Written and self-checked with GPT-6.1 Sol (OpenAI), Ultra reasoning effort. A separate exact scoped check covers the incoming comma and finite assembly arguments, their needed prerequisite interfaces and four exercise solutions. It adds no whole-lesson or whole-course review. Original exposition uses CC0 1.0; linked complete proofs retain their stated terms.

Incoming comma tests and assembly from finite pieces

Written and self-checked with GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Original exposition: CC0. Linked complete proofs retain their stated terms.

An index change is cofinal when every way of reaching an image object belongs to one connected family of choices. If its source is filtered, connectedness can be tested by an actual eventual equality of arrows. This test also controls several functors between comma categories. A second construction assembles an arbitrary small diagram from its finite pieces and explains why finite colimits together with filtered colimits suffice for all small colimits.

1. Conventions and complete prerequisites

Fix a Grothendieck universe \(\mathcal U\), ambient choice, and the enlargement axiom in Universes and small categories, Sections 1–4. All category data are ambient sets. Unless stated otherwise, categories are locally \(\mathcal U\)-small; small means \(\mathcal U\)-small. A finite category has finitely many objects and arrows, including the empty category. A product indexed by an ambient set can be a bigger category; a larger universe containing its full data permits the arguments below without asserting smallness in \(\mathcal U\).

We retain these complete interfaces.

• Filtered stages and finite limits, Sections 1–2 proves the three-axiom filteredness criterion, cocones on every finite graph, and equality of two same-stage Set elements after one actual transition arrow.
• Change an index, Sections 1–4 proves the incoming-comma definition of cofinality, inverse natural cocone factors, canonical colimit comparisons, the represented-Hom singleton test, and composition of cofinal functors. Its ambient-size argument in Section 4 is retained.
• Compatible families, Sections 2–4 gives universal-family representations, structural-map comparisons, their functor laws, formal cocone functors and opposite transfer. Section 6 makes preservation a statement about the actual image universal family.
• Products and mixed functors, Section 1 proves ambient product-category laws. Independent product indices supplies the complete coordinatewise filteredness argument, with choice and the empty product.
• Universal forks, Sections 8–9 proves the full coproduct/coequalizer presentation of arbitrary small colimits, its maps and empty case. Finite partial coproducts, formula (3.2) and its following factor proof gives their assembly by a filtered colimit.
• Outgoing-comma left extension and total assembly supply the conditional Kan construction, its maps, inverse factors and natural total comparison.

These proofs stay linked rather than being reconstructed here.

A category is filtered if it is nonempty, any two objects have a common receiver, and any two parallel arrows become equal after postcomposition. It is connected if it is nonempty and every pair of objects is joined by a finite zigzag, with either arrow direction allowed.

Three elementary consequences use the retained proofs directly. A terminal object gives filteredness by the explicit terminal-object argument in Filtered stages, Section 1. If every finite colimit exists, every finite diagram has its universal cocone, so that section's full finite-cocone criterion gives filteredness; its empty diagram supplies nonemptiness. A set-indexed product of filtered categories is filtered by the independent-coordinate proof cited above, in a larger universe if needed. The empty product is the terminal category by the product-category construction.

A filtered category is connected: for \(i,i'\), a common receiver gives

Its nonemptiness is already one of the axioms.

For a functor \(\phi:J\to I\), write

An object is \((j,s:i\to\phi(j))\). An arrow to \((k,s')\) is \(t:j\to k\) with \(\phi(t)s=s'\). Identities and composition are those of \(J\), with the complete comma-category laws retained from Change an index. Cofinality means that every \(J^i\) is connected.

A filtered inductive system is a covariant functor \(I\to C\) with \(I\) filtered. A filtered projective system is a functor \(J^{\mathrm{op}}\to C\) with \(J\) filtered: its actual domain is cofiltered. Smallness refers to the index. The complete opposite transfer in Compatible families, Section 3, supplies the dual universal-family convention; it does not identify a limit with a colimit.

2. A local arrow test for cofinality

Theorem 2.1. Suppose \(J\) is filtered and \(\phi:J\to I\). The following are equivalent.

1. \(\phi\) is cofinal.
2. Every incoming comma \(J^i\) is filtered.
3. Both of these tests hold:
   • for every \(i\), some arrow \(i\to\phi(j)\) exists;
   • for every pair \(s,s':i\rightrightarrows\phi(j)\), some \(t:j\to k\) satisfies

Whenever these conditions hold, \(I\) is filtered.

Proof from the arrow tests. Nonemptiness of \(J^i\) is the first test. For two objects \((j,s)\), \((j',s')\), choose \(a:j\to\ell\), \(b:j'\to\ell\) using filteredness of \(J\). Apply the second test to the pair \(\phi(a)s,\phi(b)s':i\rightrightarrows\phi(\ell)\). It gives \(r:\ell\to n\) with

Call their common value \(v:i\to\phi(n)\). Then \(ra,rb\) are actual comma arrows to \((n,v)\), giving a common receiver.

For parallel comma arrows \(f,g:(j,s)\rightrightarrows(k,v)\), filteredness of \(J\) gives \(e:k\to n\) with \(ef=eg\). The arrow

belongs to the comma category, and its two composites are equal as \(J\)-arrows. Thus every \(J^i\) is filtered. Filteredness implies connectedness by Section 1, proving cofinality.

Proof from cofinality. The first test follows from nonemptiness of \(J^i\). To prove the second, choose one larger universe containing the full data of \(I,J\). They are small there, and the functor

has small encoded values there. The complete represented-Hom cofinality test in Change an index, Theorem 3.1, says that its colimit is a singleton. Hence \(s,s'\in H_i(j)\) have the same colimit image. The complete same-stage equality criterion in Filtered stages, Corollary 2.3, gives one actual \(t:j\to k\) with \(H_i(t)s=H_i(t)s'\), precisely (2.1). This proves the tests, hence all three equivalent conditions.

This enlargement only licenses the Set calculation. The resulting arrows and finite witnesses belong to the original categories, so the conclusion imposes no new working-universe smallness hypothesis.

