, $\eta :1\Rightarrow GF$ and $\epsilon :FG\Rightarrow 1$ be natural transformations. If $G\epsilon {\eta }_G=1_G$, then $e={\epsilon }_{F}F\eta :F\Rightarrow F$ is an idempotent morphism in the functor category. Naturality implies $Ge\eta =\eta$ and $\epsilon e_G=\epsilon$; these absorption identities prove $e^2=e$. The omitted triangle measures exactly whether $e=1_F$.

The one-triangle adjunction idempotent splits if and only if $G$ has a left adjoint. From a splitting $F\stackrel{r}{\to }H\stackrel{s}{\to }F$, define the new unit $Gr\eta$ and counit $\epsilon s_G$; the absorption identities imply both triangle identities for an adjunction. Conversely, for $H⊣G$ with unit $\psi :1\Rightarrow GH$ and counit $\phi :HG\Rightarrow 1$, the splitting maps are $r={\epsilon }_{H}F\psi$ and $s={\phi }_{F}H\eta$. Their composites are $rs=1_H$ and $sr=e$.

For $F⊣G$, the right adjoint $G$ is full and faithful exactly when the counit $FG\to 1$ is an isomorphism. Dually, $F$ is full and faithful exactly when the unit $1\to GF$ is an isomorphism.

For $F⊣G⊣H$ with full and faithful $G$, write $\eta ,\epsilon$ for the first unit and counit and $\alpha ,\beta$ for the second. The two compositescoincide. Their common value is the double-adjoint comparison transformation $H\to F$.

For $F⊣G$, the left adjoint $F$ is faithful exactly when every unit component ${\eta }_A:A\to GFA$ is a monomorphism. Under the adjunction, equality after ${\eta }_A$ corresponds precisely to equality after applying $F$.

Suppose $\mathcal{C}$ is balanced and every morphism of $\mathcal{D}$ factors as a strong epimorphism followed by a monomorphism. For $F:\mathcal{C}\rightleftarrows \mathcal{D}:G$, both unit and counit are pointwise monic exactly when $F$ is full and faithful and its essential image is closed under strong quotients.

Let $\mathcal{C}=\{0<1\}$ and let $\mathcal{D}$ be the terminal category. The unique functor $F:\mathcal{C}\to \mathcal{D}$ is left adjoint to the functor selecting $1$. Every arrow in a poset is monic, so the unit and counit are pointwise monic, but $F$ is not full because the unique arrow $F1\to F0$ has no preimage $1\to 0$. The category $\mathcal{C}$ is not balanced.

For complete lattices, a monotone map $f:A\to B$ preserves arbitrary joins exactly when it has a right adjoint. In that case

The right Kan extension along $K:\mathcal{C}\to \mathcal{D}$ is the right adjoint to precomposition by $K$ between suitable functor categories.

This is the right-adjoint form of Kan extension; left Kan extension is its left-adjoint counterpart.

The colimit form of the Special adjoint functor theorem says that a colimit-preserving functor from a locally small, cocomplete, well-copowered category with a small generating family to a locally small category has a right adjoint.

A functor $G:\mathcal{C}\to \mathcal{D}$ from a locally small category that is complete and well-powered, with a small cogenerating family, to a locally small category has a left adjoint exactly when it preserves small categorical limits. For sufficiency, the cogenerator bound for comma-category solution sets produces a weakly initial set in each complete comma category $(B\downarrow G)$. The initial-object lemma for complete categories with a weakly initial set supplies its initial object, a universal arrow from an object to a functor. Necessity is limit preservation by a right adjoint.

Under the hypotheses of the limit form of the special adjoint functor theorem, intersect all subobjects of $C$ supporting $x:B\to GC$. Limit preservation gives a smallest supporting $(C_0,x_0)$. If $G(u)x_0=G(v)x_0$ for maps $u,v:C_0\to Q_i$, their equalizer supports $x_0$, so minimality makes it invertible and $u=v$. Thus the indicated map of hom-sets is injective. The evaluation embedding into cogenerator products embeds $C_0$ in a product of cogenerators indexed by the realized subsets of the fixed sets $\mathcal{D}(B,G{Q}_i)$. There are only a set of such products, a set of their subobjects, and a set of maps from $B$ into their images under $G$. These data give a weakly initial set in $(B\downarrow G)$. Using realized subsets avoids assuming maps into every cogenerator exist.

