Points, fibres and universal representations

Written and self-checked with GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Original text: CC0. No independent review.

A universal object collects a whole family of constructions into one object. Its maps to or from a test object describe the family, and its identity map supplies the universal datum. The same viewpoint explains why an object is determined by all its incoming maps, how representing objects vary in a diagram, and how a presheaf over another presheaf can be reconstructed from its fibres.

We assume the size conventions of Universes and small categories, the functor and transformation constructions of Natural transformations and composition of functors, and the chosen comparisons of Equivalences and chosen representatives. For the module example we use Products, disjoint unions and mixed functors. Basic references are [Yoneda], [Riehl], [Stacks] and [Schapira, Homological Algebra]. The preceding lessons supply the complete internal proof interfaces; Section 2 supplies their two-variable evaluation compatibility, and Section 6 proves both fibre reconstructions.

1. Size and the two directions of points

Fix a universe \(\mathcal U\). Write \(\mathsf{Set}_{\mathcal U}\) for the category whose objects are sets belonging to \(\mathcal U\). A category \(C\) is locally \(\mathcal U\)-small when every \(\operatorname{Hom}_C(X,Y)\) is bijective to a member of \(\mathcal U\). It is \(\mathcal U\)-small when its object set also has that size bound. The actual object and arrow labels need not belong to \(\mathcal U\).

All category data here are sets in ambient set theory. When forming categories of functors, choose a larger universe containing the data in question and \(\mathcal U\). This uses the assumed availability of sufficiently large universes; it does not assert that one fixed universe contains every category. A category can be small in that larger universe while failing to be locally small in \(\mathcal U\).

The category of presheaves and its dual version are

\[ \begin{gathered} P_{\mathcal U}(C) =\operatorname{Fun}(C^{\mathrm{op}},\mathsf{Set}_{\mathcal U}),\\ Q_{\mathcal U}(C) =\operatorname{Fun}(C^{\mathrm{op}},\mathsf{Set}_{\mathcal U}^{\mathrm{op}})\\ =\operatorname{Fun}(C,\mathsf{Set}_{\mathcal U})^{\mathrm{op}}. \end{gathered} \tag{1.1} \]

In the last identification, reverse both the domain arrows and the target arrows. A transformation in \(Q_{\mathcal U}(C)\) reverses its component functions, so it becomes a transformation in the opposite of the covariant functor category. Identities and composites are the pointwise ones from the complete functor-category construction in the preceding lesson. In particular \(Q_{\mathcal U}(C)=P_{\mathcal U}(C^{\mathrm{op}})^{\mathrm{op}}\).

For literal Hom sets belonging to \(\mathcal U\), the two point functors are

\[ \begin{gathered} h_X(T)=\operatorname{Hom}_C(T,X), \\ q_X(T)=\operatorname{Hom}_C(X,T). \end{gathered} \tag{1.2} \]

The first is contravariant in \(T\); the second is covariant. For \(u:X\to Y\), postcomposition defines \(h(u):h_X\to h_Y\). Precomposition defines \(q(u):q_Y\to q_X\). Thus \(X\mapsto q_X\), regarded in the opposite functor category, defines \(k:C\to Q_{\mathcal U}(C)\). The arrow \(k(u)\) goes from \(kX\) to \(kY\) in \(Q\); its underlying transformation goes from \(q_Y\) to \(q_X\).

For a merely locally small \(C\), use the complete Hom encoding from Products, disjoint unions and mixed functors, Section 4. Choose bijections

\[ \begin{gathered} e_{X,Y}:\operatorname{Hom}_C(X,Y) \longrightarrow H_{X,Y}, \\ H_{X,Y}\in\mathcal U. \end{gathered} \tag{1.3} \]

Set \(h_X(T)=H_{T,X}\) and \(q_X(T)=H_{X,T}\). Decode an arrow, compose in \(C\), and encode the result. For example the encoded identity is \(e_{X,X}(1_X)\), and postcomposition by \(u:X\to Y\) sends \(v\in H_{T,X}\) to \(e_{T,Y}(u e_{T,X}^{-1}(v))\). The complete cancellation proof in that lesson proves both functor laws, mixed naturality and the natural comparisons between different encodings. Restricting this Hom bifunctor gives precisely \(h\) and \(k\) above. We use this convention whenever literal membership is unavailable.

If \(C\) is \(\mathcal U\)-small, both categories in (1.1) are locally \(\mathcal U\)-small by Universes and small categories, Proposition 5.1. This is the complete component-product proof, including arbitrary labels and the empty domain. It does not bound the number of presheaves.