Proof of the final assertion. Choose \(j_0\in J\); then \(\phi(j_0)\) is an object of \(I\). Given \(i_1,i_2\), the reaching test gives \(s_a:i_a\to\phi(j_a)\). Choose \(v_a:j_a\to k\) in \(J\). The arrows \(\phi(v_a)s_a\) give a common receiver in \(I\).

For \(f,g:i\rightrightarrows i'\), choose \(s:i'\to\phi(j)\). Apply (2.1) to \(sf,sg\). If \(t:j\to k\) is its witness, the arrow \(\phi(t)s:i'\to\phi(k)\) equalizes \(f,g\). These are all three filteredness axioms. \(\square\)

The condition concerns arrows in \(J\), not merely the existence of some equalizing arrow in \(I\). Also, connected incoming commas need not be filtered when \(J\) is not filtered; Exercise 1 exhibits that boundary.

3. Receiving subcategories and incoming coslices

Proposition 3.1. Let \(I\) be filtered. For every \(i\), the incoming coslice

is filtered, and its forgetful functor \(p_i:D_i\to I\) is cofinal.

Proof. The identity functor is cofinal: its incoming comma at \(i\) has initial object \((i,1_i)\), whose unique arrow to \((k,u)\) is \(u\). Thus this comma is nonempty and connected. Theorem 2.1 applied to \(1_I\) makes \(D_i\) filtered.

Apply the arrow tests to \(p_i\). For \(x\in I\), choose a common receiver \(k\) and arrows \(u:i\to k\), \(v:x\to k\). The object \((k,u)\) of \(D_i\) receives \(v:x\to p_i(k,u)\).

Given \((k,u)\in D_i\) and \(s,s':x\rightrightarrows k\), choose \(e:k\to\ell\) with \(es=es'\). It defines a coslice arrow

whose image under \(p_i\) equalizes the given pair. Theorem 2.1 proves cofinality. \(\square\)

Proposition 3.2. Suppose \(I\) is filtered, \(\phi:J\to I\) is fully faithful, and every \(i\) has an arrow to some \(\phi(j)\). Then \(J\) is filtered and \(\phi\) is cofinal.

Proof with the lifting roles explicit. An object of \(I\) reaches an image, so \(J\) is nonempty. For \(j_1,j_2\), choose arrows \(\phi(j_a)\to\ell\) to a common receiver, then choose \(\ell\to\phi(k)\). Fullness lifts their composites to arrows \(j_a\to k\), giving the second filteredness axiom in \(J\).

Now take parallel \(f,g:j_1\rightrightarrows j_2\). Choose \(e:\phi(j_2)\to\ell\) with \(e\phi(f)=e\phi(g)\), and a receiving arrow \(r:\ell\to\phi(k)\). Fullness gives \(t:j_2\to k\) with \(\phi(t)=re\). Consequently

Faithfulness, on \(J(j_1,k)\), yields \(tf=tg\). Thus \(J\) is filtered.

For the second test of Theorem 2.1, take arbitrary \(s,s':i\rightrightarrows\phi(j)\). Equalize them by \(e:\phi(j)\to\ell\), move farther by \(r:\ell\to\phi(k)\), and use fullness to lift \(re\) to \(t:j\to k\). Then \(\phi(t)s=\phi(t)s'\). The reaching test is assumed, so the theorem proves cofinality. \(\square\)

Fullness produces arrows with specified images. Faithfulness is needed in (3.1) to obtain equality back in \(J\). No injectivity on object labels is assumed. If one prefers a full-image formulation, the complete factorization through an equivalence onto the full object image is Equivalences and chosen representatives, Corollary 3.3. Proposition 3.2 proves the entire functor version directly, including the equalization cases.

The full-subcategory declaration Stacks, Lemma 4.19.3 omits its printed proof. The argument above supplies the complete supplement rather than treating that omitted proof as an imported argument.

The binary diagonal of a filtered category,

is cofinal by the complete incoming-comma argument in Dense probes, Section 4, the paragraph establishing diagonal cofinality. That paragraph uses only the filtered axioms to construct common receivers with compatible receiving maps; the later density hypotheses play no role in this assertion. Its finite argument also applies to an ambient filtered category after enlargement.

In an iterated-comma notation, the comma of \(\Delta\) at \((i_1,i_2)\) is

An object records both arrows \(i_1\to k\), \(i_2\to k\), and its arrow has to respect both. Retaining \(p_{i_1}\) makes the notation well typed.

This is a binary result. For example, the infinite diagonal \(\mathbb N\to\mathbb N^{\mathbb N}\) is not cofinal: the incoming comma at the tuple \((0,1,2,\ldots)\) would require a natural number at least as large as every coordinate, and has no object.

4. Four functors between incoming commas

Let

be cofinal functors, with all three categories filtered. Define

Theorem 2.1 makes \(A_j,E_k\) filtered. Composition of cofinal functors is proved in Change an index, Proposition 4.1, so it also makes \(B_k\) filtered.

Theorem 4.1. The following functors are cofinal.

1. For \(j\in J\), the forgetful functor \(A_j\to I\).
2. For \(i\in I\), the functor

3. For \(k\in K\), the functor

4. For \(u:k\to\psi(j)\), the functor

The equations defining the commas verify each construction. In (4.2), applying \(\phi\) preserves a coslice equation. In (4.3), the equation \(\psi\phi(t)v=v'\) is precisely the target comma equation. In (4.4), \(\phi(t)s=s'\) implies

Their identity and composition laws are the laws of \(\phi\), or the unchanged underlying \(I\)-arrows. This verifies all four functors before checking cofinality.

Proof of 1. Fix \(i_0\in I\). The incoming comma of \(A_j\to I\) at \(i_0\) has objects

An arrow \(t:a\to a'\) satisfies both

The same triples, with the last two entries listed in the other order, are the objects of

Its arrows have exactly (4.5). The two assignments are inverse, leave the underlying arrow \(t\) unchanged, and therefore preserve identities and compositions. This is an isomorphism of whole comma categories.

Proposition 3.1 makes \(p_{i_0}\) cofinal, and the retained composition theorem makes \(\phi p_{i_0}\) cofinal. Its source is filtered by Proposition 3.1, so Theorem 2.1 makes \((j\downarrow\phi p_{i_0})\) filtered. The incoming comma just identified is consequently connected. This holds for every \(i_0\), proving 1.