A category with finite products is cartesian closed when every product functor $-\times A$ has a right adjoint $(-{)}^A$, called exponentiation by $A$.

A category with finite limits is locally Cartesian closed when every slice category is a Cartesian closed category. Covariant set-valued functor categories on small categories have this property, using the equivalence between their slices and functor categories on categories of elements.

A monoid encoding a one-object product and exponential structure: pairing is inverse to the map $z\mapsto (\pi z,{\pi }^{\prime }z)$, and currying is inverse to $y\mapsto \epsilon ⟨y\pi ,{\pi }^{\prime }⟩$. These bijections imply pairing naturality $⟨x,y⟩z=⟨xz,yz⟩$ and currying naturality $(x⟨z\pi ,{\pi }^{\prime }⟩{)}^{\ast }=x^{\ast }z$. The element $e=({\pi }^{\prime }{)}^{\ast }$ then satisfies $ez=e$ for every $z$, hence is idempotent. Splitting $e$ supplies a terminal object and a two-object Cartesian closed category whose endomorphisms at the original object are $M$. For the original object to be nonterminal, $M$ must be nontrivial.

An object isomorphic to its own exponential supplies mutually inverse folding and unfolding maps. Evaluation defines application, and folding after currying defines abstraction. The exponential universal property gives the beta rule, while the full isomorphism gives eta conversion. A mere retraction of the function object generally suffices for beta but not automatically eta.

An exponential object $B^A$ represents maps out of a product with $A$: there are natural bijectionsThe corresponding operations are currying and uncurrying, and the identity of $B^A$ corresponds to the evaluation morphism $B^A\times A\to B$.

For a small category, this formula defines the exponential object in its covariant set-valued functor category. An arrow $c\to d$ acts by precomposition with $\mathcal{C}(d,-)\to \mathcal{C}(c,-)$. Evaluation applies a natural transformation to $(1_c,z)$; currying a map $P\times G\to H$ sends $p\in P(c)$ to the transformation $(f,z)\mapsto \theta (P(f)p,z)$. The formulas prove the universal property directly, without an adjoint existence theorem.

Currying transforms a morphism of two variables into a morphism taking values in an exponential object. The evaluation map of an exponential object recovers the original morphism. Its categorical universal property requires naturality and uniqueness, not just an underlying set bijection.

The evaluation map is the counit representing application in an exponential object. For every $Z$, composition with evaluation gives the natural bijection $\mathrm{Hom}(Z,Y^X)\cong \mathrm{Hom}(Z\times X,Y)$. Its inverse is currying. Evaluation must be a morphism in the ambient category, such as an equivariant map in a category of group actions.

For left M-sets, $B^A$ consists of equivariant maps of monoid sets $M\times A\to B$ for the diagonal action on the domain. Its action is $(m\cdot f)(w,a)=f(wm,a)$ and evaluation is $f(1,a)$. The curry of an equivariant $h:C\times A\to B$ is $\hat{h}(c)(w,a)=h(w\cdot c,a)$. This construction can have noninjective action maps even when both $A$ and $B$ are decidable.

For the free monoid on $x,y$, act on $\mathbb{N}$ by adding word length and on $\{0,1\}$ trivially. Both actions are injective. The equivariant function $f(w,n)$ indicating words of length strictly greater than $n$ ending in $x$ is nonzero, but $y\cdot f=0$. Thus the exponential of monoid sets is not decidable. Strict inequality ensures invariance when the original word is empty.

If each $m$ admits $p,q$ with $pmq=p$, then $B^A$ is decidable whenever the left M-set $B$ is. For equivariant $f,g$, equality of all values at $(1,a)$ implies equality at $(m,a)$: apply $p$ and use $f(pm,pa)=pm\cdot f(1,qa)$, then cancel the injective action of $p$ on $B$. Equality after the exponential action of $m$ then gives equal traces by cancelling its action on $B$.

A functor $\mathcal{D}\times \mathcal{C}\to \mathcal{E}$ is uniquely a functor $\mathcal{D}\to [\mathcal{C},\mathcal{E}]$: fix its first coordinate and use its first-coordinate arrows as natural transformations. Uncurrying evaluates those transformations and second-coordinate arrows. The adjunction unit inserts a fixed first coordinate, and the adjunction counit is evaluation. This makes the category of small categories cartesian closed.

