Formal coproducts and economical indices

Written by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Original text: CC0. Self-checked; no independent review.

Two representables have a coproduct in presheaves, but that formal coproduct has no representing object. An arbitrary category can nevertheless be replaced by a cofinal poset with very short chains. A second construction freely adds the incoming-arrow summands to an object family; its colimit is the coproduct of that entire family. The three constructions require careful attention to tags, identities and structural maps.

Compose from right to left. Category data lie in ambient set theory. Fix a working Grothendieck universe \(\mathcal U\), with all closure and relabelling proofs retained from Universes and small categories, Sections 1–3. The presheaf assertion requires locally small Hom values, using their established encodings; the entire object set of its base need not be small. The poset construction is conditional on the ambient category data being sets and is small when the original category is small. The incoming-arrow construction uses a small index and a value category with all small colimits.

Retain the complete Yoneda identity-evaluation inverse and naturality in Points and representations, Section 2, the full arbitrary-base pointwise presheaf factors in Colimits as connected components, Section 4, and all chosen cocone factors and discrete coproduct laws in Compatible families, Section 2. Every use below preserves the specified universal family.

1. A formal coproduct of representables

Let \(X,Y\in C\), and write \(h_X(T)=C(T,X)\). In \(\widehat C\), form \[ P=h_X\amalg h_Y. \tag{1.1} \] The entire pointwise colimit proof supplies its values and restrictions: \[ \begin{gathered} P(T)=\{(0,f):f:T\to X\}\\ \amalg\{(1,g):g:T\to Y\},\\ P(a)(0,f)=(0,fa),\\ P(a)(1,g)=(1,ga)\\ (a:T'\to T). \end{gathered} \tag{1.2} \] Here the two displayed families have disjoint tags; the outer union notation retains that convention. Each presheaf map out of \(P\) is uniquely a pair of maps out of \(h_X,h_Y\), by its components and their naturality. Restrictions preserve the tags even when \(X=Y\).

Theorem 1.1. The presheaf \(P\) is not representable.

Proof using every identity map. Suppose \(\sigma:h_Z\to P\) is an isomorphism. The value \(\sigma_Z(1_Z)\) belongs to one of the two tagged families. If it is \((0,b)\), then \(b:Z\to X\). For any \(a:T\to Z\), naturality at \(a\) gives \[ \begin{aligned} \sigma_T(a) &=P(a)\sigma_Z(1_Z)\\ &=(0,ba). \end{aligned} \tag{1.3} \] This is precisely the full Yoneda inverse applied to the chosen identity value. Thus every element in the image of \(\sigma_T\) has tag \(0\). At \(T=Y\), the element \((1,1_Y)\) exists and cannot belong to that image. This contradicts the surjectivity of the isomorphism component \(\sigma_Y\).

If the chosen identity value has tag \(1\), the same naturality formula makes every image have tag \(1\), while \((0,1_X)\) at \(T=X\) has no preimage. This gives the other contradiction. Both cases exclude every \(Z\) and every representing isomorphism. No assumption that Hom sets between distinct objects are nonempty enters the proof. \(\square\)

An ordinary coproduct of \(X,Y\) in \(C\), if one exists, has its different Hom property in the target variable. It does not identify (1.1) with a represented presheaf. For the category with one object and its identity, the ordinary coproduct is that object, whereas (1.1) is a two-element presheaf. If \(C\) is empty, no \(X,Y\) exist and the assertion has no instances.

2. A cofinal poset with two object copies

For an ambient category \(C\), form the disjointly tagged set \[ \begin{gathered} \operatorname{Ob}R= \{\langle f\rangle:f\in\operatorname{Mor}C\}\\ \amalg \{X_0,X_1:X\in\operatorname{Ob}C\}. \end{gathered} \tag{2.1} \] An arrow node and either object copy are different elements, including when \(f=1_X\). Give this set the reflexive relation with additional inequalities, for each \(f:X\to Y\), \[ \langle f\rangle\le X_0, \qquad \langle f\rangle\le Y_1. \tag{2.2} \] There are no other inequalities.