Proof of 3. Take an object \((j,u:k\to\psi(j))\) of \(E_k\). An object of its incoming comma under \(F_k\) consists of \((a,v)\in B_k\) and \(s:j\to\phi(a)\) with

Thus \(v\) is determined by \((a,s)\). An arrow in this comma is \(t:a\to a'\) with \(\phi(t)s=s'\); that equation also implies the \(B_k\) triangle, by (4.6). Forgetting \(v\) gives an isomorphism of this entire comma with \(A_j\), whose inverse inserts \(v=\psi(s)u\). Both directions retain the same arrows, hence all functor laws. Since \(A_j\) is filtered, this comma is connected, proving 3.

Proof of 4. Under the isomorphism just constructed, \(U_u\) is precisely the forgetful functor

Part 3 gives a cofinal functor \(F_k\) between filtered categories. Part 1, applied to that functor, says this forgetful functor is cofinal. Transport through the displayed isomorphism proves 4.

Proof of 2. Apply part 4 to the different chain

with its middle object \(j=i\), its last-category object \(k=\phi(i)\), and \(u=1_{\phi(i)}\). The source in (4.4) is now \((i\downarrow 1_I)\), and its target is \((\phi(i)\downarrow\phi)\). The object formula sends \(r:i\to a\) to \(\phi(r)\), exactly (4.2). Both functors in this chain are cofinal, so part 4 applies. \(\square\)

Neither \(\phi\) nor \(\psi\) was assumed fully faithful. In particular, part 2 is not a full-embedding assertion. Exercise 2 makes this visible with a nonfull functor of posets.

5. How many receiving objects are enough?

Call \(I\) cofinally small if some small category \(D\) admits a cofinal functor \(D\to I\).

Proposition 5.1. A filtered, locally \(\mathcal U\)-small \(I\) is cofinally small exactly when there is a \(\mathcal U\)-small subset \(S\subseteq\operatorname{Ob}I\) such that every object of \(I\) has an arrow to an object in \(S\).

Proof using the established size interfaces. For a small cofinal \(\theta:D\to I\), let \(S\) be its object image. The complete image-as-quotient bound in Universes, Proposition 3.1, makes \(S\) small, without requiring its labels to be universe members. Every incoming comma of \(\theta\) is nonempty, so every object reaches an object of \(S\).

Conversely, take the full subcategory \(I_S\) on a small receiving \(S\). Its object set is small. Its arrows are the tagged union of \(I(s,t)\) over \(S\times S\); local smallness and the complete tagged-union bound give a small arrow set, as proved in Change an index, Section 5. The inclusion is fully faithful and has the receiving property. Proposition 3.2 makes it filtered and cofinal, so \(I_S\) is a small cofinal witness. \(\square\)

This proof uses the native local-smallness convention at a precise point. Outside it, the existing one-object monoid example at the end of Change an index, Section 5, has a small cofinal witness but no small full cofinal subcategory. Its nonsmall Hom set is the obstruction. No new proof of that size example is needed here.

For the ordered category \(\mathbb N\), the even numbers form a small full receiving subcategory: \(n\leq2n\), including \(n=0\). Its inclusion is cofinal, even though it has no terminal object. Exercise 3 tests the corresponding issue when the object collection itself is too large.

6. Two different cofinal poset replacements

For every ambient set-category \(C\), the complete construction in Formal coproducts and economical indices, Section 2 gives a poset \(R\) and cofinal \(\pi:R\to C\). Its Lemma 2.1 and Theorem 2.2 prove the relation and functor laws, every incoming-comma path, inverse cocone factors and naturality. The concluding size argument makes \(R\) small for small \(C\), finite for finite \(C\), and empty for empty \(C\). The poset need not be filtered.

A stronger theorem applies to small filtered \(I\): it has a small filtered poset with a cofinal functor to \(I\). We retain the complete free proof in Stacks, Lemma 4.21.5, already used in Profinite spaces, Section 1. It constructs a directed partial order and proves its cofinality and universal comparisons. The following checks bind its construction to our precise size and finite-witness conventions.

First, \(A=I\times\mathbb N\) is small by the product and universe proofs, and filtered by the independent-coordinate proof. The projection \(A\to I\) is cofinal: at \(i\), its incoming comma has initial object \(((i,0),1_i)\). The unique arrow from it to \(((a,n),v:i\to a)\) has components \((v,0\leq n)\), and the comma equation forces that first component.

In the retained proof, finitely generated means generated by finitely many arrows and the necessary objects and identities; the generated subcategory may have infinitely many arrows. Its finite generating graph admits a cocone by the complete finite-graph result. A generator equation \(u_x=u_y f\) propagates to any composite by repeated substitution. Thus that finite graph suffices for the generated subcategory. Postcomposing the cocone to a strictly larger natural-number coordinate prevents any arrow back to an old object.

The subcategories used by the construction form a subset of

with their category operations inherited from \(A\). This set is small by the complete power-set and product bounds. The inclusion-order arrows are a subset of its square, so the resulting poset is small as a category as well.

Each chosen subcategory has a unique terminal object. For an inclusion \(H\subseteq H'\), the terminal-object transition is the unique arrow in \(H'\) from the terminal object of \(H\) to that of \(H'\). For \(H=H'\), it is the identity. For successive inclusions, the two transition arrows compose inside the last subcategory, and terminal uniqueness identifies their composite with the direct transition. These checks give the functor laws used by the retained construction.

The directedness and incoming-comma cofinality are the full arguments of the cited lemma, not new claims based on the first rank-one poset. The product with \(\mathbb N\) is part of that proof and must remain in the construction. Thus the general cofinal-poset theorem and the filtered cofinal-poset theorem have separate complete providers and separate hypotheses.

7. Assemble an arbitrary small poset diagram

Let \(I\) be a small poset, regarded as a category, and let \(\alpha:I\to C\). There is no filteredness assumption on \(I\). Write

for its finite subsets ordered by inclusion. Each \(J\in P\) carries the full induced order, and therefore is a finite category. The power-set and subset bounds in Universes make \(P\) small, including its arrows. It is filtered: it contains \(\varnothing\), union gives upper bounds, and parallel arrows in a poset are unique.

