Extending a functor from its representables

Written by GPT-6.1 Sol (OpenAI), October 2026. Self-checked by the writing AI (GPT-6.1 Sol, Ultra). Original text public domain (CC0).

A presheaf carries its own diagram of representable pieces. When its base category is small, that entire diagram is small too. A functor on the base can therefore be extended by applying it to every piece and taking a colimit. The resulting functor preserves all small colimits, and its behavior on the original pieces determines it.

Fix a universe \(\mathcal U\), ambient choice, and a \(\mathcal U\)-small category \(C\). Smallness includes the complete category data up to the encodings in Universes and small categories. Let \(\mathcal A\) be a category whose objects and arrows form ambient sets, and which admits every \(\mathcal U\)-small colimit. No bound on its whole object set is assumed. Set \(\widehat C=\operatorname{Fun}(C^{\mathrm{op}},\mathsf{Set}_{\mathcal U})\). All representables use the complete Hom encoding from Points and representations.

Retain the whole left-Kan construction in Extending diagrams, Sections 1–3 and 5, the whole density and universe-factor arguments in Formal colimits, Sections 4–6, and the whole quotient-regrouping proof in Colimits as connected components, Section 5. We specify their actual maps in this specialization.

1. A small base makes the whole element diagram small

For \(A\in\widehat C\), write \(E_A\) for its category of elements. Its objects are \((X,a)\) with \(a\in A(X)\), and an arrow \(f:(X,a)\to(X',a')\) satisfies \(A(f)a'=a\). The full element-category proof in Points, Section 3, supplies identities, composition, the projection \(j_A:E_A\to C\), and the natural Yoneda legs \(y_{X,a}:h_X\to A\).

Here its entire object set is small: it is the tagged union of the small sets \(A(X)\) over the small object set of \(C\). The arrows are a tagged union, over the small set of object pairs, of subsets of the small Hom sets of \(C\). The complete small-union and subset proofs in Universes, Sections 2–4, bound both unions and encode the complete category. Thus \(E_A\) is \(\mathcal U\)-small, not merely locally small.

The complete density theorem gives the specified cocone

Its value factor sends a class \([(X,a),u:T\to X]\) to \(A(u)a\), with inverse \(t\mapsto[(T,t),1_T]\); retain all generator, inverse, naturality and universal-factor checks in Formal colimits, Section 4. Smallness of \(C\) now makes this particular density colimit an ordinary small-index construction.

A transformation \(v:A\to A'\) induces the functor

It keeps every underlying arrow \(f\). Naturality of \(v\) gives \(A'(f)v_{X'}a'=v_XA(f)a'=v_Xa\), so this is an element arrow. Identities, composition, \(E_{1}=1\) and \(E_{wv}=E_wE_v\) hold on each object and arrow. Yoneda gives the exact equation \(v\,y_{X,a}=y_{X,v_Xa}\).

2. Extend along the Yoneda embedding

Let \(F:C\to\mathcal A\). Define

The index is small by Section 1, so these colimits exist under the stated hypothesis on \(\mathcal A\). Yoneda identifies \(E_A\) with the incoming comma \((h\downarrow A)\), including every arrow equation, by the full natural Yoneda proof. Apply the complete left-Kan construction along \(h:C\to\widehat C\). It proves that (2.1) defines the actual functor \(L_F=\operatorname{Lan}_hF\), together with its universal transformation.

In these labels its arrow action is specified by

The family on the right is a cocone by the unchanged element arrows. The complete retained factor proof proves the unique map and both functor laws by these composites. Thus the object formula does not conceal an unproved choice of arrows.

For a transformation \(z:F\to F'\), the same construction gives

Naturality of \(z\) respects each element-arrow cocone equation. Testing (2.3) at every leg proves its naturality in \(A\), its identity law and its composition law, exactly as in the full retained Kan proof. Hence \(F\mapsto L_F\) is a functor on the ambient functor category.

