Draft lesson.

Common arrows and bounded Set comparisons

Written with GPT-6.1 Sol (OpenAI), October 2026. Original exposition: CC0. Linked complete proof interfaces retain their terms.

A compatible family can satisfy each equation eventually without satisfying them all at a single stage. The missing ingredient is one transition arrow through every witness. We construct that arrow under a cardinal bound, transport cofinality, and exchange a bounded limit with a filtered colimit of sets. The exercises distinguish a bound on objects from a bound on all arrows and ordinary filteredness from its stronger cardinal form.

1. Prerequisites and cardinal bounds

The set-theoretic prerequisites are Choice, cardinal comparison and the theorem \(\lambda\cdot\lambda=\lambda\) for every infinite cardinal \(\lambda\). The complete cardinal-foundation proof supplies the infinite square and its actual bijection; its preceding sections establish Choice, well-ordering and cardinal comparison.

For \(\pi=\aleph_0\), a finite union of finite sets is finite. For \(\pi>\aleph_0\) and two sets \(A,B\) of cardinal below \(\pi\), put \(\theta=\max(|A|,|B|,\aleph_0)<\pi\). Their tagged union injects into \(\{0,1\}\times\theta\), which injects into \(\theta\times\theta\). Thus \(|A\sqcup B|\leq\theta<\pi\); induction gives the finite-family bound. This uses infinite-cardinal multiplication and does not assume regularity of \(\pi\).

The complete universe transport and set operations fix smallness, products, tagged unions, subsets and quotients. The filtered quotient and one-arrow equality test supplies eventual equality at a fixed stage. We use the compatible-tuple limit, including its projections, factor map, naturalities and empty tuple. The incoming-comma theorem and full receiving-subcategory proof fix the direction of cofinality and its whole-arrow tests.

Identities inject a category's objects into its total morphism set. If the latter has cardinal below a working-universe cardinal, the whole category is small up to the retained encoding of labels. The Set comparison therefore lives in the working universe. An explicitly chosen enlargement licenses calculations with larger ambient category data without changing any original arrow.

Fix ambient Choice, a working Grothendieck universe and an infinite working-universe cardinal \(\pi\). We use only the cardinal facts that a finite union of sets of cardinal below \(\pi\) remains below \(\pi\), and that identities inject the object set of a category into its total morphism set. Regularity is not assumed. Category data can be ambient sets: an enlargement licenses set-valued calculations but does not replace the original arrows by arrows in another category.

The two axioms for \(\pi\)-filteredness are: every set of fewer than \(\pi\) objects has a receiver, and every set of fewer than \(\pi\) parallel arrows is equalized by a single arrow from their common target. The empty object family forces nonemptiness. The empty parallel family is equalized by the identity. These axioms imply ordinary filteredness because finite sets have cardinal below \(\pi\).

2. One arrow through many witnesses

Let \(t_a:x\to y_a\), for \(a\in A\), be fewer than \(\pi\) arrows in a \(\pi\)-filtered category. If \(A\) is empty, take \(h=1_x\). Otherwise receive the objects \(y_a\) at \(z\), with arrows \(r_a:y_a\to z\). Equalize the parallel family \(r_at_a:x\to z\) by one \(q:z\to w\). Fix \(a_0\in A\) and put \(h=qr_{a_0}t_{a_0}\). Then \(s_a=qr_a\) satisfies \(s_at_a=h\) for every \(a\). Repeated object labels and repeated composites only reduce cardinality; there is no transfinite iteration or regularity assumption.