Lemma 2.1. This is a poset, and its associated category admits a functor \(\pi:R\to C\) sending \[ \begin{gathered} \pi(X_0)=\pi(X_1)=X,\\ \pi(\langle f:X\to Y\rangle)=X,\\ \pi(\langle f\rangle\to X_0)=1_X,\\ \pi(\langle f\rangle\to Y_1)=f. \end{gathered} \tag{2.3} \]

Proof of all relation and functor laws. Reflexivity was inserted explicitly. Every nonidentity comparison starts at an arrow node and ends at an object copy, so two nonidentity comparisons cannot compose. Thus transitivity adds nothing, and no pair of opposite nonidentity comparisons exists; antisymmetry follows. In particular, the category has just its identities and the two declared edges per arrow node. Their targets remain different for an endomorphism, since \(X_0\ne X_1\). Each assigned edge in (2.3) has the required domain and codomain. Assigning each identity to the identity of its image preserves identities. Every composable pair includes an identity, so the identity laws in \(C\) prove preservation of every composition. This proves the entire functor construction. \(\square\)

Theorem 2.2. The functor \(\pi\) is cofinal.

Proof with explicit comma paths. Use the full incoming-comma construction and nonempty connected convention from Change an index, Section 1. Fix \(Z\in C\). In \((Z\downarrow\pi)\), write \[ \begin{gathered} U_n(X,q)=(X_n,q:Z\to X),\\ A(f,q)=(\langle f:X\to Y\rangle,\\ q:Z\to X). \end{gathered} \tag{2.4} \] The comma equations give exactly the two arrows \[ \begin{gathered} A(f,q)\longrightarrow U_0(X,q),\\ A(f,q)\longrightarrow U_1(Y,fq). \end{gathered} \tag{2.5} \] Their maps are the identity and \(f\) from (2.3). All remaining comma arrows are identities, since the index has no other nonidentity edges.

The object \(U_0(Z,1_Z)\) supplies nonemptiness. For every \(q:Z\to X\), the following is a valid zigzag: \[ \begin{gathered} U_0(Z,1_Z)\longleftarrow A(q,1_Z)\\ \longrightarrow U_1(X,q)\\ \longleftarrow A(1_X,q) \longrightarrow U_0(X,q). \end{gathered} \tag{2.6} \] Each arrow is an instance of (2.5), and every comma equation has the displayed composite \(q\). Thus either object copy over any \(q\) is connected to the common object. A general \(A(f,q)\) is connected to \(U_0(X,q)\) by the first arrow in (2.5). We have exhausted all comma objects, so this category is connected as well as nonempty. This holds for every \(Z\), proving cofinality. \(\square\)

The full cocone factors can also be written without a path choice. Let \(D:C\to B\). For an \(R\)-cocone \(b\) of \(D\pi\) to \(W\), its identity-node edges give \[ b_{X_0}=b_{\langle1_X\rangle}=b_{X_1}. \tag{2.7} \] Its edges at \(f:X\to Y\) then give \[ \begin{aligned} b_{Y_0}D(f)&=b_{Y_1}D(f)\\ &=b_{\langle f\rangle}\\ &=b_{X_0}. \end{aligned} \tag{2.8} \] Hence \(a_X=b_{X_0}\) is a \(C\)-cocone of \(D\). Conversely restrict any such \(a\) by setting \(b_p=a_{\pi p}\). The \(C\)-cocone equation proves both required edge equations. Starting with \(a\), extension of its restriction is \(a\). Starting with \(b\), (2.7) and the first edge at \(f\) recover both object copies and every arrow node. Thus the two operations are inverse on entire cocones. Postcomposing the target map, or precomposing the component maps of a diagram transformation, commutes with both formulas. The correspondence is natural in \(D,W\).

Consequently either chosen colimit of \(D\) or \(D\pi\) gives the other by the complete representing-cocone proof in Change an index, Section 2. If their legs are \(c_X,d_p\), the specified comparisons satisfy \[ \begin{gathered} e:\operatorname{colim}_R D\pi \to\operatorname{colim}_C D,\\ e d_p=c_{\pi p},\\ x:\operatorname{colim}_C D \to\operatorname{colim}_R D\pi,\\ x c_X=d_{X_0}. \end{gathered} \tag{2.9} \] The second family is compatible by (2.8) for \(d\). At every \(c_X\), \(ex c_X=c_X\). At either object-copy leg, (2.7) gives \(xe d_{X_n}=d_{X_n}\); at every arrow-node leg, the first edge gives \(xe d_{\langle f\rangle}=d_{X_0}=d_{\langle f\rangle}\). Universal uniqueness proves both inverse identities and the retained leg laws prove naturality.