Actual colimit objects

Theorem 7.1. Suppose every finite restriction \(\alpha|J\) has a colimit \(L_J\) in \(C\), including \(J=\varnothing\). These objects form a canonical \(P\)-diagram. If its colimit \(L\) exists, then \(L\) is a colimit of \(\alpha\). Its comparison with any other chosen \(\operatorname{colim}_I\alpha\) is the canonical natural isomorphism respecting all original coprojections.

Proof of the diagram and factor. Write the finite-stage legs as \(\lambda_j^J:\alpha(j)\to L_J\). For \(J\subseteq J'\), define \(r_{JJ'}\) by

The target family is a cocone on \(\alpha|J\), so the map exists uniquely. Testing its composites with every \(\lambda_j^J\) proves

For \(J=\varnothing\), the same uniqueness is initial-object uniqueness, so there is no exception to either law.

Let \(l_J:L_J\to L\) be the outer colimit legs. Put

For every \(J\) containing \(i\), (7.1) and the outer cocone equation give \(c_i=l_J\lambda_i^J\). For an arrow \(i\leq i'\), use \(J=\{i,i'\}\): the finite-stage cocone equation gives

Thus \(c\) is an \(I\)-cocone.

Given any cocone \(a_i:\alpha(i)\to T\), each finite restriction has a unique factor \(h_J:L_J\to T\) with \(h_J\lambda_j^J=a_j\). For \(J\subseteq J'\), the maps \(h_{J'}r_{JJ'}\) and \(h_J\) have the same finite-stage legs, hence are equal. This includes \(J=\varnothing\), where there is a unique map out of the initial \(L_\varnothing\). Outer universality gives the unique \(h:L\to T\) with \(hl_J=h_J\), and (7.2) gives \(hc_i=a_i\).

If \(g:L\to T\) has \(gc_i=a_i\), then \(gl_J\lambda_j^J=a_j\). Finite-stage uniqueness, also for the empty stage, gives \(gl_J=h_J\). Outer uniqueness yields \(g=h\). This proves the full \(I\)-colimit property.

If \(B=\operatorname{colim}_I\alpha\) has legs \(b_i\), the finite factors into \(B\) give the canonical map \(\Phi:L\to B\), characterized by

Both \(L\) and \(B\) are \(I\)-colimits with their specified legs, so the complete universal-family uniqueness theorem makes \(\Phi\) invertible, with inverse characterized by \(b_i\mapsto c_i\).

For naturality, take \(z:\alpha\to\alpha'\), with corresponding chosen stages and outer colimits. Define

The right side is a finite cocone by naturality of \(z\). Finite-stage uniqueness gives \(r'_{JJ'}z_J=z_{J'}r_{JJ'}\), so these maps induce \(z_L:L\to L'\) with \(z_Ll_J=l'_Jz_J\). Their identity and composition laws follow from the same leg tests. If \(z_B:B\to B'\) is the original-diagram colimit map, (7.3)–(7.4) give

Finite-stage uniqueness gives equality after each \(l_J\), including the empty \(J\); outer uniqueness gives \(\Phi'z_L=z_B\Phi\). This is naturality of the actual canonical comparison. \(\square\)

If \(I=\varnothing\), \(P\) has the single object \(\varnothing\). The inner stage is an initial object, and the outer diagram has that same colimit. Thus Theorem 7.1 correctly requires an initial object. Conversely, knowing only that one particular \(I\)-colimit exists does not supply missing colimits of its finite restrictions.

Formal colimits without represented objects

For any small diagram, retain the formal cocone functor

It is an object of \(C^\vee=\operatorname{Fun}(C,\mathsf{Set})^{\mathrm{op}}\), using the complete small Hom-family encodings of Compatible families. A represented colimit is additional data, not part of this definition.

For \(J\subseteq J'\), restriction gives a covariant-functor map \(Q_{\alpha|J'}\to Q_{\alpha|J}\), hence a map \(Q_{\alpha|J}\to Q_{\alpha|J'}\) in \(C^\vee\). Thus the formal finite stages form a \(P\)-diagram there.

Proposition 7.2. Their formal colimit is \(Q_\alpha\):

No inner or outer colimit object in \(C\) is presumed to exist.

Proof with inverse compatible-family assignments. At \(T\), the left side's underlying covariant value is \(\lim_{P^{\mathrm{op}}}Q_{\alpha|J}(T)\), by the complete pointwise functor-category limit theorem and its opposite form in Compatible families, Section 4. Its element is a compatible family of finite cocones \(a^J\).

Restrict an \(I\)-cocone to all \(J\). Conversely set \(a_i=a_i^{\{i\}}\). For \(i\leq i'\), the finite cocone on \(\{i,i'\}\), and its restrictions to the singletons, give \(a_{i'}\alpha(i\leq i')=a_i\). This is an \(I\)-cocone. Compatibility of restrictions also gives \(a_j^J=a_j\) for every \(j\in J\), so reconstructing and then restricting recovers every finite cocone. At \(J=\varnothing\), there is only the empty cocone, also recovered. The reverse composite recovers each original \(a_i\).

Postcomposition by \(T\to T'\) commutes with both assignments. Precomposing the legs by a diagram transformation \(z:\alpha\to\alpha'\) also commutes with them, by the same component formula. The bijection is therefore natural in the target and in the diagram. Opposite variance turns this precise restriction isomorphism into (7.5). \(\square\)

Pointwise small Set limits provide the formal object in (7.5). They do not assert that it is representable in \(C\). Theorem 7.1 states sufficient actual existence hypotheses and proves its representing cocone.

Why two comma directions occur

The assembly has a useful index description. Define the subposet

with both projections. Its category laws and functor laws are inherited from the complete product/subcategory construction.

Proposition 7.3. The projection \(q\) is cofinal. For each \(J\in P\), the functor

is cofinal.

Proof of \(q\). The incoming comma at \(i_0\) consists of pairs \((i,A)\in K\) with \(i_0\leq i\). It contains \((i_0,\{i_0\})\). For any such \((i,A)\), its two arrows

belong to that comma: their \(I\)-coordinates preserve the incoming map from \(i_0\). Every object is therefore connected to the displayed fixed object. The comma is nonempty and connected. If \(I\) is empty, there are no incoming commas to test, so cofinality remains valid.

