Representable Functors and Universal Elements

Intuition Beginner

Some objects in mathematics are easiest to describe not by what they are made of, but by how everything else relates to them. A product of two sets is best described by saying: to give a map into it is the same as giving a pair of maps, one to each factor. A free group is best described by saying: to give a homomorphism out of it is the same as choosing where the generators go. In both cases you pin the object down by the answers to a single repeated question, asked of every other object.

That repeated question is what a representable functor packages. Fix one object, and form the rule that sends any other object to the set of arrows between them. This rule is a translation, in the sense of the previous unit: it carries objects to sets and arrows to functions between those sets. A functor that looks exactly like one of these arrow-counting rules is called representable, and the fixed object doing the counting is the thing it represents.

Here is the everyday picture. Imagine you want to identify a person you cannot see directly. You can still pin them down by listing, for every location, the routes that reach them there. If two people have the same routes to them from everywhere, they are the same person. An object is determined by the arrows into it from everywhere, and a representable functor is the record of those arrows.

Visual Beginner

Picture a single fixed dot, the representing object, sitting at the center. Around it are all the other objects of the category. For each one, draw the bundle of arrows running from it to the center dot. The representable functor is the bookkeeping that, for every outer dot, hands you the set of arrows in its bundle.

Now one special arrow stands out: the arrow from the center dot to itself that does nothing, the identity. This loop at the center is the universal element. Every other arrow in every bundle is recovered by starting from this loop and pushing it outward. So the whole web of bundles is generated by one loop at the middle.

feature	a plain functor to sets	a representable functor
sends each object to	some set	the set of arrows to a fixed object
sends each arrow to	some function	composition with that arrow
special extra data	none required	one identity loop at the fixed object
what it captures	any assignment	the fixed object's "address book"

The picture says the central self-loop is the seed. One loop, pushed outward in every direction, regrows the entire collection of arrow-bundles. That seed is the universal element, and storing it is the same as storing the whole functor.

Worked example Beginner

Work in the category of sets, where objects are sets and arrows are functions. Fix the one-element set, written $\{\ast \}$. Form the rule that sends any set $X$ to the collection of functions from $\{\ast \}$ to $X$. A function from a single point into $X$ just picks out one element of $X$, so this rule hands back, for each set, a copy of the set's own elements.

Step 1. Take a concrete set, $X=\{a,b,c\}$. A function from $\{\ast \}$ to $X$ is a choice of where to send the single point, so there are three of them: send $\ast$ to $a$, to $b$, or to $c$. The rule outputs a three-element collection, matching $X$ itself.

Step 2. Take an arrow, a function $g:X\to Y$ with $Y=\{p,q\}$ given by $g(a)=p$, $g(b)=p$, $g(c)=q$. The functor turns $g$ into a function on the collections: a point-picking map into $X$ followed by $g$ becomes a point-picking map into $Y$. Picking $a$ then applying $g$ lands on $p$.

Step 3. Spot the universal element. The fixed object is $\{\ast \}$ itself, and the identity function on $\{\ast \}$ is the seed. Pushing it forward by any point-picking map $\{\ast \}\to X$ just gives that same map back, which is the element it names. So every element of every $X$ is the seed pushed along one arrow.

What this tells us: the rule "functions out of a point" is the same as the rule "elements of a set", and the whole forgetful, element-listing functor on sets is represented by the single point, with the identity on the point as its universal element. Pinning the functor down took one object and one arrow.

Check your understanding Beginner

Formal definition Intermediate+

Let $\mathcal{C}$ be a locally small category, so that for objects $c,d$ the arrows form a set $\mathcal{C}(c,d)$. Each object $c$ gives two hom-functors. The covariant $\mathcal{C}(c,-):\mathcal{C}\to \mathbf{Set}$ sends $d$ to $\mathcal{C}(c,d)$ and an arrow $g:d\to d^{\prime }$ to post-composition $g\circ (-)$. The contravariant $\mathcal{C}(-,c):{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$ sends $d$ to $\mathcal{C}(d,c)$ and $g:d\to d^{\prime }$ to pre-composition $(-)\circ g$ ^{[Riehl 2016]}.