This poset has \(|\operatorname{Mor}C|+2|\operatorname{Ob}C|\) objects and one identity at each of them, plus two nonidentity edges per original arrow. The full small finite-union and transported encoding proof therefore bounds its objects and arrows when \(C\) is small, and the same formulas give finiteness when \(C\) is finite. It need not be directed. If \(C\) is empty, \(R\) is empty, cofinality is vacuous, and the cocone correspondence is the empty one; a colimit then exists exactly when the value category supplies the corresponding initial object.

3. An incoming-arrow coproduct diagram

Let \(I\) be small, let \(B\) admit small colimits, and let \(A_i\in B\) be an object for each \(i\in I\). No maps between these objects are needed yet. For each incoming arrow \(a:k\to i\), give a chosen coproduct its summand map \[ \begin{gathered} \Psi A(i)=\coprod_{a:k\to i} A_k,\\ \iota_a^i:A_k\to\Psi A(i). \end{gathered} \tag{3.1} \] The family is indexed by the actual arrows, including identities. Distinct arrows retain distinct summands even if their domains and codomains agree. Smallness of \(I\) bounds this incoming-arrow set as a subset of its small morphism set, by the full universe subset and encoding proof.

For \(s:i\to j\), the entire coproduct universal property gives the unique map with \[ \Psi A(s)\iota_a^i=\iota_{sa}^j. \tag{3.2} \] Both \(\Psi A(1_i)\) and the identity have the same composite with every \(\iota_a^i\). For successive \(s,t\), both \(\Psi A(t)\Psi A(s)\) and \(\Psi A(ts)\) have composite \(\iota_{tsa}\) on each summand. Coproduct uniqueness proves both functor laws.

Proposition 3.1 (the full free-diagram factor). For a diagram \(E:I\to B\), there are inverse assignments \[ \begin{gathered} \operatorname{Nat}(\Psi A,E)\\ \simeq\prod_{k\in\operatorname{Ob}I}B(A_k,E(k)),\\ v\longmapsto (v_k\iota_{1_k}^k)_k. \end{gathered} \tag{3.3} \] The right side is the ambient set of component families; use a larger universe if necessary.

Proof on all summands, with both naturalities. Given maps \(z_k:A_k\to E(k)\), define \(v_i\) by the unique coproduct factor with \[ \begin{gathered} v_i\iota_a^i=E(a)z_k\\ (a:k\to i). \end{gathered} \tag{3.4} \] For \(s:i\to j\), the composites of \(E(s)v_i\) and \(v_j\Psi A(s)\) with \(\iota_a^i\) are both \(E(sa)z_k\). Uniqueness makes \(v\) natural. Evaluating at \(1_k\) recovers \(z_k\), because \(E(1_k)=1\). Conversely, for natural \(v\), its square at \(a:k\to i\) and (3.2) give \[ v_i\iota_a^i=E(a)v_k\iota_{1_k}^k. \tag{3.5} \] Thus reconstruction from its identity-summand values gives exactly the original \(v_i\), on every summand; uniqueness proves the other inverse equation.

A family map \(h_k:A'_k\to A_k\) induces the unique natural \(\Psi h\) with \((\Psi h)_i\iota'{}^i_a=\iota^i_a h_k\). Testing summands proves naturality and both functor laws in \(h\). For a diagram transformation \(r:E\to E'\), the two routes applied to \(v\Psi h\) and \(rv\) respectively give the families \(z_kh_k\) and \(r_kz_k\). Conversely (3.4) gives their factors, since \(r_iE(a)=E'(a)r_k\). These are both-variable naturality squares for (3.3), with their specified inverse assignments. \(\square\)

Let \(I_{\mathrm d}\) be the discrete category on \(\operatorname{Ob}I\), and let \(\delta:I_{\mathrm d}\to I\) be the identity on objects. Then \(A\) is a diagram on \(I_{\mathrm d}\), and (3.3) says precisely that \(\Psi A\) is its left Kan extension along \(\delta\). In the complete incoming-comma construction of Extending diagrams, Section 3, \((\delta\downarrow i)\) is literally discrete on the arrows \(a:k\to i\): its only candidate morphisms come from identities, and their triangle equation forces the same \(k,a\). Its colimit is therefore the coproduct (3.1), with exactly the maps (3.2). The full retained Kan factor proof agrees with (3.3)–(3.5).