3. Receive every arrow of a bounded diagram

Suppose \(D:K\to I\), with fewer than \(\pi\) total morphisms of \(K\), and assume the two axioms above for \(I\). If \(K\) is empty choose any object of \(I\); the empty family is a cocone. Otherwise receive all \(D(k)\) at \(z\), writing \(r_k:D(k)\to z\). For each actual arrow \(f:k\to k'\), ordinary filteredness gives \(t_f:z\to z_f\) equalizing \(r_{k'}D(f)\) and \(r_k\). The star lemma gives one \(h:z\to w\) factoring through every \(t_f\). Consequently \(hr_{k'}D(f)=hr_k\) for every arrow, including identities and composites. Thus \((hr_k)_k\) is a cocone on the given functor. In this criterion the bound is on the total morphism set, including identities and composites. Counting generators alone does not verify that stated bound.

Conversely, a cocone on a discrete category with fewer than \(\pi\) objects gives the object-family axiom. For a parallel-arrow set \(B\), form a category with two distinct objects, one arrow for each \(b\in B\) from the first to the second, and the two identities. Its total arrow count remains below infinite \(\pi\). A cocone gives a single postarrow from the second object equalizing all the given arrows. This also works for empty \(B\). The empty test category supplies nonemptiness. The criterion therefore holds for every infinite \(\pi\), not just regular cardinals.

The ordinary and monotone cases. If \(\pi'\leq\pi\) are infinite cardinals, every \(\pi'\)-bounded object or parallel-arrow test is a \(\pi\)-bounded test. Thus \(\pi\)-filteredness implies \(\pi'\)-filteredness. At \(\pi=\aleph_0\), the two axioms require receivers for finite object families and one equalizer for finite parallel-arrow families, including the empty object family. The complete finite filteredness criterion identifies these with ordinary filteredness in both directions. Nothing here asserts a terminal object.

4. Transfer the bound through a cofinal functor

Let \(\phi:J\to I\) be cofinal and \(J\) \(\pi\)-filtered. The complete incoming-comma theorem supplies nonempty filtered \((i\downarrow\phi)\) for every \(i\), and ordinary filteredness of \(I\). It is used with its full arrow test, rather than as a title-level citation.

For a bounded family of objects \(i_a\) choose arrows \(u_a:i_a\to\phi(j_a)\). Receive the bounded \(j_a\) in \(J\), then apply \(\phi\) and compose the given \(u_a\). For an empty family use any object of \(J\) and its image.

For a nonempty bounded set \(B\subseteq I(i_1,i_2)\), fix \(u_1:i_1\to\phi(j_1)\) and \(u_2:i_2\to\phi(j_2)\) once. For each \(s\in B\), filteredness of \((i_1\downarrow\phi)\) gives arrows \(a_s:j_1\to k_s\) and \(b_s:j_2\to k_s\) with

Receive the \(k_s\) at \(k\), with \(r_s:k_s\to k\). Equalize the entire parallel family \(r_sa_s:j_1\to k\) by \(q:k\to k'\). Next equalize the entire family \(qr_sb_s:j_2\to k'\) by \(q':k'\to k''\). Their final common values are \(a:j_1\to k''\) and \(b:j_2\to k''\), respectively. Applying the same postcompositions to the displayed equations yields \(\phi(a)u_1=\phi(b)u_2s\) for every \(s\). Therefore the single arrow \(v=\phi(b)u_2:i_2\to\phi(k'')\) equalizes \(B\). For empty \(B\) take \(1_{i_2}\).

Synchronizing only the \(j_1\) family would not prove this conclusion: the \(i_2\) leg must be fixed for all \(s\). The second equalization is essential.

5. Keep a small full receiving subcategory

Call \(I\) cofinally small if some small category \(A\) admits a cofinal functor \(\theta:A\to I\). By the complete receiving-set criterion, the object image \(S=\{\theta(a):a\in\operatorname{Ob}A\}\) is small and receives an arrow from every object of \(I\). Indeed, \(S\) is bijective to the quotient of \(\operatorname{Ob}A\) by equality of image, so the proved universe quotient and transport bounds give smallness. For each \(i\), nonemptiness of \((i\downarrow\theta)\) supplies \(i\to\theta(a)\). This argument does not require \(A\) to be \(\pi\)-filtered. For a cofinally small, locally small, \(\pi\)-filtered \(I\), choose this \(S\) and apply the following construction.