A full subcategory $\mathcal{D}$ of a Cartesian closed category $\mathcal{C}$ is an exponential ideal when $B^A$ belongs to $\mathcal{D}$ for every $B\in \mathcal{D}$ and every $A\in \mathcal{C}$.

For a reflective subcategory $\mathcal{D}$ of a Cartesian closed category $\mathcal{C}$, with reflector $L$, the subcategory $\mathcal{D}$ is an exponential ideal if and only if the canonical comparisonis an isomorphism for every $A,B$. The proof repeatedly curries a map into an object of $\mathcal{D}$ and factors it through the unit of the reflection.

An object $X$ in a category with finite products is exponentiable when $-\times X$ has a right adjoint $[X,-]$. The terminal object is exponentiable, and exponentiable objects are closed under binary products because right adjoints compose:

In a complete category that is a locally small category with a coseparator $S$ and all monomorphisms regular, an object $A$ is an exponentiable object exactly when $\mathcal{C}(-\times A,S)$ is a representable functor. A representing object $E$ gives representations $E^I$ for maps into powers $S^I$. An equalizer presentation by powers of a coseparator then constructs a representation for maps into every target. The Yoneda lemma makes these representations functorial and supplies the right adjoint.

If $A$ and $B$ are exponentiable objects, thenhas the composite of their exponential functors as a right adjoint. The terminal object supplies the empty product, so exponentiable objects are closed under finite products.

If the terminal object is also initial and $E$ is exponentiable, the left adjoint $-\times E$ preserves the initial object. But $0\times E\cong E$ because $0$ is terminal, while preservation of initiality gives $0\times E\cong 0$. Hence $E\cong 0$.

A $T_0$ space $E$ is exponentiable in ${\mathbf{Top}}_0$ if and only if the functor ${\mathbf{Top}}_0(-\times E,S)$ to sets is representable, where $S$ is the Sierpiński space. Every $T_0$ space is an equalizer of maps between powers of $S$, so a representing object for maps into $S$ constructs exponentials for every target by products and equalizers.

The exponential $B^A$ in the category of posets is the poset of monotone maps $A\to B$ ordered pointwise. Evaluation is monotone, and currying gives the cartesian-closed adjunction.

An object $A$ of a Cartesian closed category is tiny when exponentiation $(-{)}^A$ itself has a right adjoint.

The extra adjunction defining a tiny object should not be confused with the filtered-colimit condition for a compact object (mathematics).

Let $\mathcal{C}$ be a small Cauchy-complete category with initial object $0$. The terminal object of $[\mathcal{C},\mathbf{Set}]$ is $h_0$, so $\mathrm{Nat}(P,F)\cong (F^P)(0)$. If $P$ is tiny, exponentiation by $P$ is itself a left adjoint and preserves all colimits. Evaluation at zero also preserves colimits. Consequently $\mathrm{Nat}(P,-)$ preserves coproducts and epimorphisms, since every epimorphism in a set-valued functor category is the coequalizer of its kernel pair. Thus $P$ is an irreducible projective in a set-valued functor category, hence representable.

For a small category $\mathcal{C}$ with binary coproducts, exponentiation by $h_A=\mathcal{C}(A,-)$ in $[\mathcal{C},\mathbf{Set}]$ is naturally $F\mapsto F(-⨿A)$: the exponential at $B$ is $\mathrm{Nat}(h_B\times h_A,F)\cong \mathrm{Nat}(h_{B⨿A},F)\cong F(B⨿A)$ by the Yoneda lemma. This precomposition functor has a right adjoint given by Right Kan extension, so $h_A$ is a tiny object.

In a Cartesian closed category, exponentiation satisfies $(-{)}^{A\times B}\cong ((-{)}^B{)}^A$. If the two exponential functors have right adjoints, their composite has the composite of those right adjoints in reverse order. Therefore a product of tiny objects is tiny. The terminal object is tiny since exponentiation by it is the identity, covering the empty product.

If a small finite-product category $\mathcal{C}$ splits idempotents, the tiny objects of its presheaf topos are exactly the representable presheaves. Exponentiation by $yA$ is precomposition with $-\times A$ and therefore has a right adjoint given by Right Kan extension. Conversely, if $P$ is tiny, then $\mathrm{Nat}(P,-)$ preserves all colimits; idempotent completeness makes every such presheaf representable.