Theorem 3.2. If \(P=\coprod_k A_k\), with legs \(j_k:A_k\to P\), then \(P\) is the colimit of \(\Psi A\), with the cocone defined by \[ \begin{gathered} c_i\iota_a^i=j_k\\ (a:k\to i). \end{gathered} \tag{3.6} \]

Proof of the complete cocone property. Equation (3.2) makes \(c_j\Psi A(s)\) and \(c_i\) agree on every original summand, so \(c\) is compatible. Given a cocone \(b_i:\Psi A(i)\to W\), set \[ z_k=b_k\iota_{1_k}^k. \tag{3.7} \] Its cocone equation at \(a:k\to i\) gives \[ \begin{aligned} b_i\iota_a^i &=b_i\Psi A(a)\iota_{1_k}^k\\ &=b_k\iota_{1_k}^k=z_k. \end{aligned} \tag{3.8} \] There is a unique coproduct factor \(f:P\to W\) with \(fj_k=z_k\). Equations (3.6) and (3.8) show \(fc_i=b_i\) on every summand, hence as arrows. If \(g\) also factors \(b\), then \(gj_k=gc_k\iota_{1_k}^k=b_k\iota_{1_k}^k=z_k\), so coproduct uniqueness gives \(g=f\). This proves the colimit with its actual legs. Postcomposition in \(W\) commutes with (3.7) and the factor formulas, giving the full natural target correspondence. \(\square\)

For the original diagram \(D:I\to B\), take \(A_i=D(i)\) to obtain the incoming-arrow diagram \(\Psi D\). A transformation \(z:D\to D'\) induces the just-defined maps \(\Psi z\) and the coproduct map \(Pz\) with \(Pz\,j_k=j'_kz_k\). Testing every \(\iota_a\) gives \[ Pz\,c_i=c'_i(\Psi z)_i. \tag{3.9} \] Thus the colimit–coproduct comparison is natural in the original diagram. The construction of \(\Psi D\) uses its object values; its original arrow maps instead define the natural counit \(\varepsilon_i:\Psi D(i)\to D(i)\) by \(\varepsilon_i\iota_a^i=D(a)\). The functor law \(D(sa)=D(s)D(a)\) proves that counit is natural. It can impose new identifications on the colimit, as the examples show.

If \(I\) is empty, \(P\) is the empty coproduct, an initial object of \(B\). Both the family and diagram in (3.3) are empty, so there is one component family and one transformation. The cocone factor proof says exactly that this initial object is the empty colimit. All required small coproducts and colimits exist by the original hypothesis; no filteredness or nonemptiness is inserted.

4. Four graded exercises with full solutions

Exercise 1 — compare the two coproduct properties

Introductory. Let \(C=\mathsf{Pt}\), the category with one object \(t\) and its identity. Compute its ordinary coproduct \(t\amalg t\), the presheaf coproduct \(h_t\amalg h_t\), and the comparison induced by the ordinary coprojections. Verify both universal properties and explain why the comparison cannot be invertible.

Solution. The only object \(t\), with both coprojections its identity, is the ordinary coproduct. For its only target, there is one pair of maps out of \(t\) and one map out of the candidate coproduct, and the factor assignment is the unique bijection. A presheaf on \(\mathsf{Pt}\) is a set, with its identity action. Thus \(h_t\) is a singleton and its formal coproduct has the two tags \((0,1_t),(1,1_t)\). For any target presheaf \(W\), a map out of this coproduct uniquely specifies its independent values at the two tags, exactly a pair of maps from the singleton. This is its full universal property. The comparison to \(h_{t\amalg t}=h_t\) sends both tags to \(1_t\), by the ordinary identity coprojections. It is not injective and therefore not invertible.

Exercise 2 — keep all incoming summands

Intermediate. Let \(I=\{0<1\}\). Let \(D(0)=\{a,b\}\), \(D(1)=\{c,d\}\), with both \(a,b\) sent to \(c\). Compute \(\Psi D\), its colimit cocone, its counit to \(D\), and the induced map to \(\operatorname{colim}D\). Show every universal factor.

Solution. There is one incoming arrow at \(0\) and two at \(1\). Hence \(\Psi D(0)=\{a,b\}\) and \(\Psi D(1)=\{(0,a),(0,b),(1,c),(1,d)\}\), with \(\Psi D(0\to1)\) the injection into the first two points. The tags \(0,1\) here specify the domains of the two incoming arrows. Its terminal index \(1\) makes its colimit the displayed four-element set: any cocone is determined by its component from that value, and its component from stage \(0\) is the composite with the injection. This gives the unique factor to every target, including an empty target when such a cocone does not exist.