Proof of \(\xi_J\). Its target is an outgoing comma for \(p\). An object is \(((i,A),A\subseteq J)\), with \(i\in A\); an arrow has \(i\leq i'\), \(A\subseteq A'\subseteq J\). This differs from the literal fiber of \(p\), where \(A=J\).

For an object \(((i,A),A\subseteq J)\), an object of its incoming comma under \(\xi_J\) is exactly \(j\in J\) with \(i\leq j\). The morphisms are the order relations among these \(j\)'s. Indeed \(A\subseteq J\) already supplies the other coordinate, and the outgoing-comma triangle is unique. Thus this whole comma is the upper interval of \(i\) inside \(J\). Since \(i\in A\subseteq J\), it has initial object \(i\); the unique arrow \(i\to j\) is its initial-arrow factor. It is connected and nonempty.

The assignment (7.7) sends \(j\leq j'\) to the pair \((j\leq j',J=J)\), so it satisfies identities and composition. If \(J=\varnothing\), both \(J\) and \((p\downarrow J)\) are empty; cofinality is vacuous. This proves the proposition in every case. \(\square\)

Under Theorem 7.1's actual hypotheses, let \(B(J)\) be the outgoing-comma colimit of \(\alpha q\) over \((p\downarrow J)\). It exists: \(\xi_J\) is cofinal and the restricted diagram is \(\alpha|J\), so the complete either-direction existence theorem in Change an index, Corollary 2.2, transfers the existing \(L_J\).

Write its legs as \(\beta^J_{i,A}:\alpha(i)\to B(J)\), for \(i\in A\subseteq J\). Its canonical isomorphism \(e_J:B(J)\to L_J\) has

This family is a cocone because every \(i\leq i'\) lies in the full \(J\). On the \(\xi_J\) legs, the inverse cofinal comparison has the original \(\lambda_j^J\) as its test family, proving (7.9) for the stated canonical isomorphism.

The conditional outgoing-comma Kan construction makes \(B\) a \(P\)-diagram. If \(J\subseteq J'\), its map sends \(\beta^J_{i,A}\) to \(\beta^{J'}_{i,A}\). Testing these legs and (7.1) proves \(e_{J'}B(J\subseteq J')=r_{JJ'}e_J\). Thus \(e\) is natural.

Theorem 7.1 has already established \(\operatorname{colim}_I\alpha\). Cofinality of \(q\) now supplies \(\operatorname{colim}_K\alpha q\), and \(e\) transports the existing outer \(L\) to \(\operatorname{colim}_P B\). All pointwise Kan colimits and both total colimits needed by the retained total-assembly proof therefore exist. Its Hom-factor proof uses only those objects and the conditional left extension, so its exact argument applies here without requiring all small colimits of \(C\).

We obtain the canonical isomorphisms

The middle arrow is the total Kan unit comparison; the right arrow is the cofinal comparison for \(q\). At \((i,\{i\})\in K\), the first two maps send its coprojection to \(l_{\{i\}}\lambda_i^{\{i\}}=c_i\), whereas the right map sends it to \(b_i\). Thus the composite from \(L\) to the final object is exactly \(\Phi\) of (7.3), by universal uniqueness. The retained natural Kan and cofinal comparisons give the same naturality already checked directly.

The formal version can instead apply these constructions in \(C^\vee\), where the relevant small colimits exist pointwise after opposites. Its compatibility identity is (7.5). In both versions, \((p\downarrow J)\) is outgoing, and the commas testing \(\xi_J\)'s cofinality are incoming.

8. Existence and preservation from finite operations

Theorem 8.1. The following assertions hold.

1. If \(C\) has finite coproducts and small filtered colimits, it has all small coproducts.
2. If \(C\) has finite colimits and small filtered colimits, it has all small colimits.
3. In the setting of 1, a functor \(F:C\to C'\) preserving finite coproducts and small filtered colimits preserves all small coproducts.
4. In the setting of 2, a functor preserving finite colimits and small filtered colimits preserves all small colimits.

Finite operations include the empty ones. Preservation means that the image of the specified universal cocone is universal, so it includes existence of the corresponding image colimit. No independent hypothesis that \(C'\) has all small colimits is needed.

Existence, retaining the complete construction. For a small family \((X_s)_{s\in S}\), let \(P_S=\operatorname{Fin}(S)\) and choose

with legs \(\iota_s^J\). The maps \(r_{JJ'}\) are characterized by \(r_{JJ'}\iota_s^J=\iota_s^{J'}\); their laws are the complete universal-family laws in Compatible families. The empty stage is initial. The small filtered colimit

exists by assumption, with legs \(l_J\).

The full factor argument immediately after formula (3.2) in Limits and colimits of formal objects, Section 3, identifies compatible maps from finite partial coproducts with arbitrary maps \(X_s\to T\). Its proof uses only the finite coproduct properties and this outer colimit property, so it applies verbatim as an interface to the present arbitrary category \(C\), without an ind-category hypothesis. It says that (8.1) is the coproduct with individual legs

For \(S=\varnothing\), it is the initial empty stage. This proves 1 using that complete existing factor.

If finite colimits exist, coequalizers and finite coproducts exist by Universal forks, Section 9. Part 1 now gives small coproducts, and that lesson's complete object/arrow coproduct and coequalizer construction in Section 8 gives every small diagram colimit. Both presentation families are small because both the object and arrow sets of its index category are small. The construction includes the empty diagram. This proves 2.

There is also a finite-poset route to the general reduction. For \(D:A\to C\) with \(A\) small, the complete general poset replacement of Section 6 gives a small cofinal \(\pi:R\to A\). It need not be filtered. Apply Theorem 7.1 to \(D\pi\): finite colimits give its finite stages, and small filtered colimits give their outer colimit. It is \(\operatorname{colim}_R D\pi\). The complete cofinal existence theorem supplies \(\operatorname{colim}_A D\) and the canonical isomorphism

Its composite with each original finite-stage leg is the \(D(\pi r)\) leg of the \(A\)-colimit. Theorem 7.1 and the retained cofinal naturality prove naturality of this precise map. This supplies the general-small-diagram bridge without replacing \(R\) by a filtered poset.