Definition (representable functor). A functor $F:\mathcal{C}\to \mathbf{Set}$ is representable if there is an object $c$ of $\mathcal{C}$ and a natural isomorphism $\Phi :\mathcal{C}(c,-)\cong F$. The object $c$ is a representing object and $\Phi$ a representation. A contravariant $F:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$ is representable when $F\cong \mathcal{C}(-,c)$ for some $c$; such an $F$ is also called a functor of points of $c$. When $\mathcal{C}$ is enriched or the target is not $\mathbf{Set}$ the same definition is taken with the appropriate enriched hom; here the target is $\mathbf{Set}$.

Definition (universal element). Given a representation $\Phi :\mathcal{C}(c,-)\cong F$, the universal element of $F$ is $u={\Phi }_c({\mathrm{id}}_c)\in F(c)$, the image of the identity. For the contravariant case $u={\Phi }_c({\mathrm{id}}_c)\in F(c)$ likewise. An element $u\in F(c)$ is called universal when for every object $d$ and every element $x\in F(d)$ there is a unique arrow $f:c\to d$ (covariant case) with $F(f)(u)=x$. The phrase "$F$ is represented by $c$ with universal element $u$" packages object and element together.

The notation introduced here — $\mathcal{C}(c,-)$ and $\mathcal{C}(-,c)$ for the covariant and contravariant hom-functors, ${\int }_{\mathcal{C}}F$ for the category of elements, $u\in F(c)$ for the universal element — is recorded in _meta/NOTATION.md.

Definition (category of elements). For $F:\mathcal{C}\to \mathbf{Set}$ the category of elements ${\int }_{\mathcal{C}}F$ has objects the pairs $(d,x)$ with $x\in F(d)$; a morphism $(d,x)\to (d^{\prime },x^{\prime })$ is an arrow $f:d\to d^{\prime }$ in $\mathcal{C}$ with $F(f)(x)=x^{\prime }$. The projection ${\int }_{\mathcal{C}}F\to \mathcal{C}$ forgets the element. For a contravariant $F:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$ one takes ${\int }_{\mathcal{C}}F$ with morphisms $f:d\to d^{\prime }$ satisfying $F(f)(x^{\prime })=x$.

The element-listing functor on $\mathbf{Set}$ is $\mathbf{Set}(\{\ast \},-)$, represented by the singleton. The forgetful functor $U:\mathbf{Grp}\to \mathbf{Set}$ is represented by $\mathbb{Z}$, since a homomorphism $\mathbb{Z}\to G$ is the choice of the image of $1$, an element of $G$; its universal element is $1\in U(\mathbb{Z})$. The contravariant powerset functor $\mathcal{P}:{\mathbf{Set}}^{\mathrm{op}}\to \mathbf{Set}$, $X\mapsto \mathcal{P}(X)$ with $f\mapsto f^{-1}$, is represented by the two-element set $\Omega =\{0,1\}$: subsets of $X$ correspond to characteristic functions $X\to \Omega$, and the universal element is the subset $\{1\}\subseteq \Omega$, the subobject classifier of $\mathbf{Set}$.

Counterexamples to common slips Intermediate+

• Not every functor to sets is representable. The functor sending each set to its set of finite subsets is not representable: there is no set $c$ with finite-subset structure naturally isomorphic to maps out of $c$, because finite subsets do not compose with arbitrary functions to give a hom-functor. Representability is a real restriction.

• The representing object is data, but it is determined. A representable functor can have many representations, but any two representing objects are uniquely isomorphic (the theorem below). Saying "the" representing object is legitimate once one fixes that this is up to a unique isomorphism, not on the nose.

• Covariant and contravariant representability differ. A covariant $\mathcal{C}(c,-)$ encodes maps out of $c$ (free-object-style universal properties); a contravariant $\mathcal{C}(-,c)$ encodes maps into $c$ (product-style and limit-style universal properties). Conflating the two reverses every universal arrow.

Key theorem with proof Intermediate+

The signature result is that a representation, a universal element, and a terminal object of the category of elements are three names for one piece of data, and that the representing object is therefore unique up to a unique isomorphism.