Suppose \(I\) is locally small and \(\pi\)-filtered, and a working-universe-small object set \(S\) receives an outgoing arrow from every object of \(I\). The full subcategory \(I_S\) is small in total arrows: its arrow set is the tagged union of the small Hom sets over \(S\times S\). An ambient receiver of any bounded family in \(S\) can be moved farther into \(S\); fullness retains all resulting legs. Similarly an ambient equalizer of a bounded parallel family can be moved farther into \(S\), and its composite remains an equalizer. Nonemptiness follows by moving one ambient object into \(S\). Thus \(I_S\) is \(\pi\)-filtered. Its ordinary cofinality follows from the complete full-receiving-subcategory proof. This reasoning does not apply the transfer lemma in its invalid reverse direction.

6. Exchange bounded limits with filtered colimits

Let \(K\) have fewer than \(\pi\) total arrows and let \(I\) be a small \(\pi\)-filtered category. Work in a universe \(\mathcal U\) containing the cardinal representative \(\pi\), with both index categories \(\mathcal U\)-small, and let \(D:K^{\mathrm{op}}\times I\to\mathsf{Set}_{\mathcal U}\) have \(\mathcal U\)-small values. If an enlargement is needed, choose it to contain the complete index, value and map data; then use the proved universe transport on their encodings. The tuple limits and quotient colimits constructed below are small in that same universe. Denote \(L_i=\lim_{k\in K^{\mathrm{op}}}D(k,i)\) and \(Q_k=\operatorname{colim}_{i\in I}D(k,i)\). The compatible-tuple limit and the filtered quotient construction give the canonical map

Every structural transport in \(I\) preserves the tuple equations by functoriality in both variables, so \(i\mapsto L_i\) is a functor. The quotient cocone equations make the displayed formula well-defined. For an arrow in \(K^{\mathrm{op}}\), the tuple compatibility and commutation of the two functor variables give target compatibility. A natural transformation of \(D\) commutes with \(\Lambda\) coordinatewise. Restricting the limit along a functor of shapes gives the same commutative comparison; no claim about unrestricted changes of the colimit index is needed.

For injectivity, receive the stages of two given representatives at one stage \(i_0\). Their images agree in the target precisely when each pair of coordinates has equal classes in \(Q_k\). The complete same-stage equality theorem then gives one outgoing witness \(t_k:i_0\to i_k\) for each coordinate. There are fewer than \(\pi\) coordinates, because their identities are distinct total arrows of \(K\). Apply the star lemma to obtain one \(h:i_0\to i_1\) factoring through every \(t_k\). The two transported tuples are equal in every coordinate at \(i_1\), hence their colimit classes are equal.

For surjectivity, start with an actual compatible target tuple \((\xi_k)_k\). Choose representatives \(\xi_k=[i_k,x_k]\), and receive the bounded family of stages at \(i_0\). Replace all \(x_k\) by their transports to \(i_0\). For each actual arrow \(f:k\to k'\) of \(K^{\mathrm{op}}\), compatibility means the classes of \(D(f,i_0)(x_k)\) and \(x_{k'}\) agree in \(Q_{k'}\). The same-stage theorem supplies an outgoing witness \(t_f:i_0\to i_f\) for that equality. Apply the star lemma to the bounded TOTAL arrow family to obtain one \(h:i_0\to i_1\) through every \(t_f\). At \(i_1\) the transported coordinates satisfy every actual arrow equation, so they define an element of \(L_{i_1}\) mapped to the given tuple. Injectivity proves independence of all representative and witness choices, so the inverse of this canonical bijection is natural.

If \(K\) is empty, every \(L_i\) is a singleton and the target is a singleton. The filtered index \(I\) is nonempty and connected, so the colimit of these singletons is a singleton; this proves the empty-shape case directly. Empty coordinate sets cause no hidden choice: lifts are selected only after an actual tuple or class is given. No regularity of \(\pi\), finite limits in \(I\), additive structure or abelian hypotheses enter the proof.