Preservation of small coproducts, with the actual maps. Choose finite partial coproducts \(Q'_J\) of \(F(X_s)\) in \(C'\). They exist because \(F\) preserves finite coproducts. Their legs are \(\iota'{}^J_s\), and their canonical transitions are \(r'_{JJ'}\). The finite preservation comparisons are the isomorphisms

For \(J\subseteq J'\), testing all these legs gives

For the empty \(J\), both sides are the unique map from its initial object, so (8.5) still holds. Thus \(e\) is a natural isomorphism of the finite-partial diagrams.

Filtered preservation says that \(F(Q)\), with legs \(F(l_J)\), is a colimit of \(J\mapsto F(Q_J)\). Equations (8.4)–(8.5) transport this to a colimit of \(J\mapsto Q'_J\), with legs \(F(l_J)e_J\). The same complete finite-partial-coproduct factor used for 1 therefore makes \(F(Q)\) a coproduct of the family \(F(X_s)\). Its individual legs are

Hence the image of the original coproduct cocone is universal. The actual canonical map

is invertible, proving 3. For a family map \(X_s\to Y_s\), testing (8.7) on every original summand proves that its square commutes. Thus these canonical comparisons are natural in the family.

Preservation of arbitrary small colimits. Use the exact presentation already proved in Universal forks, Section 8. For \(D:A\to C\), write

The resulting original cocone has legs \(b_x=q\iota_x\). Its full colimit factor and empty case are the retained proof, not a new construction here.

Part 3 makes \(F(U),F(V)\) the corresponding coproducts of the image values, with their actual \(F(\kappa)\) and \(F(\iota)\) legs. For chosen image-family coproducts \(U',V'\), let \(e_U:U'\to F(U)\), \(e_V:V'\to F(V)\) be their canonical isomorphisms. Form the presentation maps \(a',b':U'\rightrightarrows V'\) for \(FD\). Testing every arrow summand gives

For the second equation, functoriality gives \(F(\iota_yD(x\to y))=F(\iota_y)F(D(x\to y))\), as required by (8.8); every diagram arrow is retained.

Finite-colimit preservation makes \(F(q)\) a coequalizer of \(F(a),F(b)\). The map \(F(q)e_V\) is a coequalizer of \(a',b'\): for \(v:V'\to T\) with \(va'=vb'\), (8.9) says that \(ve_V^{-1}\) equalizes \(F(a),F(b)\). It factors uniquely through \(F(q)\), and hence \(v\) factors uniquely through \(F(q)e_V\). Conversely such a factor equalizes the pair. This proves the actual transported coequalizer property.

The complete presentation factor now makes \(F(B)\) the colimit of \(FD\), with legs

The canonical comparison from any chosen \(\operatorname{colim}_A FD\) to \(F(B)\), characterized by these legs, is therefore invertible. This proves 4.

For a transformation \(z:D\to D'\), both routes in the naturality square, tested on an image-diagram leg, equal \(F(b'_xz_x)\). Universal uniqueness proves the square. Empty object or arrow families cause no exception: their coproduct factors are unique initial-object maps, and the empty diagram has its initial colimit.

The same last argument proves the useful unnumbered recall: finite colimits and small coproducts already imply all small colimits, and preserving those two sorts of operations implies preserving all small colimits. The filtered hypothesis is needed to obtain the small coproducts above; it is not needed once those coproducts are available. \(\square\)

9. Four graded exercises with full solutions

Exercise 1 — introductory: which arrows are missing?

(a) For \(\phi:\mathbb N\to\mathbb N\), \(\phi(n)=2n\), describe its incoming comma at \(i\), and check the two arrow tests.

(b) Let \(G\) be the one-object category with arrow group \(\{1,g\}\), \(g^2=1\), and let \(T\) be the terminal category. Examine \(G\to T\): cofinality, the two tests without the filtered-source hypothesis, and the fully faithful receiving criterion.

(c) Repeat the last comparison for the functor from the discrete two-object category to \(T\). State which of fullness and faithfulness fails in each example.

Solution. In (a), \((i\downarrow\phi)\) is the ordered tail

It has least element \(\lceil i/2\rceil\). It is filtered: the maximum of two integers is a receiver and parallel arrows are unique. For reaching, choose \(n=\lceil i/2\rceil\). Any pair \(i\rightrightarrows2n\), when present, consists of the same unique order arrow, so \(t=1_n\) satisfies the second test. Since \(\mathbb N\) is filtered, Theorem 2.1 applies. The functor is also fully faithful: \(2n\leq2m\) exactly when \(n\leq m\). Every \(i\) reaches an even image, so Proposition 3.2 applies with its hypotheses checked.

In (b), the sole incoming comma is \(G\) itself: there is a unique structural map to the sole target object and both group arrows satisfy its triangle. This category is nonempty and connected, so the functor is cofinal. The reaching test holds. Every parallel pair in \(T\) is its identity repeated, so its second test is satisfied by the identity of \(G\). Nevertheless \(G\) is not filtered. If an arrow \(h\) equalized \(1,g\), then \(h=hg\); composing by \(h^{-1}\) would give \(1=g\), a contradiction. Thus the filtered-source assumption in Theorem 2.1 cannot be omitted.

The Hom map \(\{1,g\}\to\{1\}\) is surjective and not injective. This functor is full and not faithful. Its target is filtered and has the receiving property, but its source is not filtered. Faithfulness in the lifted equality (3.1) is therefore a necessary hypothesis of Proposition 3.2.

In (c), every Hom map to \(T\) is injective: a singleton maps injectively to a singleton, and an empty set maps injectively to a singleton. Thus the functor is faithful. It is not full on the two distinct objects, because their empty Hom set does not surject onto the target singleton. Its sole incoming comma is the discrete two-object category, which is nonempty and disconnected. Hence it is not cofinal; its source also has no common receiver for those objects. Every target object still reaches an image. This shows why the fullness hypothesis is needed to lift receiving arrows. The two counterexamples isolate the roles of fullness and faithfulness, rather than substituting object reachability for either one.