Without a small object set, even local smallness can fail. Take the discrete category on the ambient set \(\mathcal U\), with singleton Hom sets on each object. A transformation between its two constant two-element presheaves is a choice of one of four functions at every object. Its transformation set contains a copy of \(\mathcal P(\mathcal U)\), by choosing the identity or the transposition at each object. This power set is not \(\mathcal U\)-small: a bijection to \(S\in\mathcal U\), followed by \(S\subseteq\mathcal U\), would inject \(\mathcal P(\mathcal U)\) into \(\mathcal U\), contradicting Cantor's theorem. Hence the transformation set is not \(\mathcal U\)-small. The same example applies to \(Q\) by taking opposites.

2. Evaluation at an identity

Theorem 2.1 (Yoneda). For a presheaf \(A\), a covariant set functor \(B\), and \(X\in C\), there are bijections

\[ \begin{gathered} \operatorname{Nat}(h_X,A)\simeq A(X),\\ \operatorname{Nat}(q_X,B)\simeq B(X). \end{gathered} \tag{2.1} \]

They are natural in both variables. The second transformation set is \(\operatorname{Hom}_{Q}(B,kX)\), when \(B\) is regarded as an object of \(Q\). Evaluation uses the identity at \(X\).

Proof. Retain the complete identity-evaluation bijection in Internal groups, Section 2. That proof constructs the inverse by \(a\mapsto(f\mapsto A(f)a)\), checks its naturality in the test object using \(A(ft)=A(t)A(f)\), and checks both inverse equations by the identity and the naturality square at \(f\). It applies to every set-valued presheaf, before the group operations in that section. Applying this complete construction to \(C^{\mathrm{op}}\) gives the covariant assertion: its inverse sends \(b\) to \(f\mapsto B(f)b\).

Here are the additional two-variable compatibility checks. For \(u:X\to Y\) and \(\alpha:h_Y\to A\), evaluate the composite \(\alpha h(u)\) at the identity of \(X\). Naturality of \(\alpha\) at \(u\) gives

\[ \begin{aligned} (\alpha h(u))_X(1_X) &=\alpha_X(u)\\ &=A(u)\bigl(\alpha_Y(1_Y)\bigr). \end{aligned} \tag{2.2} \]

For \(\delta:A\to A'\), evaluation of \(\delta\alpha\) at \(1_X\) is \(\delta_X(\alpha_X(1_X))\). These are precisely the squares for the presheaf variable and for the object variable. Their commuting composites prove naturality on the product category.

For the covariant assertion, precomposition gives \(q(u):q_Y\to q_X\). If \(\beta:q_X\to B\), its naturality gives

\[ \begin{aligned} (\beta q(u))_Y(1_Y) &=\beta_Y(u)\\ &=B(u)\bigl(\beta_X(1_X)\bigr). \end{aligned} \tag{2.3} \]

Postcomposition by \(\delta:B\to B'\) is evaluated as \(\delta_X(\beta_X(1_X))\). Thus both covariant-variable squares commute too. The already proved inverses are natural: apply the bijection on both ends of each square to its equality.

For the encoded version, evaluate \(\alpha_X\) at \(e_{X,X}(1_X)\), and send \(a\) to the transformation taking \(v\in H_{T,X}\) to \(A(e_{T,X}^{-1}(v))(a)\). Decoding and encoding cancel in each identity, composite and naturality equation just used. The complete natural Hom comparison from Section 1 identifies these maps with the proved bijection. The covariant version uses \(H_{X,T}\) with the same cancellation. This supplies the entire proof at the stated size convention. \(\square\)

The first naturality variable is contravariant for \(\operatorname{Nat}(h_X,A)\), and covariant for \(\operatorname{Nat}(q_X,B)\). In the second variable it is covariant for \(A\); for the second assertion, viewing \(B\) in \(Q\) makes it contravariant in \(B\). These are the bifunctor variances \(C^{\mathrm{op}}\times P\) and \(Q^{\mathrm{op}}\times C\).

The bijections in (2.1) are between ambient sets. Their left sides are \(\mathcal U\)-small, even when general transformation sets in \(P\) are not. They need not literally belong to \(\mathcal U\); the complete Hom transport in Section 1 supplies an encoding if a literal \(\mathsf{Set}_{\mathcal U}\)-valued bifunctor is required.

Corollary 2.2. Both \(h:C\to P\) and \(k:C\to Q\) are fully faithful. A morphism \(u:X\to Y\) is invertible if all functions \(\operatorname{Hom}(T,u)\) are bijective, or if all functions \(\operatorname{Hom}(u,T)\) are bijective.

Proof. Set \(A=h_Y\) in (2.1). Evaluation sends postcomposition by \(u\) to \(u\), or to its code, so the actual Hom map of \(h\) is bijective. For \(k\), apply the covariant assertion to \(B=q_X\) at \(Y\); evaluation identifies \(\operatorname{Nat}(q_Y,q_X)\) with \(\operatorname{Hom}_C(X,Y)\), and the actual Hom map is precomposition by that arrow. Under either probe hypothesis, the corresponding transformation has invertible components. Use the complete component-inverse proof in Natural transformations, Lemma 2.1 and the complete reflection proof in Relations and cancellation, Proposition 3.2. They imply that \(u\) is invertible. \(\square\)

When an object is identified with its point functor, the notation \(X(T)\) means \(\operatorname{Hom}_C(T,X)\), with the encoding understood. The notation \(A(B)\) for presheaves means \(\operatorname{Nat}(B,A)\). For arbitrary \(B\) this is not necessarily a small evaluation set.