7. Four graded exercises with full solutions

Exercise 1. (foundation) Retain the starting leg

Let \(I\) be \(\pi\)-filtered and \(x\in I\). Prove that \((x\downarrow I)\) is \(\pi\)-filtered and its projection to \(I\) is cofinal. Give actual arrows for both conclusions.

Solution. An object is \(s:x\to i\). An arrow from \(s:x\to i\) to \(t:x\to j\) is \(f:i\to j\) with \(fs=t\). Identities and composition satisfy these equations by the laws in \(I\).

For a bounded family \(s_a:x\to i_a\), receive the \(i_a\) at \(z\) by arrows \(r_a:i_a\to z\). If the family is nonempty, equalize the bounded parallel family \(r_as_a:x\to z\) by \(q:z\to w\). Its common value \(u:x\to w\) is a coslice object, and each \(qr_a\) is an actual coslice arrow \(s_a\to u\). For an empty family the object \(1_x\) supplies the receiver.

For parallel coslice arrows \(f_b:(i,s)\to(j,t)\), use one ambient \(h:j\to k\) equalizing their bounded underlying family. It is a coslice arrow \((j,t)\to(k,ht)\), and its composites are the equal \(hf_b\). Empty families use the target identity. These are the two cardinal-filtered axioms in the coslice.

For the projection \(p\), any \(y\in I\) has a common receiver with \(x\). Choose \(u:x\to z\) and \(v:y\to z\); then \(v:y\to p(z,u)\) supplies the first incoming-comma test. Given \((j,s)\) and arbitrary \(a,b:y\rightrightarrows j\), choose \(e:j\to j'\) with \(ea=eb\). The actual coslice arrow \(e:(j,s)\to(j',es)\) gives the second test. The source coslice is filtered, so the retained whole-arrow cofinality theorem proves \(p\) cofinal. No terminal object or bound on the entire ambient category was used.

Exercise 2. (intermediate) Coordinatewise stability at different stages

Let \(K\) be small with \(|\operatorname{Ob}K|<\pi\), let \(I\) be small and \(\pi\)-filtered, and let \(D,D':K^{\mathrm{op}}\times I\to\mathsf{Set}\). Let \(\theta:D\to D'\) be natural. For each \(k\), suppose there is \(i_k\) such that \(\theta_{k,i}\) is bijective whenever an arrow \(i_k\to i\) exists. Prove that

is bijective. Does this argument require \(|\operatorname{Mor}K|<\pi\)?

Solution. Receive the bounded family \(i_k\) at \(i_0\). For empty \(K\), choose any \(i_0\in I\). At every target \(i\) of an arrow \(i_0\to i\), the composites \(i_k\to i_0\to i\) show that all \(\theta_{k,i}\) are bijective. For an arrow \(f:k\to k'\) in \(K^{\mathrm{op}}\), naturality and cancellation give

Thus the coordinatewise inverse sends compatible tuples to compatible tuples and inverts the limit map at \(i\). Naturality in every arrow of \(I\) makes these inverse tuple maps commute with stage transitions.

Restrict the index along the cofinal projection \((i_0\downarrow I)\to I\) from Exercise1. The two limit diagrams become naturally isomorphic on this coslice, with the actual inverse maps just constructed. Their colimits are isomorphic by the inverse cocone-factor property. Cofinality identifies these with the original colimits and identifies their comparison with the displayed map. Hence that map is bijective.

Only the object family \(i_k\) needed cardinal synchronization. Naturality of the given transformation already handles every arrow of \(K\), so this assertion needs no total-arrow bound. Smallness of \(K\) still gives its Set limits. In the empty case those limits are singletons throughout.