Full faithfulness of Yoneda makes \((X,1_{h_X})\) terminal in \((h\downarrow h_X)\). The entire terminal-comma and inverse-factor proof in Extending diagrams, Section 5, therefore makes the specified unit

an isomorphism. Its inverse sends the leg at \((T,u:T\to X)\) to \(F(u)\). The cocone equations prove both inverse equations, and the retained unit proof proves naturality in \(X\) and \(F\). No arbitrary objectwise isomorphism replaces this specified comparison.

3. Why every small colimit is preserved

Fix \(Y\in\mathcal A\). Use a universe \(\mathcal V\) containing the total category data, the selected constructions and \(\mathcal U\). Form the encoded presheaf \(H_Y(X)=\operatorname{Hom}_{\mathcal A}(F(X),Y)\) with values in \(\mathsf{Set}_{\mathcal V}\). Its \(C\)-arrow action is precomposition by \(F(f)\); its \(Y\)-arrow action is postcomposition. The complete encoded Hom proof verifies the contravariant laws and the commuting target action. This enlargement allows arbitrary target Hom bounds.

The colimit factors of (2.1), Yoneda, and the complete density theorem in that value universe give

Here \(A\) is included in the larger-valued presheaf category. The forward map first composes with each \(\iota^A_{X,a}\). The inverse is the unique cocone factor corresponding to the compatible family. The complete density/Yoneda factors prove both inverse equations. Formula (2.2) and target postcomposition show naturality in \(A\) and \(Y\). We do not assert that \(H_Y\) has values in the original \(\mathcal U\).

Now let \(D:I\to\widehat C\) be small, with colimit \(A\) and legs \(d_i:D(i)\to A\). The complete universe-preservation theorem in Formal colimits, Section 6, preserves this specified colimit after value enlargement. Therefore

Every bijection is natural in \(Y\). By the just-checked \(A\)-naturality of (3.1), the composite sends \(b:L_F(A)\to Y\) to the maps \(bL_F(d_i)\). Bijectivity thus gives the full factor and uniqueness property of that actual cocone. This proves preservation of every small presheaf colimit with its structural maps.

Equivalently, since \(\mathcal A\) has the small colimit of \(L_FD\), its canonical map to \(L_F(A)\) is an isomorphism. The full Hom-probe inverse criterion in Points, Section 2, applies to the identified factor bijections. Empty \(I\) is included: \(L_F\) sends the empty presheaf to an initial object of \(\mathcal A\).

4. Determine an extension from its original values

Suppose \(G:\widehat C\to\mathcal A\) has a specified isomorphism \(\delta:Gh\to F\). Assume only that \(G\) preserves every small colimit of a diagram of original representables. In particular it preserves the actual density diagram (1.1), since its entire \(E_A\) is small.

Its density legs, transported through \(\delta\), make \(G(A)\) a colimit of the same \(Fj_A\) as (2.1). Define its comparison by

Naturality of \(\delta\) and the element-arrow equations make the right side a cocone. Because both displayed objects are colimits of that diagram, the complete structural-map uniqueness proof supplies an inverse commuting with every leg. Both composites are identities by the same uniqueness.

For \(v:A\to A'\), the two maps \(G(v)\theta_A\) and \(\theta_{A'}L_F(v)\) have identical composites with every \(\iota^A_{X,a}\): use \(v y_{X,a}=y_{X,v_Xa}\), then (2.2) and (4.1). Colimit uniqueness proves naturality. At \(A=h_X\), testing the terminal element gives \(\theta_{h_X}\eta^F_X=\delta_X^{-1}\). Thus the comparison respects the specified original-value isomorphism.

Any natural comparison having this property is forced by naturality at every \(y_{X,a}\); (4.1) then forces all its components. This is uniqueness with the given original-value comparison. It does not rule out different comparisons arising from different automorphisms of \(F\).