3. Elements and universal data

For \(A:C^{\mathrm{op}}\to\mathsf{Set}_{\mathcal U}\), let \(E_A\) have objects \((X,s)\), with \(s\in A(X)\), and arrows

\[ \begin{gathered} u:(X,s)\longrightarrow(Y,t),\\ u:X\to Y,\qquad A(u)(t)=s. \end{gathered} \tag{3.1} \]

Identities satisfy the condition because \(A(1_X)=1\). If \(A(u)(t)=s\) and \(A(v)(w)=t\), then \(A(vu)(w)=A(u)A(v)(w)=s\). Composition is therefore inherited from \(C\), as are associativity and both units. Each Hom is a subset of the corresponding Hom in \(C\), so \(E_A\) is locally small.

For \(F:C\to D\) and a presheaf \(A\) on \(D\), the pulled-back element category has objects \((X,s\in A(FX))\) and condition \(A(Fu)(t)=s\). Yoneda identifies these objects with transformations \(h_{FX}\to A\), and its naturality identifies the arrows with the comma-category triangles. For a covariant \(B:D\to\mathsf{Set}_{\mathcal U}\), use \(B(Fu)(s)=t\). This is the same dual comma construction, because \(\operatorname{Hom}_Q(B,kFX)=B(FX)\). The covariant functor laws give its category laws in the same way.

If \(C\) is \(\mathcal U\)-small, both pulled-back categories are \(\mathcal U\)-small. Their objects form the tagged union of the small sets \(A(FX)\), or \(B(FX)\), over the small object set of \(C\). The complete small-family union proof in Universes and small categories, Proposition 3.1 applies even when the object labels themselves are outside \(\mathcal U\). Their arrows form a tagged union, over the small family of object pairs, of subsets of the small Hom sets of \(C\). The same closure proof bounds that union. These statements bound the whole category data up to bijections, rather than asserting literal membership for its original labels.

A representation of \(A\) is an object \(T\) and a specified natural isomorphism \(\rho:h_T\to A\). Its universal element is \(a=\rho_T(1_T)\), with the encoded identity used when necessary. The corresponding functions are

\[ \begin{gathered} \operatorname{Hom}_C(Y,T)\longrightarrow A(Y),\\ f\longmapsto A(f)(a). \end{gathered} \tag{3.2} \]

The arrow goes from the test object \(Y\) to \(T\). For a covariant representation \(q_T\to B\), it goes from \(T\) to \(Y\), and the formula is \(f\mapsto B(f)(b)\).

Proposition 3.1. A presheaf is representable exactly when its element category has a terminal object. A covariant set functor is representable exactly when its covariant element category has an initial object.

Proof interface. Retain the complete universal-element and terminal-object argument in Solution sets and universal representations, Section 1. Its Hom condition is exactly (3.1), and its representation map is (3.2). The target of that map is \(A(Y)\). Applying the proved statement to \(C^{\mathrm{op}}\) gives the covariant initial-object assertion. Naturality and invertibility of the component maps are included in that proof and the component criterion of Section 2. \(\square\)

Two representations \((T,\rho)\), \((T',\rho')\) have a unique isomorphism \(v:T\to T'\) with \(\rho'h(v)=\rho\). Indeed full faithfulness lifts \((\rho')^{-1}\rho\) uniquely, and the proved reflection of isomorphisms makes its lift invertible. This uniqueness holds with both representation maps fixed. The bare object \(T\) may have many automorphisms.

4. Representatives in a diagram

Theorem 4.1. Suppose \(K:J\to P_{\mathcal U}(C)\) takes every object to a representable presheaf. There are a functor \(T:J\to C\) and a natural isomorphism \(\rho:hT\to K\). A second pair \((T',\rho')\) has a unique natural isomorphism \(v:T\to T'\) with \(\rho'(hv)=\rho\).

Proof interface and construction. Choose \((Tj,\rho_j:h_{Tj}\to Kj)\) for each ambient object of \(J\). The full subcategory of \(P\) on the objects of the form \(h_X\) is the full image of \(h\). Corollary 2.2 makes \(h\) an equivalence with this image by the complete equivalence criterion in Equivalences and chosen representatives, Theorem 1.1. The complete chosen-object transport of its Proposition 3.1, followed by a quasi-inverse of this equivalence, provides the asserted diagram and comparison.