Theorem (representability via universal elements). Let $F:\mathcal{C}\to \mathbf{Set}$ be a functor and $c$ an object of $\mathcal{C}$. The following are equivalent ^{[Riehl 2016]}:

1. a natural isomorphism $\Phi :\mathcal{C}(c,-)\cong F$;
2. an element $u\in F(c)$ that is universal, that is, for every $d$ and every $x\in F(d)$ there is a unique $f:c\to d$ with $F(f)(u)=x$;
3. the pair $(c,u)$ is an initial object of ${\int }_{\mathcal{C}}F$ (terminal, for $F$ contravariant).

Moreover any two universal elements are connected by a unique isomorphism of their underlying objects compatible with the elements, so the representing object is unique up to unique isomorphism.

Proof. (1 $\Rightarrow$ 2) Given $\Phi$, set $u={\Phi }_c({\mathrm{id}}_c)$. For $f:c\to d$, naturality of $\Phi$ at $f$ gives a commuting square: $F(f)\circ {\Phi }_c={\Phi }_d\circ \mathcal{C}(c,f)$. Evaluating at ${\mathrm{id}}_c$ and using $\mathcal{C}(c,f)({\mathrm{id}}_c)=f$ yields $F(f)(u)={\Phi }_d(f)$. Since ${\Phi }_d$ is a bijection, the assignment $f\mapsto F(f)(u)$ from $\mathcal{C}(c,d)$ to $F(d)$ is a bijection; that is precisely the statement that for each $x\in F(d)$ there is a unique $f$ with $F(f)(u)=x$. So $u$ is universal.

(2 $\Rightarrow$ 1) Given a universal $u$, define ${\Phi }_d:\mathcal{C}(c,d)\to F(d)$ by ${\Phi }_d(f)=F(f)(u)$. Universality says each ${\Phi }_d$ is a bijection. Naturality: for $g:d\to d^{\prime }$, $F(g)({\Phi }_d(f))=F(g)(F(f)(u))=F(g\circ f)(u)={\Phi }_{d^{\prime }}(g\circ f)={\Phi }_{d^{\prime }}(\mathcal{C}(c,g)(f))$, using functoriality of $F$. So $\Phi$ is a natural isomorphism.

(2 $\iff$ 3) An object of ${\int }_{\mathcal{C}}F$ is a pair $(d,x)$, and a morphism $(c,u)\to (d,x)$ is an arrow $f:c\to d$ with $F(f)(u)=x$. The existence of a unique such $f$ for every $(d,x)$ is exactly the statement that $(c,u)$ is initial in ${\int }_{\mathcal{C}}F$. This is the same clause as universality of $u$, so the two are identical.

For uniqueness, an initial object of any category is unique up to a unique isomorphism: if $(c,u)$ and $(c^{\prime },u^{\prime })$ are both initial, the unique maps between them compose to forced identities, so the underlying $\theta :c\to c^{\prime }$ is an isomorphism with $F(\theta )(u)=u^{\prime }$, and it is the only such map. $\square$

Bridge. This equivalence builds toward the entire functorial language of universal properties and appears again in 41.04.02, where the Yoneda lemma supplies the bijection $[{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}](\mathcal{C}(-,c),F)\cong F(c)$ that makes the universal element ${\Phi }_c({\mathrm{id}}_c)$ a theorem rather than a definition. The foundational reason a single element $u$ controls the whole functor is that naturality forces $F(f)(u)={\Phi }_d(f)$, so the value at every object is the seed pushed forward; this is exactly the mechanism by which a limit represents the cone functor in 41.02.01, where the universal cone is the universal element of $C\mapsto \mathrm{Cone}(C,F)$. The reformulation as an initial object generalises the uniqueness-up-to-unique-isomorphism already met for terminal objects, and putting these together shows that "$c$ represents $F$" is the same data as "${\int }_{\mathcal{C}}F$ has an initial object", which is the bridge from the abstract definition to the concrete universal constructions across mathematics catalogued in the Master tier.

Exercises Intermediate+

Advanced results Master

Theorem (universal arrows and representability). Let $U:\mathcal{D}\to \mathcal{C}$ be a functor and $c$ an object of $\mathcal{C}$. A universal arrow from $c$ to $U$ is an object $d$ of $\mathcal{D}$ together with $\eta :c\to U(d)$ such that every $f:c\to U(d^{\prime })$ factors as $U(g)\circ \eta$ for a unique $g:d\to d^{\prime }$. Such an arrow exists exactly when the functor $\mathcal{C}(c,U(-)):\mathcal{D}\to \mathbf{Set}$ is representable, with representing object $d$ and universal element $\eta$. Choosing a universal arrow for every $c$ is precisely the data of a left adjoint to $U$, with the universal elements assembling into the unit of the adjunction; this is the representability route into the theory of 41.03.01. The free group, free module, Stone-Čech compactification, and abelianization are universal arrows of this form, hence representability statements.

