Draft lesson.

Small predecessors and maps into filtered colimits

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

A well-order can have a long upper tail even when its small initial segments form a useful indexing category. A different indexing category consists of small subsets of a set. Regularity supplies the bounds for the two filteredness proofs. The preservation property studied afterward is defined for every infinite cardinal: it asks whether a map into a filtered colimit already comes from one stage, with equality detected at a common later stage.

1. The size conventions and retained constructions

We work in ZF with set-indexed Choice, the universe-enlargement axiom stated in the cardinal companion, and an explicitly chosen Grothendieck universe \(\mathcal U\). The complete universe operations and transport maps supply subsets, products, tagged unions, quotients and encodings. A set is \(\mathcal U\)-small if it is bijective to a member of \(\mathcal U\); this does not assert that its original labels belong there. A small category has a small total arrow set, including identities and composites. Its object set is small because \(i\mapsto1_i\) is injective. A locally small category need only have small Hom sets.

Let \(\pi\in\mathcal U\) be an infinite cardinal represented by its initial ordinal. We retain the complete well-order and cardinal comparison arguments, the infinite square and union bounds, and the definition of regularity from the cardinal companion. In particular, finite unions of sets of cardinal below \(\pi\) still have cardinal below \(\pi\), for every infinite \(\pi\). Regularity is the stronger assertion

The ordinary union has the same bound: well-order \(S\), and send each point of the union to its copy at the least index containing it. This is the explicit ordinary-union injection retained in the cardinal companion. An empty family gives an empty union. We use regularity only where it is stated below.

We also retain the complete bounded filteredness criterion. A category is \(\pi\)-filtered precisely when every set of fewer than \(\pi\) objects has a receiver and every set of fewer than \(\pi\) parallel arrows is equalized by one arrow from their common target. The empty object family forces nonemptiness; an empty parallel family can use the target identity. The criterion is equivalent to receiving every diagram whose total arrow set has cardinal below \(\pi\). It holds for arbitrary infinite \(\pi\). At \(\aleph_0\) it is ordinary filteredness, and \(\pi\)-filteredness implies \(\pi'\)-filteredness when \(\pi'\leq\pi\).

The size of a \(\pi\)-filtered category itself need not be below \(\pi\). The cardinal bound concerns the test diagrams it must receive. Whenever we use a small index, its entire arrow set is \(\mathcal U\)-small. For set-valued calculations, values and structural functions are encoded in the same universe. If an enlargement is required, choose it explicitly to contain the whole index, value and map data. The original category and its arrows stay fixed.

2. Small predecessors in an arbitrary well-order

Assume in this section that \(\pi\) is regular. Let \(J\) be any well-ordered set with \(|J|\geq\pi\). We do not require \(J\) to be an initial ordinal or to be \(\mathcal U\)-small. Define

Give \(I\) its inherited order. As a category its arrows are exactly the pairs \((x,y)\) with \(x\leq y\); identity is \((x,x)\), and composition sends \((x,y),(y,z)\) to \((x,z)\). Thus every pair of objects has at most one arrow, and the category laws follow from the order laws.

Proposition 2.1. The subset \(I\) is downward closed, is well-ordered by its inherited order, is \(\pi\)-filtered, and has no supremum in itself.

Proof. If \(x\leq y\) and \(y\in I\), then \(P_x\subseteq P_y\), so \(|P_x|<\pi\). This proves downward closure. Every nonempty subset of \(I\) is also a nonempty subset of \(J\), whose least point lies in that subset; hence the inherited order is a well-order.

For a family \(S\subseteq I\) with \(|S|<\pi\), put

Regularity and the retained tagged-union injection give \(|T|<\pi\). Since \(|J|\geq\pi\), the complement is nonempty, so its least point \(b\) exists. If \(b<s\) for some \(s\in S\), then \(b\in P_s\subseteq T\), a contradiction. Therefore every \(s\leq b\). Moreover any \(y<b\) must belong to \(T\), by minimality of \(b\) in its complement. Thus \(P_b\subseteq T\), which proves \(b\in I\). The actual cocone legs are the order arrows \((s,b)\). For \(S=\varnothing\), \(T=\varnothing\) and \(b\) is the first point of \(J\), with empty predecessor set. This supplies the required empty-family witness.