Equivalently, the arrow determined by these chosen identifications is the unique lift satisfying

\[ \begin{gathered} h(Tu)=\rho_{j'}^{-1}K(u)\rho_j,\\ u:j\to j'. \end{gathered} \tag{4.1} \]

This is exactly the conjugated-arrow rule in the proved transport construction. Its identity and composition checks, transferred through the faithful \(h\), give the two functor laws. Equation (4.1) is naturality of \(\rho\). For another pair, the transformation \((\rho')^{-1}\rho:hT\to hT'\) has a unique natural lift by the complete postcomposition-lifting proof in Natural transformations, Theorem 2.2. Its inverse lifts too, so the lift is a natural isomorphism. Faithfulness gives its stated uniqueness. The empty \(J\) has empty choices and the same assertions. \(\square\)

For \(F:C\to D\), precomposition defines \(F^*:P(D)\to P(C)\), with \((F^*A)(X)=A(FX)\) and transformation component \((F^*\alpha)_X=\alpha_{FX}\). The complete precomposition proof in Natural transformations, Section 2 verifies all laws. Restricting along \(h_D\) gives a diagram \(K:D\to P(C)\) whose values are

\[ K(Y)(X)=\operatorname{Hom}_D(FX,Y). \tag{4.2} \]

Use the Hom encoding of Section 1 for its literal presheaf values. Its action in \(Y\) is postcomposition. If every \(K(Y)\) is representable, Theorem 4.1 gives \(G:D\to C\) and bijections

\[ \begin{gathered} \operatorname{Hom}_D(FX,Y) \\ \simeq\operatorname{Hom}_C(X,GY). \end{gathered} \tag{4.3} \]

They are natural in \(X\) by each representation, and in \(Y\) by the diagram comparison. Thus \(G\) is a right adjoint to \(F\). Given two identifications \(\theta_i:K\to hG_i\), the unique compatible comparison satisfies \(\theta_1=(h\theta)\theta_0\). There is no assertion that every natural isomorphism between the bare functors is this comparison.

5. Universal objects and algebraic operations

Let \(R\) be any unital ring, \(N\) a right \(R\)-module, and \(M\) a left \(R\)-module. For an abelian group \(L\), let \(\mathsf{Bal}_R(N,M;L)\) be the set of maps \(c:N\times M\to L\) additive in each variable and satisfying \(c(nr,m)=c(n,rm)\). Postcomposition by a group homomorphism preserves all three conditions. It preserves identities and composites because function composition does, so these sets form a covariant functor of \(L\).

The complete arbitrary-ring construction in Products, disjoint unions and mixed functors, Section 5 gives the natural representation

\[ \begin{gathered} \operatorname{Hom}_{\mathsf{Ab}}(N\otimes_R M,L)\\ \simeq\mathsf{Bal}_R(N,M;L),\\ t\longmapsto ((n,m)\mapsto t(n\otimes m)). \end{gathered} \tag{5.1} \]

Its proof specializes the complete central-algebra tensor construction to \(\mathbb Z\), so it applies to noncommutative \(R\) and arbitrary modules. The representing object is initial in the covariant element category: its universal element is the balanced map \((n,m)\mapsto n\otimes m\). Balance is essential; the assertion does not concern all unbalanced biadditive maps.

There is also a precise distinction for algebraic operations. If a contravariant group-valued functor has underlying presheaf represented by \(T\), transport its operations through the specified representation. The complete transport proof in Internal groups, Section 2 gives a group object of \(C\). When finite products exist, it supplies the multiplication, identity and inverse morphisms, with every group law.

For a covariant group-valued functor represented by \(T\), apply that same complete result to \(C^{\mathrm{op}}\). It gives a group object of \(C^{\mathrm{op}}\), called a cogroup object of \(C\). If \(C\) has finite coproducts, its structure maps have directions \(T\to T\sqcup T\), \(T\to0\), and \(T\to T\); the laws are the opposite group-object diagrams.

The distinction can be seen in sets. The covariant functor \(\operatorname{Hom}(\varnothing,-)\) is the constant singleton functor, carrying its unique group structure naturally. Its representative \(\varnothing\) is a cogroup object of sets. It cannot be a group object of sets: there is no identity map from a singleton to the empty set, and its incoming maps from a singleton form the empty set, which cannot be a group.

Remark. If the functor represented by an object in the covariant direction takes values in groups, the object is a cogroup object, not a group object; the empty-set example shows the difference. The contravariant conclusion is the group-object statement proved above.

6. Reconstructing a presheaf from its fibres

Fix \(A\in P_{\mathcal U}(C)\). An object of \(P_{\mathcal U}(C)/A\) is a transformation \(t:G\to A\). An arrow \(r:(G,t)\to(G',t')\) is a transformation \(r:G\to G'\) satisfying \(t'r=t\). These triangles are closed under identities and composition by their equations.

Theorem 6.1. There is an equivalence

\[ P_{\mathcal U}(C)/A \simeq P_{\mathcal U}(E_A). \tag{6.1} \]

It sends a presheaf over \(A\) to its fibres, and reconstructs it by summing those fibres. The assertion allows any ambient locally \(\mathcal U\)-small \(C\).