Since \(G\) is naturally isomorphic to \(L_F\), it also preserves every small presheaf colimit. Indeed its canonical colimit comparison is conjugate, via the components of \(\theta\), to the invertible canonical comparison for \(L_F\); naturality of \(\theta\) checks every original leg. Preservation for all presheaf diagrams is a conclusion here, while preservation for representable-valued diagrams was the hypothesis.

5. Restriction is an equivalence on colimit-preserving functors

Write \(\operatorname{Fun}_{\mathrm{colim}}(\widehat C,\mathcal A)\) for the full ambient subcategory consisting of functors preserving every \(\mathcal U\)-small colimit. “Full” means that every natural transformation between such functors is included. Neither its object collection nor its transformation sets are asserted to be \(\mathcal U\)-small.

Restriction along Yoneda gives

Section 3 makes \(F\mapsto L_F\) a functor into this domain, with the complete transformation action (2.3). The specified unit \(F\to h^*L_F\) is (2.4), a natural isomorphism.

For \(G\) in the domain, the Kan counit \(\varepsilon_G:L_{Gh}\to G\) has the exact density-leg formula \((\varepsilon_G)_A\iota^A_{X,a}=G(y_{X,a})\). Section 4, with \(\delta=1_{Gh}\), proves that it is invertible and natural in \(A\). For \(z:G\to G'\), naturality at each \(y_{X,a}\) and (2.3) show \(z\varepsilon_G=\varepsilon_{G'}L_{h^*z}\); the cocone factors prove the equality on every component. Thus this is a natural isomorphism in the functor \(G\) too.

The complete Kan adjunction retains both triangle equations for these specified unit and counit. Restricting its two functors to the displayed categories keeps those equations. The full adjunction equivalence proof in Passing maps, Section 3 therefore proves that (5.1) is an equivalence, with quasi-inverse \(F\mapsto L_F\).

Its full faithfulness can also be seen directly. A transformation \(t:Gh\to G'h\) extends to the unique maps \(\bar t_A:G(A)\to G'(A)\) satisfying \(\bar t_A G(y_{X,a})=G'(y_{X,a})t_X\). The source density cocone is universal, so these maps exist uniquely. Testing at the same legs proves naturality in \(A\), recovery of \(t\) on \(h_X\), preservation of identities and compositions, and uniqueness among natural extensions. Restricting and extending a given natural transformation therefore gives it back. These are the exact inverse Hom maps.

If \(C\) is empty, \(\widehat C\) is the terminal category. Its sole presheaf is its empty colimit. The left extension of the unique \(F\) is an initial object of \(\mathcal A\). A colimit-preserving functor from this terminal category must send that presheaf to an initial object, and every map between two initial objects is the unique isomorphism. Hence both sides of (5.1) are equivalent to the terminal category, with the same specified comparisons.

6. Extend a map between bases

Let \(C'\) also be small and let \(F:C\to C'\). Define

The small pointwise presheaf theorem supplies all colimits in the target. Sections 2–3 give the functor, preservation and the specified isomorphism \(\widehat F h_C\simeq h_{C'}F\). Its value at \(A\in\widehat C\), evaluated at \(V\in C'\), is

A class is represented by the triple \((X,a,u:V\to F(X))\). For \(v:V'\to V\), restriction sends it to \((X,a,uv)\). For \(z:A\to A'\), the value map sends it to \((X,z_Xa,u)\). The full quotient and Hom laws check every generator, identities, composition and the commuting two actions. These are the actual presheaf and functor maps.

Proposition 6.1. There is a natural isomorphism \(\widehat F\simeq\operatorname{Lan}_{F^{\mathrm{op}}}\) from \(\widehat C\) to \(\widehat{C'}\).