Theorem (limits and colimits as representations). A limit of $F:\mathcal{J}\to \mathcal{C}$ is a representing object for the contravariant cone functor $C\mapsto \mathrm{Cone}(C,F)$, with universal element the limiting cone; dually a colimit represents the covariant cocone functor. This recovers the result of 41.02.01 inside the representability language: the universal cone is a universal element, and uniqueness of the limit up to unique isomorphism is the uniqueness of the representing object proved above. Products represent the pairing functor, equalizers the agreement functor, and pullbacks the compatible-pair functor, so each finite limit shape is a separate representability statement assembled from the same theorem.

Theorem (the functor of points and moduli; the geometric reading). In algebraic geometry a scheme $S$ is determined by its functor of points $h_S=\mathbf{Sch}(-,S):{\mathbf{Sch}}^{\mathrm{op}}\to \mathbf{Set}$, and many geometric objects are defined as functors and then proved representable. The functor sending a scheme $T$ to the set of closed subschemes of ${\mathbb{P}}_T^n$ flat over $T$ with fixed Hilbert polynomial is represented by the Hilbert scheme, Grothendieck's theorem of FGA; the Picard scheme represents the relative-line-bundle functor, and Quot schemes represent quotient-sheaf functors. The universal element of the Hilbert functor is the universal flat family over the Hilbert scheme, from which every other family is pulled back uniquely. Representability is the precise content of the statement that a moduli problem "has a fine moduli space"; non-representability (the presence of automorphisms) is what forces the passage to stacks. The cross-reference 04.02.01 develops the functor-of-points philosophy on the geometric side.

Theorem (the fundamental group and classifying spaces as representations). For pointed spaces up to homotopy, the functor $X\mapsto [S^n,X{]}_{\ast }$ of based homotopy classes is corepresented by the $n$-sphere, so ${\pi }_n(X)=[S^n,X{]}_{\ast }$ exhibits homotopy groups as a corepresentability statement. Dually, for a topological group $G$ the functor sending a paracompact space $X$ to the set of isomorphism classes of principal $G$-bundles over $X$ is represented in the homotopy category by the classifying space $BG$, with universal element the universal bundle $EG\to BG$; every bundle is a pullback of $EG$ along a unique-up-to-homotopy classifying map. Singular cohomology $H^n(-;A)$ is represented by the Eilenberg-MacLane space $K(A,n)$, the Brown representability theorem characterising which contravariant homotopy functors are representable. Group cohomology $H^n(G;M)$ is the cohomology of $BG$, so the same representing space governs both.

Synthesis. Putting these together, representability is the single mechanism by which a universal property is stated: an object is named not intrinsically but as the representing object of a functor, and the universal element is the seed from which the natural isomorphism is reconstructed. The foundational reason one theorem covers products, free objects, tensor products, quotients, limits, the fundamental group, classifying spaces, and Hilbert schemes is that each is the assertion that a particular set-valued functor is representable, and the equivalence proved above turns every such assertion into the existence of an initial or terminal object in a category of elements. This is exactly the uniqueness pattern that makes "the" product or "the" free group well-defined up to unique isomorphism, and it is dual throughout: covariant representability encodes maps out and yields free constructions and colimits, contravariant representability encodes maps in and yields limits and functors of points. The central insight is that the Yoneda lemma of 41.04.02 makes the universal element ${\Phi }_c({\mathrm{id}}_c)$ a computed quantity, so representability and the embedding of a category into its presheaves are two faces of one fact, and the bridge is that adjunctions in 41.03.01 are exactly families of representations varying over the source — the unit and counit being the universal elements — which is why the whole apparatus of universal constructions reduces to this one definition.