Fibre functor. Define

\[ \begin{gathered} \Lambda(G,t)(X,s) \\ =\{v\in G(X):t_X(v)=s\}. \end{gathered} \tag{6.2} \]

For \(u:(X,s)\to(Y,a)\), its contravariant restriction sends \(v\) to \(G(u)(v)\). This is in the required fibre because

\[ \begin{aligned} t_X(G(u)v)&=A(u)t_Y(v)\\ &=A(u)(a)=s. \end{aligned} \tag{6.3} \]

The identity and composition laws are those of \(G\), restricted to these subsets. For an arrow \(r\) over \(A\), define \((\Lambda r)_{(X,s)}(v)=r_X(v)\). It lies in the same fibre since \(t'_Xr_X=t_X\). Naturality is the restriction of \(r_XG(u)=G'(u)r_Y\); pointwise identity and composite equations prove that \(\Lambda\) is a functor. Every fibre belongs to \(\mathcal U\), being a subset of the universe member \(G(X)\).

Sum functor. For \(H:E_A^{\mathrm{op}}\to\mathsf{Set}_{\mathcal U}\), set

\[ \begin{gathered} (\Sigma H)(X)= \coprod_{s\in A(X)} H(X,s),\\ p_X(s,v)=s. \end{gathered} \tag{6.4} \]

For \(u:X\to Y\), the element-category arrow \(u_a:(X,A(u)a)\to(Y,a)\) defines

\[ \begin{gathered} (\Sigma H)(u)(a,v) \\ =\bigl(A(u)a,H(u_a)(v)\bigr). \end{gathered} \tag{6.5} \]

For \(u=1_X\), the element-category arrow is an identity, so this is the identity function. For \(X\xrightarrow{u}Y\xrightarrow{w}Z\) and \((b,v)\) at \(Z\), the first coordinate after the two restrictions is \(A(u)A(w)b=A(wu)b\). The relevant element arrows compose to \((wu)_b=w_bu_{A(w)b}\). Contravariance of \(H\) makes the second coordinate \(H(u_{A(w)b})H(w_b)v=H((wu)_b)v\). Thus (6.5) satisfies the composition law. Also \(p_X(\Sigma H)(u)=A(u)p_Y\), proving that \(p:\Sigma H\to A\) is natural.

For \(\gamma:H\to H'\), send \((s,v)\) to \((s,\gamma_{(X,s)}v)\). This preserves the projection. Naturality follows from \(\gamma_{(X,A(u)a)}H(u_a)=H'(u_a)\gamma_{(Y,a)}\). Identity and composite equations hold in each labelled summand. This makes \(\Sigma\) a functor into the slice. The sum belongs to \(\mathcal U\): its index \(A(X)\) and all its summands belong to \(\mathcal U\), so the complete universe-member indexed-sum closure applies. No smallness assumption on the object set of \(C\) entered this argument.

The two comparisons. For \((G,t)\), define

\[ \begin{gathered} \epsilon_X:(\Sigma\Lambda G)(X)\\ \longrightarrow G(X),\\ (s,v)\longmapsto v, \\ v\longmapsto(t_X(v),v) \\ \quad\text{for its inverse}. \end{gathered} \tag{6.6} \]

Both composites are identities because the labelled fibre already satisfies \(t_X(v)=s\). The map is over \(A\). Restriction takes \((s,v)\) to \((A(u)s,G(u)v)\), so (6.6) commutes with restriction. An arrow \(r\) over \(A\) takes that pair to \((s,r_Xv)\); forgetting the label before or after it gives \(r_Xv\). Thus \(\epsilon\) is natural in the slice object as well as in \(X\).

For \(H\), the fibre of \(\Sigma H(X)\) over \(s\) consists exactly of the pairs \((s,v)\) with \(v\in H(X,s)\). The maps

\[ \begin{gathered} \eta_{(X,s)}:H(X,s)\longrightarrow \\(\Lambda\Sigma H)(X,s), \\ v\longmapsto(s,v) \end{gathered} \tag{6.7} \]

and \((s,v)\mapsto v\) are inverse. Restriction along an element arrow sends the pair to the pair with its forced source label and restricted second coordinate, so it commutes with \(\eta\). A transformation \(\gamma\) changes only the second coordinate, proving naturality in \(H\). Equations (6.6) and (6.7) give both natural quasi-inverse comparisons, proving (6.1). Empty fibres and the empty base category are included. \(\square\)

This fibre construction is the presheaf specialization of Stacks, localized topos site. Its stated inverse uses these same fibres; the complete restricted actions and two inverse comparisons needed here are given above.