Proof. Apply the whole pointwise Kan construction in sets to \(F^{\mathrm{op}}:C^{\mathrm{op}}\to(C')^{\mathrm{op}}\). At \(V\), its incoming comma is \((F^{\mathrm{op}}\downarrow V)=(V\downarrow F)^{\mathrm{op}}\). An object is \((X,u:V\to F(X))\). An arrow in the unreversed comma is \(f:X\to X'\) with \(u'=F(f)u\); after reversal it runs from \((X',u')\) to \((X,u)\). The diagram value \(A(X)\) is covariant on this opposite comma, because the reversed arrow acts by \(A(f):A(X')\to A(X)\). Both its objects and its arrows are small by the complete small-comma proof.

Retain the entire mixed-Hom triple-regrouping proof in Colimits as connected components, Section 5, specialized to \(h_{C'}F\) and \(h_C\). Yoneda identifies its incoming comma over \(A\) with \(E_A\). Its exact inverse class maps give

In explicit labels both quotients impose the same generated relation:

for every \(f:X\to X'\), \(a'\in A(X')\) and \(u:V\to F(X)\). On the left it is the forward element arrow; on the right it is the reversed comma arrow acting by \(A(f)\). Thus the regrouping respects every relation in both directions, and its composites fix every representative.

For \(z:A\to A'\), both routes replace \(a\) by \(z_Xa\); naturality of \(z\) respects (6.4). For \(v:V'\to V\), both routes replace \(u\) by \(uv\); associativity respects (6.4). Therefore the comparison is natural in both variables and defines the stated isomorphism of functors.

At \(A=h_T\), the specified represented-value comparison sends the triple \((X,a:X\to T,u)\) to \(F(a)u:V\to F(T)\). Both regrouped forms have this same composite. Hence the comparison agrees with the structural maps on representables. Empty triple sets give empty quotients, and if an empty base occurs the existing empty-index construction gives exactly the initial presheaf. \(\square\)

7. Four graded exercises with full solutions

Exercise 1 — no original objects

Take \(C=\varnothing\). Determine its presheaf category, its Yoneda extension of \(F:C\to\mathcal A\), and the colimit-preserving functors from that presheaf category.

Solution. There is one empty presheaf and one empty natural transformation, so \(\widehat C\) is terminal. Its element category is empty. Formula (2.1) gives an initial object \(0\) of \(\mathcal A\). Every endomorphism of \(0\) is its identity, so this defines the functor on the unique presheaf arrow. For any \(Y\), there is exactly one map \(0\to Y\), as required by the empty cocone factor.

A functor from \(\widehat C\) preserves the empty colimit precisely when its value is initial. Such a functor then preserves every small colimit: a diagram of initial objects has exactly one cocone to every \(Y\), so its colimit is initial with the specified unique legs. There is exactly one natural transformation between any two of these functors, and it is invertible. Thus their full category is equivalent to the terminal category \(\operatorname{Fun}(\varnothing,\mathcal A)\).

Exercise 2 — two kinds of freely chosen pieces

Let \(C\) be discrete on \(x,y\), and let \(F(x)=P,F(y)=Q\) in sets. Compute \(L_F(A)\) for \(A(x)=S,A(y)=T\). Describe every natural transformation \(L_F\to L_{F'}\), and count them for the cardinalities

Solution. The element category is discrete on the tagged elements of \(S\) and \(T\). Its colimit is \((S\times P)\sqcup(T\times Q)\). The legs insert the corresponding copy of \(P\) or \(Q\); a cocone is a function from each such copy, and its unique factor is the function on this tagged union defined by those functions.

The representables have pairs of values \((1,\varnothing)\) and \((\varnothing,1)\). Restricting a transformation therefore gives maps \(p:P\to P'\) and \(q:Q\to Q'\). Its component at \((S,T)\) sends \((s,a)\) to \((s,p(a))\) and \((t,b)\) to \((t,q(b))\). These maps commute with every function on \(S,T\), so they are natural. Naturality at every insertion of a represented copy forces precisely these values, proving uniqueness. The number is \(3^2 2^3=72\).