The coproduct \(P=D(0)\amalg D(1)\) has those same four elements. The cocone \(c_1\) is the identity and \(c_0\) injects \(a,b\) into their tags, as (3.6) states. The counit at \(0\) is the identity on \(a,b\); at \(1\) it sends \((0,a),(0,b),(1,c)\) to \(c\), and \((1,d)\) to \(d\). Its square commutes on both original points. The colimit of \(D\) is \(\{c,d\}\), again by the terminal index and the full endpoint factor. The induced colimit map has exactly those counit values, so it makes the two original points \(a,b\) coincide with the \(c\) point while retaining \(d\). Thus the colimit of the free diagram is the entire object coproduct; the counit is responsible for the additional relations.

Exercise 3 — a group is replaced by a short poset

Advanced. Let \(C\) have one object \(t\) and the group \(\{1,g\}\), \(g^2=1\). List the poset \(R\), its functor \(\pi\), and every object and nonidentity arrow of \((t\downarrow\pi)\). For the action diagram on \(\{a,b\}\) where \(g\) swaps the two points, compute the two colimits and their comparison factors.

Solution. The poset has four objects \(\langle1\rangle,\langle g\rangle,t_0,t_1\). Each arrow node is below each object copy, giving four nonidentity edges; there are also four identities. Every object maps to \(t\). Both edges to \(t_0\) map to \(1\). The edges to \(t_1\) map respectively to \(1,g\). These assignments satisfy all composition laws since there are no composable nonidentity edges.

The comma has four upper objects \(U_0(t,1),U_0(t,g),U_1(t,1),U_1(t,g)\), and four lower objects \(A(1,1),A(1,g),A(g,1),A(g,g)\). For each \(f,q\in\{1,g\}\), its two nonidentity edges go to \(U_0(t,q)\) and \(U_1(t,fq)\). These eight edges are all of its nonidentity arrows, together with the eight identities. The underlying graph is an eight-vertex cycle, so there is one component. The explicit paths of (2.6) can alternatively verify every vertex's connection to \(U_0(t,1)\).

In the original action, an invariant map to any set \(W\) must take \(a,b\) to the same value; conversely any such map is invariant. Its unique factor is from a singleton, so the original colimit is a singleton. In the restricted diagram, all four index values have two points. The identity-node edges identify both object-copy values point by point, and the \(\langle g\rangle\) edges then identify \(a\) with \(b\). Thus all eight tagged points lie in one class. A restricted cocone is determined by one value, by those same edge equations, and factors uniquely from the singleton. Both colimits have their one point, \(e,x\) fix that point, and the complete factor formulas of (2.7)–(2.9) recover all four original cocone components.

Exercise 4 — free an action without applying it to the values

Challenge. For the same two-element group \(I\), let \(D(t)=\{a,b\}\) with \(g\) swapping \(a,b\). Compute \(\Psi D\), all its action maps, its colimit and counit. Verify directly the correspondence (3.3) for every equivariant target \(E\).

Solution. The only value of \(\Psi D\) has four points \((1,a),(1,b),(g,a),(g,b)\). Postcomposition by \(g\) exchanges the labels \(1,g\) while retaining \(a\) or \(b\). The identity fixes all four points, and applying this label exchange twice gives the identity, so every functor law is checked.

Its two quotient classes are \(\{(1,a),(g,a)\}\) and \(\{(1,b),(g,b)\}\). An invariant cocone function is exactly a choice of one value for each class, so it factors uniquely from the two-element set \(\{a,b\}\). The cocone forgets the arrow label, and the coproduct over the single object is \(D(t)=\{a,b\}\), as Theorem 3.2 asserts.

The counit sends \((s,x)\) to \(D(s)x\), so its values at \((1,a),(1,b),(g,a),(g,b)\) are \(a,b,b,a\). It intertwines the \(g\) actions because \(D(gs)=D(g)D(s)\). The colimit of \(D\) is a singleton, and the induced colimit map sends both free classes to that point.

For any diagram \(E\) on this group and any function \(z:\{a,b\}\to E(t)\), define \(v(s,x)=E(s)z(x)\). Then \(v(gs,x)=E(g)v(s,x)\), proving equivariance. Its values at \((1,x)\) are \(z(x)\). Conversely, equivariance of \(v\) forces \(v(g,x)=E(g)v(1,x)\), so the identity-label values uniquely determine every component. These two assignments are inverse, even for empty values, and postcomposition by any equivariant target map preserves them. Precomposition by any function of the free family likewise acts on \(z\) before the same formula. This verifies the full factor correspondence on these actual objects.

5. References and retained proof interfaces