The equivalence also respects the point functors. An element \(s\in A(X)\) determines \(\alpha_s:h_X\to A\) by Yoneda. For \((Y,a)\in E_A\), its fibre is

\[ \begin{gathered} \Lambda(h_X,\alpha_s)(Y,a)\\ \simeq \{f:Y\to X:A(f)(s)=a\}\\ =\operatorname{Hom}_{E_A}((Y,a),(X,s)). \end{gathered} \tag{6.8} \]

The bijection decodes the Hom value. Restriction along an element arrow is precomposition on both sides. A target arrow \(u:(X,s)\to(X',s')\) has \(A(u)s'=s\), so \(\alpha_{s'}h(u)=\alpha_s\); on (6.8) it acts by postcomposition on both sides. Thus (6.8) is natural in both element-category variables. With the compatible Hom encoding inherited from \(C\), the fibre itself is an encoded Hom value, so it belongs to \(\mathcal U\). Consequently \(\Lambda\) carries the diagram \((X,s)\mapsto(h_X,\alpha_s)\) to the Yoneda diagram of \(E_A\), up to this specified natural isomorphism.

There is a useful compatibility when the base changes. A map \(a:A\to B\) gives \(E_a:E_A\to E_B\), \((X,s)\mapsto(X,a_Xs)\), keeping the underlying arrows; their element condition follows from naturality of \(a\). For \(G\xrightarrow{t}B\), form \(G\times_B A\to A\) pointwise. At \((X,s)\), its fibre is naturally the fibre of \(G(X)\) over \(a_Xs\). The comparison forgets the already fixed second coordinate of a pair \((v,s)\), with inverse \(v\mapsto(v,s)\). Restrictions act on both coordinates, and maps of \(G\) act on the first; these formulas prove naturality and identify this base change with precomposition by \(E_a\).

7. Enlarging the universe and changing the base

Let \(\mathcal U\subseteq\mathcal V\). Inclusion \(\mathsf{Set}_{\mathcal U}\to\mathsf{Set}_{\mathcal V}\) is fully faithful: its functions between fixed sets are exactly the same functions. Postcomposition therefore gives a fully faithful \(P_{\mathcal U}(C)\to P_{\mathcal V}(C)\), by the complete postcomposition proof in Natural transformations, Theorem 2.2.

A \(\mathcal U\)-valued presheaf is representable before enlargement exactly when it is representable afterwards. Keep the \(\mathcal U\)-coded Hom functors as \(\mathcal V\)-valued functors. If different Hom encodings are used in \(\mathcal V\), their complete natural comparison from Section 1 identifies them with these. A representation in \(\mathcal V\) is then an isomorphism between the images of two \(\mathcal U\)-valued presheaves. Full faithfulness lifts it and its inverse; faithfulness lifts the inverse equations. This gives the original representation. The forward implication is postcomposition of its representation map. The category \(C\) and its objects have stayed fixed. No claim is made that every \(\mathcal V\)-valued presheaf is \(\mathcal U\)-valued.

Representations also transfer across equivalences. Let \(E:C\to D\) have quasi-inverse \(G\) and natural comparison \(\beta:EG\to1_D\). Suppose a covariant \(H:C\to\mathsf{Set}\) is represented by \(T\). For \(Y\in D\), full faithfulness of \(E\), followed by postcomposition with \(\beta_Y\), gives

\[ \begin{gathered} H(GY)\simeq\operatorname{Hom}_C(T,GY)\\ \longrightarrow\operatorname{Hom}_D(ET,Y),\\ b\longmapsto\beta_YE(b). \end{gathered} \tag{7.1} \]

This is bijective by the complete fully faithful criterion for equivalences. For \(v:Y\to Y'\), naturality of \(\beta\) gives \(v\beta_YE(b)=\beta_{Y'}E(G(v)b)\). Thus (7.1) is natural and represents \(HG\) by \(ET\). Applying the same argument to opposite categories gives the contravariant transfer; its map is \(b:GY\to T\mapsto E(b)\beta_Y^{-1}:Y\to ET\). Naturality follows from the inverse naturality equation for \(\beta\). Decode or encode the Hom sets as in Section 1 if their labels require it.

8. Empty tests and constant values

For the empty category, there is exactly one functor to \(\mathsf{Set}_{\mathcal U}\) and exactly one transformation between it and itself. Thus its presheaf category is the terminal category. For the category with one object and its identity only, evaluation gives an isomorphism of its presheaf category with \(\mathsf{Set}_{\mathcal U}\): a set determines the identity functor diagram, and a function determines the sole transformation component. These assignments preserve every identity and composite and are inverse on both objects and arrows.

On any \(C\), the constant empty presheaf is initial, because it has exactly one component function to each presheaf; those functions automatically commute with restriction. The constant singleton presheaf is terminal for the dual reason. This includes empty \(C\).

An object \(Z\) is terminal in \(C\) exactly when \(h_Z\) is terminal in \(P(C)\). In one direction every Hom into \(Z\) is a singleton, so its point functor is naturally isomorphic to the constant singleton presheaf. In the other direction, terminality of \(h_Z\) makes every \(\operatorname{Nat}(h_W,h_Z)\) a singleton. The proved Yoneda bijection makes every \(\operatorname{Hom}_C(W,Z)\) a singleton, which is the definition of terminality.