Exercise 2 — intermediate: all four induced functors

Take \(I=J=K=\mathbb N\) with the usual order, and set

Check cofinality of both functors. For \(j=2\), \(k=3\), and the arrow \(u:3\leq\psi(2)=4\), describe each of the four functors of Theorem 4.1, taking \(i=3\) in part 2. Compute their incoming commas and exhibit their least elements. Explain why \(\phi\) need not be full, and identify the chain that derives part 2 from part 4.

Solution. Both functions are monotone, so they define functors of posets. The incoming comma for \(\phi\) at \(j\) is the tail \(n\geq2j\), and for \(\psi\) at \(k\) it is the tail \(m\geq\lceil k/2\rceil\). Each is nonempty with its stated least element and is connected; hence both functors are cofinal.

Here \(A_2\) consists of \(n\geq4\), because \(2\leq\lfloor n/2\rfloor\). Part 1 is its inclusion into \(I=\mathbb N\). At \(x\in I\), the incoming comma is the tail \(n\geq\max\{4,x\}\), with least element \(\max\{4,x\}\).

For part 2, \((3\downarrow I)\) is the tail \(n\geq3\). The target \(A_{\phi(3)}=A_1\) is the tail \(n\geq2\). The functor retains \(n\), so it is the inclusion of the first tail into the second. At \(n_0\geq2\), its incoming comma is the tail \(n\geq\max\{3,n_0\}\), with least element \(\max\{3,n_0\}\).

For part 3, \(B_3\) consists of \(n\geq4\): \(3\leq2\lfloor n/2\rfloor\) is equivalent to that bound. The category \(E_3\) consists of \(m\geq2\), and \(F_3(n)=\lfloor n/2\rfloor\). At \(m_0\geq2\), its incoming comma has objects \(n\) with \(m_0\leq\lfloor n/2\rfloor\), hence exactly \(n\geq2m_0\), with least element \(2m_0\). Its arrow relation is the same inherited order; no extra morphisms appear.

For part 4, \(U_u:A_2\to B_3\) retains \(n\). Its structural inequality is the composite

Both source and target are the tail \(n\geq4\), so this is their identity functor. At \(n_0\geq4\), its incoming comma is the tail \(n\geq n_0\), with least element \(n_0\). All four functors are cofinal.

The functor \(\phi\) is not full: there is no arrow \(3\to2\) in its source, but its two image objects both equal \(1\), so the target has their identity arrow. The theorem only requires cofinality and filtered categories, which we checked.

To obtain part 2 from part 4, use

choose the middle object \(3\), the last-category object \(\phi(3)=1\), and \(u=1_1\). This gives exactly the tail inclusion from \(n\geq3\) to \(n\geq2\) computed above. Using the chain \(I\to J\xrightarrow{1_J}J\) would instead give an identity on \(A_{\phi(3)}\), and would miss this nonidentity inclusion.

Exercise 3 — advanced: a large family of finite subsets

Let \(\mathcal U\) be the fixed universe and let \(I=\operatorname{Fin}(\mathcal U)\), ordered by inclusion, in a larger ambient universe.

(a) Check filteredness and local \(\mathcal U\)-smallness. Prove that \(I\) is not cofinally \(\mathcal U\)-small.

(b) Explain what changes for \(\operatorname{Fin}(A)\) when \(A\) is \(\mathcal U\)-small.

(c) In the receiving-subset criterion, identify the bound on the full arrow set and the hypothesis that supplies it. Can smallness of the receiving object set replace that hypothesis?

Solution. The category in (a) is an ambient set-category: its objects are a subset of the power set of the set \(\mathcal U\), and its arrows are the inclusion relation. It contains the empty subset, and union gives a finite common upper bound. Parallel arrows are unique, so it is filtered. Every Hom set is empty or a singleton, and hence \(\mathcal U\)-small.

Suppose \(I\) were cofinally \(\mathcal U\)-small. Proposition 5.1 would give a \(\mathcal U\)-small receiving family \(S\) of finite subsets of \(\mathcal U\). For each \(u\in\mathcal U\), the object \(\{u\}\) would have an arrow to some \(A_u\in S\), so \(u\in A_u\). Consequently

Every finite \(A\) is small. The complete small-indexed-union bound in Universes, Proposition 3.1, would make the left side of (9.1) small. But Universes, Lemma 1.1, proves by the full Cantor argument that \(\mathcal U\) is not \(\mathcal U\)-small. This contradiction proves (a). Enlargement of the ambient universe makes the construction legitimate but does not change this working-universe size failure.

In (b), the complete transported power-set bound makes \(\operatorname{Fin}(A)\) small as an object set. Its arrows are a subset of the square of that small set, so the category is small. It is filtered by the same empty-subset, union and thinness checks. Its identity is a small cofinal witness. This conclusion also holds for empty \(A\), when the sole object is \(\varnothing\).

For (c), the full arrow set on \(S\) is the disjointly tagged family

Smallness of \(S\times S\) supplies the index bound. Local \(\mathcal U\)-smallness supplies the bound on every summand, and the existing tagged-union theorem supplies the total bound. A small object set alone does not bound these summands. The complete one-object nonsmall-monoid counterexample in Change an index, Section 5, already proves that even a small cofinal witness need not produce a small full cofinal subcategory when local smallness is removed. Thus the local-size convention is an essential part of the receiving-subset criterion.

Exercise 4 — challenge: disconnected indices and actual coprojections

Let \(I=\{0,1\}\times\mathbb N\), with \((\epsilon,n)\leq(\epsilon',n')\) exactly when \(\epsilon=\epsilon'\) and \(n\leq n'\). Define a set diagram by

with inclusion transition maps.

(a) Show that \(I\) is not filtered. Compute the finite-stage colimits \(L_J\), all maps \(r_{JJ'}\), and the outer colimit over finite subsets. Verify its original \(I\)-coprojections and its universal property, including the empty stage.

(b) Put \(a=(0,0)\), \(b=(1,2)\), and \(J_0=\{a,b\}\). List the objects and nonidentity arrows of the outgoing comma \((p\downarrow J_0)\) from Section 7. Compare it with the literal fiber of \(p\). Check every incoming comma of \(\xi_{J_0}\).