Full proof set Master

Proposition 1 (the Yoneda bijection specialises to the universal element). For $F:\mathcal{C}\to \mathbf{Set}$ and an object $c$, the map $\mathrm{Nat}(\mathcal{C}(c,-),F)\to F(c)$, $\Phi \mapsto {\Phi }_c({\mathrm{id}}_c)$, is a bijection, and under it a natural isomorphism corresponds to a universal element.

Proof. Define the inverse sending $u\in F(c)$ to the family ${\Phi }_d^u(f)=F(f)(u)$. This is natural: for $g:d\to d^{\prime }$, $F(g)({\Phi }_d^u(f))=F(g)(F(f)(u))=F(g\circ f)(u)={\Phi }_{d^{\prime }}^u(g\circ f)$. The two assignments are mutually inverse: starting from $\Phi$, naturality at $f$ applied to ${\mathrm{id}}_c$ gives ${\Phi }_d(f)=F(f)({\Phi }_c({\mathrm{id}}_c))={\Phi }_d^u(f)$ with $u={\Phi }_c({\mathrm{id}}_c)$; starting from $u$, evaluating ${\Phi }_c^u$ at ${\mathrm{id}}_c$ returns $F({\mathrm{id}}_c)(u)=u$. So the map is a bijection. The transformation ${\Phi }^u$ is a natural isomorphism precisely when each ${\Phi }_d^u$ is a bijection, which is the definition of $u$ being universal. $\square$

Proposition 2 (uniqueness of the representing object up to unique isomorphism). If $(c,u)$ and $(c^{\prime },u^{\prime })$ both represent $F$, there is a unique isomorphism $\theta :c\to c^{\prime }$ with $F(\theta )(u)=u^{\prime }$.

Proof. Both $(c,u)$ and $(c^{\prime },u^{\prime })$ are initial objects of ${\int }_{\mathcal{C}}F$ by the key theorem. Initiality of $(c,u)$ gives a unique morphism $(c,u)\to (c^{\prime },u^{\prime })$, an arrow $\theta :c\to c^{\prime }$ with $F(\theta )(u)=u^{\prime }$; initiality of $(c^{\prime },u^{\prime })$ gives a unique ${\theta }^{\prime }:c^{\prime }\to c$ with $F({\theta }^{\prime })(u^{\prime })=u$. The composite ${\theta }^{\prime }\circ \theta$ is an endomorphism of the initial object $(c,u)$, and initiality forces a unique such endomorphism, which must be ${\mathrm{id}}_c$; symmetrically $\theta \circ {\theta }^{\prime }={\mathrm{id}}_{c^{\prime }}$. So $\theta$ is the unique isomorphism with the stated compatibility. $\square$

Proposition 3 (representable functors preserve limits). If $F:\mathcal{C}\to \mathbf{Set}$ is representable then $F$ preserves every limit that exists in $\mathcal{C}$.

Proof. Write $F\cong \mathcal{C}(c,-)$; natural isomorphisms preserve and reflect limit cones, so it suffices to treat $\mathcal{C}(c,-)$. Let $(L,\pi )$ be a limit of $D:\mathcal{J}\to \mathcal{C}$. A cone over $\mathcal{C}(c,-)\circ D$ with apex a set $S$ assigns to each $s\in S$ a compatible family $({\lambda }_j(s){)}_j$ with ${\lambda }_j(s):c\to D_j$ and $D(u)\circ {\lambda }_j(s)={\lambda }_k(s)$, that is, a cone over $D$ with apex $c$. By the universal property of $L$ each such cone factors as ${\pi }_j\circ h(s)$ for a unique $h(s):c\to L$, and $s\mapsto h(s)$ is the unique function $S\to \mathcal{C}(c,L)$ inducing the given cone. So $\mathcal{C}(c,L)$ with ${\pi }_j\circ (-)$ is the limit of $\mathcal{C}(c,-)\circ D$ in $\mathbf{Set}$, and $\mathcal{C}(c,-)$ preserves the limit. $\square$

Proposition 4 (universal arrows are representations of $\mathcal{C}(c,U-)$). Let $U:\mathcal{D}\to \mathcal{C}$ and $c\in \mathcal{C}$. A universal arrow $(d,\eta :c\to U(d))$ from $c$ to $U$ is the same data as a representation of the functor $\mathcal{C}(c,U(-)):\mathcal{D}\to \mathbf{Set}$ with universal element $\eta$.