More generally, suppose \(\rho:h_Z\to\Delta S\) represents a constant presheaf. For any \(u:Z\to Z\), naturality at \(u\), evaluated at the identity, gives

\[ \rho_Z(u)=\rho_Z(1_Z), \tag{8.1} \]

because the constant functor acts by the identity of \(S\). Its bijective component therefore identifies \(S\) with a singleton: the nonempty set \(\operatorname{End}(Z)\) maps constantly and bijectively onto \(S\). Then each \(\operatorname{Hom}(W,Z)\) is a singleton, so \(Z\) is terminal. Conversely a singleton \(S\) and a terminal \(Z\) give this representation. An empty \(S\) cannot be represented, because the identity of a representative would have nowhere to go. On an empty category there is no representing object, although all its constant diagrams coincide with its unique presheaf.

9. Exercises with full solutions

Exercise 1 (beginner: five element objects)

Let \(C\) be the walking arrow \(0\xrightarrow{u}1\). Define a presheaf \(A\) by \(A(0)=\{a,b,c\}\), \(A(1)=\{x,y\}\), and \(A(u)(x)=a\), \(A(u)(y)=b\). List its element category. Find its representable presheaves and decide whether \(A\) is one of them. Count all subpresheaves of \(A\).

Solution. The element objects are \((0,a),(0,b),(0,c),(1,x),(1,y)\). Besides their identities there are exactly the two arrows \((0,a)\to(1,x)\) and \((0,b)\to(1,y)\), each lying over \(u\). These are the only pairs satisfying (3.1). There are no composable pairs of nonidentity arrows, so the list gives the complete category.

\[ \begin{aligned} (0,a)&\xrightarrow{\ u\ }(1,x),\\ (0,b)&\xrightarrow{\ u\ }(1,y),\\ (0,c)&\quad\text{is isolated}. \end{aligned} \]

The projection sends the left column to \(0\), the right column to \(1\), and both displayed arrows to \(u\). Identities are understood; the diagram displays every nonidentity arrow.

The representable \(h_0\) has values of sizes \((1,0)\), and \(h_1\) has values \((1,1)\), with its sole restriction the identity between singleton sets. Every representative must be either \(0\) or \(1\), so these are all representable isomorphism classes. The sizes \((3,2)\) exclude a representation of \(A\). Its element category also has no terminal object: no object receives arrows from both \((0,c)\) and \((1,x)\).

A subpresheaf is exactly a pair \(B_0\subseteq A(0)\), \(B_1\subseteq A(1)\), with \(A(u)(B_1)\subseteq B_0\). For \(B_1\) empty there are \(2^3=8\) choices for \(B_0\). For either singleton \(B_1\), there are \(2^2=4\) choices; for both elements there are \(2\) choices. Thus the answer is \(8+4+4+2=18\). Every counted pair has the inherited restriction and identity maps, so all eighteen are actual subpresheaves.

Exercise 2 (intermediate: an idempotent action)

Let \(C\) have one object and endomorphism monoid \(\{1,e\}\), with \(e^2=e\). A presheaf \(S\) has underlying set \(\{0,1,2,3\}\); the action of \(e\) fixes \(0,1\) and sends \(2\) to \(0\), \(3\) to \(1\). Count \(\operatorname{Nat}(h_*,S)\), \(\operatorname{Nat}(S,h_*)\), the endomorphisms of \(S\), and its automorphisms. Decide whether \(S\) is representable.

Solution. The representable \(h_*\) has two elements \(1,e\); its restriction by \(e\) sends both to \(e\). Write \(r\) for the idempotent action on \(S\). A transformation \(h_*\to S\) is determined by the value \(z\) at \(1\); its value at \(e\) must be \(r(z)\). Idempotence proves the remaining equivariance equation at \(e\). Hence there are four transformations, in agreement with Yoneda.

A transformation \(f:S\to h_*\) must send the fixed elements \(0,1\) to \(e\), since \(e\) is the only fixed point in \(h_*\). Each of \(2,3\) may go to \(1\) or \(e\), because either has restriction \(e\). These choices give all the equivariance equations, so there are four such transformations.

For an endomorphism \(g\) of \(S\), each of \(g(0),g(1)\) must be one of the two fixed points. Once these two images are chosen, \(g(2)\) can be either of the two points in the fibre of \(r\) over \(g(0)\), and \(g(3)\) either point over \(g(1)\). Thus there are \(2^2\cdot2\cdot2=16\) endomorphisms. An automorphism permutes the two fixed points. In each case it must send each nonfixed point to the unique nonfixed point over the corresponding image fixed point, so there are exactly two automorphisms. Their inverses also commute with \(r\), either directly or by the component-inverse criterion. Finally every representable has two elements, whereas \(S\) has four, so \(S\) is not representable.