Exercise 3. (intermediate) A countable family defeats ordinary filteredness

Take \(I=\mathbb N\) with its usual order and the discrete category \(K=\mathbb N\). Define

with inclusions along \(i\leq j\). Compute the image of the canonical product comparison. Explain its failure for \(\pi=\aleph_1\), despite ordinary filteredness of \(I\) and \(|\operatorname{Mor}K|<\pi\).

Solution. The discrete shape has only identities, so its total arrow set is countable. At stage \(i\), a tuple has arbitrary bits in coordinates \(0,\ldots,i\) and zero in larger coordinates. Transition maps include it into the next such product and are injective. The filtered colimit is therefore all binary sequences with finite support: every stage tuple has finite support, and any finite support fits in one initial segment. Equality is literal equality after inclusion, by the retained filtered quotient test.

For each fixed \(n\), the diagram eventually becomes the constant set \(\{0,1\}\), so its colimit is \(\{0,1\}\). The target product is all binary sequences. The canonical comparison includes the finite-support sequences into that product. The constant sequence of ones has no preimage.

The category \(I\) is nonempty, finite subsets have upper bounds, and parallel arrows in this thin category are already equal. Hence it is ordinarily filtered. It is not \(\aleph_1\)-filtered: its countable family of all objects has no receiver. This violates exactly the stronger filteredness hypothesis while retaining the total-arrow bound on the discrete shape.

Exercise 4. (advanced) Two objects can hide infinitely many equations

Let \(K\) have objects \(a,b\), countably many arrows \(u_n:a\to b\), and only the two identities besides them. Let \(I\) be the finite subsets of \(\mathbb N\), ordered by inclusion. Put \(E=\{\ast\}\sqcup\mathbb N\). At \(F\in I\), let \(D(a,F)\) be the quotient identifying exactly \(\ast\) and the members of \(F\). Put \(D(b,F)=\{t\}\), and let \(D(u_n^{\mathrm{op}},F)\) send \(t\) to the class of \(n\). Use quotient maps when \(F\subseteq G\). Prove that these data give a functor and that its limit–colimit comparison is not surjective, despite the finite object set of \(K\).

Solution. Quotient maps send the class of each \(n\) at \(F\) to its class at \(G\), and therefore commute with every \(u_n^{\mathrm{op}}\) map. Identity arrows act identically. No two nonidentity arrows are composable in \(K\); thus these equations verify every functor law and every commutation between its two variables.

At a finite \(F\), a limit element would be a pair \((x,t)\) with \(x=[n]_F\) for all \(n\). Two distinct numbers outside \(F\) retain distinct classes, so no such \(x\) exists. Every stage limit and their colimit are empty.

The colimit of \(D(a,-)\) is a singleton: a class of \(n\) at \(F\) becomes the class of \(\ast\) at \(F\cup\{n\}\). Any two stage elements therefore agree after a common finite union and these identifications. The diagram has elements, so this colimit is nonempty. The colimit of \(D(b,-)\) is the singleton \(\{t\}\). All induced maps between these singleton colimits are unique, so their \(K^{\mathrm{op}}\)-limit is a singleton. The comparison is the empty function into that singleton and is not surjective.

The category \(I\) is small and ordinarily filtered, hence \(\aleph_0\)-filtered: finite unions receive finite families, and thinness equalizes parallel arrows. The two objects of \(K\) have cardinal below \(\aleph_0\), but its TOTAL arrow set is countably infinite. Every \(u_n\) imposes its own equation. The theorem's total-arrow hypothesis cannot be replaced by an object-count hypothesis.

8. Mathematical prerequisites

Section1 links complete proofs of the universe constructions, filtered quotient and same-stage equality, compatible-tuple limit, and ordinary incoming-comma and full receiving-subcategory theorems. Sections2–6 prove cardinal synchronization, two-family cofinal transfer, cardinal closure of a small full receiving subcategory, and the bounded Set comparison, using the stated maps and hypotheses. The cardinal prerequisites are logically separate from those categorical arguments.