Proof. The functor $G=\mathcal{C}(c,U(-))$ sends $d^{\prime }\in \mathcal{D}$ to $\mathcal{C}(c,U(d^{\prime }))$ and $g:d^{\prime }\to d^{\prime \prime }$ to $U(g)\circ (-)$. An element $\eta \in G(d)=\mathcal{C}(c,U(d))$ is universal for $G$ iff for every $d^{\prime }$ and every $f\in G(d^{\prime })=\mathcal{C}(c,U(d^{\prime }))$ there is a unique $g:d\to d^{\prime }$ with $G(g)(\eta )=f$, that is $U(g)\circ \eta =f$. That is verbatim the defining property of a universal arrow from $c$ to $U$. By the key theorem the universal element $\eta$ is equivalent to a representation $\mathcal{D}(d,-)\cong G$, completing the identification. $\square$

Connections Master

• The Yoneda lemma and density 41.04.02. The Yoneda lemma is the unconditional version of Proposition 1: it computes $\mathrm{Nat}(\mathcal{C}(c,-),F)\cong F(c)$ for every $F$, of which representability is the special case where the natural transformation is invertible. The co-produced unit 41.04.02 uses this to show the Yoneda embedding $y:\mathcal{C}\hookrightarrow [{\mathcal{C}}^{\mathrm{op}},\mathbf{Set}]$ is full and faithful and that every presheaf is a colimit of representables — the density theorem — so representable functors are the generators this unit previews. The universal element ${\Phi }_c({\mathrm{id}}_c)$ defined here is exactly the element the Yoneda bijection there names.

• Limits and colimits as universal cones 41.02.01. Proposition 1 of 41.02.01 already recast a limit as a representing object for the cone functor; this unit supplies the general theorem that a representation, a universal element, and an initial or terminal object of ${\int }_{\mathcal{C}}F$ coincide, so the universal cone of 41.02.01 is recovered as the universal element here. Products, equalizers, and pullbacks are the contravariant representability statements that 41.02.01 treats one shape at a time.

• Adjunctions via representability 41.03.01. The co-produced unit 41.03.01 builds adjunctions as families of universal arrows: by Proposition 4 a left adjoint to $U$ is a choice of representation of $\mathcal{C}(c,U(-))$ for every $c$, with the universal elements assembling into the unit of the adjunction. So the representability theorem of this unit is the local statement whose globalisation over the source category is an adjunction, and the unit and counit there are the universal elements catalogued here.

• The functor of points in algebraic geometry 04.02.01. The geometric unit 04.02.01 develops the functor-of-points philosophy under which a scheme is its representable presheaf $\mathbf{Sch}(-,S)$ and moduli problems are functors to be proved representable; the Hilbert and Picard schemes of the Advanced results are the load-bearing examples. The universal element on the geometric side is the universal family, and the obstruction to representability — automorphisms forcing a stack — is the geometric reading of the uniqueness clause proved here.

Historical & philosophical context Master

Representable functors crystallised in the late 1950s as the categorical formulation of universal mappings. Mac Lane's treatment of universal arrows and the comma-category description of universality in Categories for the Working Mathematician (1971; 2nd ed. 1998) ^{[Mac Lane 1998]} fixed the language in which a universal property is the representability of a set-valued functor, and the Yoneda lemma, attributed to Nobuo Yoneda from a 1954 conversation reported by Mac Lane, supplied the bijection that makes the universal element the image of the identity. Riehl's Category Theory in Context (2016) ^{[Riehl 2016]} organises representability, universal elements, and the category of elements as the through-line of its second chapter, the presentation followed here.

The functor-of-points philosophy was Grothendieck's. In the Fondements de la géométrie algébrique seminars (1957-62) ^{[Grothendieck 1960]} he reconceived a scheme as the functor it represents on the category of schemes and proved the representability of the Hilbert and Picard functors, constructing the Hilbert scheme as the fine moduli space carrying a universal flat family. This reframing — that the points of a space are the maps into it from all test objects, and that a geometric object is whatever represents a chosen moduli functor — became the organising principle of modern algebraic geometry and the entry point to algebraic stacks, where the failure of representability caused by automorphisms is taken as data rather than as an obstruction.

Bibliography Master

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}

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}

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