Exercise 3 (advanced: maps over a finite base)

Let \(C\) be the discrete category on \(0,1\). Take \(A(0)=\{a,b\}\), \(A(1)=\{c,d,f\}\), and a presheaf \(G\to A\) whose five fibre sizes, in that order, are \(1,2,0,1,2\). Describe (6.1) in this case. Count all endomorphisms and automorphisms of \(G\) over \(A\), and compare them with all endomorphisms of its underlying presheaf. Describe the effect of swapping \(a,b\) and cyclically permuting \(c,d,f\) in the base.

Solution. The category \(E_A\) is discrete on five objects, one per labelled base element. A presheaf on it is five independent sets. The fibre functor lists these sets; the sum functor groups the first two at \(0\) and the last three at \(1\). Their inverse maps are exactly (6.6) and (6.7), so the equivalence here is \(\mathsf{Set}^2/A\simeq\mathsf{Set}^5\).

An endomorphism over \(A\) is an independent function on each fibre. A set of size \(n\) has \(n^n\) endomorphisms, with the empty set contributing its one empty function. Thus the count is \(1\cdot4\cdot1\cdot1\cdot4=16\). The automorphism count is \(1!\,2!\,0!\,1!\,2!=4\), using the one permutation of the empty set. Each total set \(G(0),G(1)\) has three elements. Without the maps to \(A\), these two sets have \(3^3\cdot3^3=729\) endomorphisms. Most do not preserve the fibre labels.

For the specified base automorphism \(a:A\to A\), the element functor \(E_a\) is the stated permutation of the five discrete objects. Pulling back \(G\to A\) gives, at a label \(s\), the old fibre at \(a(s)\), by the last paragraph of Section 6. Thus the two fibres at \(0\) swap. At \(1\), each old fibre appears at the inverse image of its old label under the cycle. Maps on fibres undergo this same permutation. This is an actual functor on the five-set category, with inverse induced by the inverse base permutation.

Exercise 4 (expert: natural operations on arbitrary modules)

Let \(R\) be any unital ring and \(U:R\text{-}\mathsf{Mod}\to\mathsf{Set}\) its underlying-set functor. For every integer \(n\geq0\), classify all natural transformations \(U^n\to U\). Determine which have \(R\)-linear components. Compute composition of the operations. For \(R=M_2(\mathbb F_2)\), give a natural unary operation whose component on the regular left module is not \(R\)-linear.

Solution. The finite free left module \(R^n\) represents \(U^n\): a linear map \(R^n\to M\) is uniquely determined by the images of its standard basis vectors, and arbitrary \(n\)-tuples give the map \((a_i)\mapsto\sum_i a_i m_i\). This includes \(n=0\), with the zero module and the unique empty tuple. The finite free-module construction and its universal property are the ordinary-module specialization in Operators and internal tensor–Hom, Section 4.

Apply the covariant Yoneda bijection to \(U\) and this representative. A natural operation corresponds to a vector \((r_1,\ldots,r_n)\in U(R^n)\), and evaluation under the representing map gives exactly

\[ \tau_M(m_1,\ldots,m_n)=\sum_{i=1}^n r_i m_i. \tag{9.1} \]

Every such operation is natural, since an \(R\)-linear \(a:M\to N\) satisfies \(a(\sum_i r_i m_i)=\sum_i r_i a(m_i)\). The Yoneda bijection proves that there are no other natural operations. It also proves uniqueness of the coefficient vector; directly, evaluating on the standard tuple in \(R^n\) recovers that vector. For \(n=0\) this is the zero element in every module.

These functions are additive. They are \(R\)-linear on \(M^n\) for every \(M\) exactly when every \(r_i\) belongs to the centre of \(R\). Central coefficients suffice by commuting them past every scalar. Conversely, use \(M=R\) and a tuple with \(1\) in coordinate \(i\) and zeros elsewhere. Linearity after multiplying this tuple by \(a\in R\) requires \(r_i a=a r_i\) for every \(a\).

If an operation with coefficients \(r_i\) receives operations with coefficients \(s_{ij}\), substitution gives the coefficient \(\sum_i r_i s_{ij}\) in coordinate \(j\). The order of the factors follows from \(r_i(s_{ij}m_j)=(r_i s_{ij})m_j\); commutativity has not been assumed. Unary composition is therefore multiplication \(r,s\mapsto rs\), and the identity unary operation has coefficient \(1\). The nullary operation is zero, consistently with substitution into an empty sum.

For the requested ring, use coefficient \(r=E_{12}\). The function \(m\mapsto E_{12}m\) is natural in all left modules. On the regular module put \(a=E_{21}\), \(m=I\). Then \(r(am)=E_{11}\), whereas \(a(rm)=E_{22}\). These matrices are distinct, so the component is not \(R\)-linear. This distinguishes natural operations on the underlying sets from natural transformations of module-valued functors.

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