(c) Let \(E=\{\mathrm{red},\mathrm{blue}\}\), and let \(F(X)=E\times X\). Prove that \(F\) preserves finite coproducts and small filtered colimits. For an arbitrary small family \((X_s)_{s\in S}\), write the finite-partial coproduct comparison, its transition squares, and the resulting full coproduct comparison on actual elements. Include empty families and naturality in the family.

(d) For the discrete category \(C\) with distinct objects \(A,B\), let a discrete two-object index send its objects to \(A,B\). Explain precisely which formal finite-stage statement still holds and which actual colimit object does not exist.

Solution. In (a), \((0,0)\) and \((1,0)\) have no common receiver, so \(I\) is not filtered. For each finite \(J\), let \(M_\epsilon(J)\) be the greatest \(n\) for which \((\epsilon,n)\in J\), when such an \(n\) exists. Then

The original finite-stage legs are inclusions. In each nonempty branch of \(J\), its greatest index is terminal, so a cocone on that branch is uniquely determined by its map on the largest prefix. The two branches have no connecting arrows. Hence the maps on their disjoint prefixes combine uniquely into a map from (9.2), proving the finite-stage colimit property. For \(J=\varnothing\), this formula gives \(\varnothing\), the initial set, with its unique empty cocone.

If \(J\subseteq J'\), each existing maximum can only increase, and a previously absent branch may appear. Thus \(r_{JJ'}\) is the inclusion \(L_J\subseteq L_{J'}\); it agrees with every finite-stage leg. These inclusions satisfy the identity and composition laws, including maps out of the empty stage.

The outer colimit is \(L=\{0,1\}\times\mathbb N\), with inclusions \(l_J:L_J\to L\). Every \((\epsilon,m)\) belongs to the stage for \(J=\{(\epsilon,m)\}\). If maps \(h_J:L_J\to T\) are compatible, define \(h(\epsilon,m)\) using that singleton stage. Any other stage containing the point agrees after passing to its union with the singleton stage, so this prescription gives \(hl_J=h_J\). Every point occurs in a stage, proving uniqueness. Its original \(I\)-legs are exactly the inclusions \(\alpha(\epsilon,n)\to L\), since they are \(l_{\{(\epsilon,n)\}}\lambda^{\{(\epsilon,n)\}}_{(\epsilon,n)}\). For an original \(I\)-cocone, the finite-stage factorizations are compatible by their legs; the outer factorization just proved gives its unique map from \(L\). This checks the comparison of Theorem 7.1 directly for a nonfiltered index.

In (b), an outgoing-comma object is \(((i,H),H\subseteq J_0)\), where \(i\in H\). There are exactly four:

Their only nonidentity arrows are \((a,\{a\})\to(a,J_0)\) and \((b,\{b\})\to(b,J_0)\). The coordinates \(a,b\) are incomparable, so there are no other arrows. The literal fiber \(p^{-1}(J_0)\) has only the two upper objects; it is a different category.

The functor \(\xi_{J_0}\) sends \(a\) to \((a,J_0)\) and \(b\) to \((b,J_0)\). At either object with first coordinate \(a\), its incoming comma contains just the source object \(a\) with its unique structural arrow. At either object with first coordinate \(b\), it contains just \(b\). Each of the four commas is terminal, so \(\xi_{J_0}\) is cofinal. The value \(L_{J_0}\) has the four elements \((0,0),(1,0),(1,1),(1,2)\). Nothing in this calculation uses filteredness of \(I\).

For (c), realize a coproduct as a disjointly tagged union. The finite comparison is

Its inverse sends \((e,(s,x))\) to \((s,(e,x))\). This also treats \(J=\varnothing\), when both sets are empty, and it identifies each coprojection with its image under \(F\). Thus \(F\) preserves finite coproducts with their actual legs.

For a small filtered set diagram \(D\), the canonical comparison is

Transitions retain the color \(e\), so this is well defined. A representative of \([i,x]\) gives a preimage of \((e,[i,x])\). If two images agree, their colors agree and their second coordinates have a common witness in the diagram \(D\), by the complete result in Filtered stages, Section 2. At that witness, the pairs with this same color agree as well. The same common-witness criterion for the filtered diagram \(E\times D\) proves equality of their source classes. Hence (9.4) is bijective. Its inverse is \((e,[i,x])\mapsto[i,(e,x)]\); the same witness check proves independence of the representative. This is an application of the retained equality theorem, and establishes filtered-colimit preservation without an additional equality construction.

For \(J\subseteq J'\), both routes in the finite-partial transition square send \((s,(e,x))\), \(s\in J\), to \((e,(s,x))\) in the larger tagged union. Therefore

The outer comparison of Section 8 consequently has the actual formula

with the inverse already displayed. On the \(s\)-coprojection it is \(F(j_s)\), exactly as required for the canonical comparison. For a family of maps \(z_s:X_s\to Y_s\), both naturality routes send \((s,(e,x))\) to \((e,(s,z_s(x)))\). This checks naturality, every transition square, and the original legs. Empty \(S\), empty \(X_s\), and the empty partial stage all give the appropriate empty sets and unique maps. Applied to (a), the comparison for the assembled diagram retains the color and the point \((\epsilon,m)\), agreeing with every image-diagram leg.

Finally, in (d) an actual coproduct of \(A,B\) cannot exist. Its underlying object would have to be either \(A\) or \(B\), and the other object has no arrow to it, so there is no cocone with that vertex. Even the empty-stage actual colimit is unavailable: the discrete two-object category has no initial object.

Nevertheless its formal cocone functor is defined. For either \(T=A\) or \(T=B\),

The empty-stage formal functor is the constant singleton, and a singleton stage is the corresponding representable \(C(A,-)\) or \(C(B,-)\). In \(\operatorname{Fun}(C,\mathrm{Set})^{\mathrm{op}}\), the filtered formal colimit over finite stages is the pointwise inverse limit of these functors. Since the full two-object stage belongs to the indexing poset and is its greatest element, that inverse limit is \(Q_\alpha\), the constant empty functor. Its existence in the functor completion makes no assertion that it, or the empty stage, is represented by an object of \(C\). This is exactly the formal conclusion of Proposition 7.2 and the reason for the separate existence hypotheses of Theorem 7.1.