For example, \(S=2,T=1,|P|=2,|Q|=3\) gives a seven-element extension value. Empty \(S\) or \(T\) removes exactly that tagged part, and both empty give the empty set.

Exercise 3 — one piece attaches to another

Let \(C=(x\xrightarrow{a}y)\), and let \(F(a)=p:P\to Q\) in sets. Take \(A(x)=\{u,v\}\), \(A(y)=\{t\}\), and \(A(a)t=u\). Compute \(L_F(A)\), its structural maps, and all cocone factors.

Solution. The element category has \((x,u),(x,v),(y,t)\), and its one nonidentity arrow runs from \((x,u)\) to \((y,t)\). The \(F\)-diagram is a copy of \(P\xrightarrow{p}Q\) and a separate copy of \(P\). Its colimit is \(Q\sqcup P\).

The leg at \((y,t)\) inserts \(Q\); the leg at \((x,u)\) is that insertion composed with \(p\); the leg at \((x,v)\) inserts the separate \(P\). A cocone to \(Z\) consists of \(b:Q\to Z\), \(c:P\to Z\), and \(d:P\to Z\) with \(d=bp\). Its unique factor \(Q\sqcup P\to Z\) uses \(b\) on \(Q\) and \(c\) on the separate \(P\). Its composites are all three stated legs, and those first two inclusions force uniqueness. No injectivity or surjectivity of \(p\) is required; the \(u\)-copy attaches through that actual map.

Exercise 4 — check the opposite comma

Let \(C\) have one object and its identity, and let \(C'=(0\to1)\). For the functor selecting \(0\), compute \(\widehat F(S)\) at both \(C'\)-objects and its restriction arrow. Repeat for the functor selecting \(1\), and verify the left-Kan comma indices.

Solution. The input presheaf is a set \(S\), and its elements index a discrete family of copies of the selected representable. For \(F(*)=0\), the Hom sets \(\operatorname{Hom}(0,0)\) and \(\operatorname{Hom}(1,0)\) are \(1\) and \(\varnothing\). Thus the values are \(S\) at \(0\) and \(\varnothing\) at \(1\), with restriction the unique map \(\varnothing\to S\).

The comma \((0\downarrow F)\) has one object and its identity, while \((1\downarrow F)\) is empty. Their opposite categories give the same singleton and empty indices; colimits of the constant value \(S\) are \(S\) and \(\varnothing\). This checks the evaluation formula and its empty boundary.

For \(F(*)=1\), both Hom sets \(\operatorname{Hom}(0,1)\) and \(\operatorname{Hom}(1,1)\) are singletons. Both values are \(S\), and precomposition with \(0\to1\) sends each labelled singleton arrow to its unique counterpart, so the restriction is \(1_S\). Both comma indices are singleton categories and give that same result. Precomposition along \(F^{\mathrm{op}}\) instead goes from presheaves on \(C'\) to sets; its direction differs from the extension just computed.

8. References and retained proof interfaces

• Emily Riehl, Category Theory in Context, Sections 6.1, 6.2 and 6.5, for Kan extensions and extension along the Yoneda embedding.
• Extending diagrams by universal maps, full Sections 1–3 and 5, supplies the left-Kan adjunction, both-variable factor naturality, small-comma proof, and specified fully faithful unit.
• Formal colimits and their test objects, full Sections 4–6, supplies the complete density factors, all empty cases, target-Hom universe enlargement and preservation of specified small presheaf colimits after enlargement. Its linked canonical density proof retains its stated GFDL licence; no canonical text is reproduced here.
• Colimits as connected components, full Sections 4–5, supplies pointwise presheaf factors and the entire opposite-comma triple-regrouping proof. Points and representations, Sections 1–3, supplies encoded Hom/Yoneda, full probes and the small element category.
• Passing maps across an adjunction, full Theorem 3.1, supplies the specified-unit/counit equivalence criterion. Universes and small categories, Sections 1–5, supplies all size encodings, component-product bounds and the enlargement axiom.