Every parallel family consists of equal arrows whenever it is nonempty, because the category is thin. Postcomposition by the target identity equalizes it, and also equalizes the empty family. The retained two-axiom criterion proves \(\pi\)-filteredness.

Suppose \(I\) had a greatest point \(m\). If \(m\) were also greatest in \(J\), then \(J=P_m\cup\{m\}\) would have cardinal below \(\pi\), by the finite-union bound, contradicting \(|J|\geq\pi\). Otherwise the nonempty upper tail has a least point \(n>m\). There is no point strictly between them, so \(P_n=P_m\cup\{m\}\). The same bound gives \(n\in I\), contradicting maximality of \(m\). Thus \(I\) has no greatest point. A supremum of the whole set \(I\) lying in \(I\) would be a greatest point, so no such supremum exists. \(\square\)

There is also an exact size description without restricting \(J\). The complete ordinal collapse of a well-order gives an order-isomorphism \(r:J\to\beta\). Since \(|J|\geq\pi\), the ordinal \(\beta\) is at least \(\pi\). The map \(r\) sends \(P_x\) bijectively to the ordinal \(r(x)\). By initiality of \(\pi\),

Consequently \(r|_I:I\to\pi\) is an order-isomorphism, with inverse the restriction of \(r^{-1}\). Encode an arrow \((x,y)\) as \((r(x),r(y))\). All arrows then form a subset of \(\pi\times\pi\) belonging to \(\mathcal U\); source, target, identity and composition transport through these bijections. Identities inject \(\pi\) into that arrow set, and the retained square theorem bounds it above by \(\pi\). Thus \(I\) is small and its total arrow cardinal is exactly \(\pi\), not below \(\pi\). This size calculation uses the square theorem, not regularity.

For example, let \(J\) be a copy of \(\omega\) followed by one new point. With \(\pi=\aleph_0\), \(I\) is the copy of \(\omega\). Its supremum exists in \(J\), at the new point, but does not belong to \(I\). This is compatible with Proposition 2.1. The construction does not assert that \(I\to J\) is cofinal.

3. Small subsets with every inclusion arrow

Again assume \(\pi\) regular. For a \(\mathcal U\)-small set \(A\), let

Its order is inclusion, and its category includes every inclusion arrow, not only a selected generating family. Its identities and compositions have the pair formulas of Section 2.

Proposition 3.1. This is a small \(\pi\)-filtered category. The empty subset is its least object.

Proof. First suppose \(A\in\mathcal U\). Objects form a subset of \(\mathcal P(A)\), and total arrows form the subset

The complete universe closure results put both sets in \(\mathcal U\). The source, target, identity and composition graphs are subsets of products of these member sets, so the whole category data have universe-member encodings.

If only a bijection \(a:A\to A_0\in\mathcal U\) is given, send \(B\) to \(a(B)\), and \((B,C)\) to \((a(B),a(C))\). Their inverses use \(a^{-1}\). Cardinalities and inclusions are preserved; identities and composition commute with these formulas. This is an isomorphism of the whole categories, giving the claimed smallness even for external labels.

For \(S\subseteq\mathcal P_{<\pi}(A)\) of cardinal below \(\pi\), the union

has cardinal below \(\pi\), by regularity and the ordinary-union bound. It is an object and receives every \(B\) by the actual inclusion \(B\subseteq W\). If \(S\) is empty, \(W=\varnothing\) still is an object. Thinness makes the target identity a common equalizer for any parallel family, including the empty one. Apply the retained filteredness criterion. The inclusions from \(\varnothing\) show that it is least. \(\square\)

Smallness here does not imply a cardinal bound below \(\pi\). If \(|A|\geq\pi\), the singleton assignment \(a\mapsto\{a\}\), followed by the identity assignment, injects \(A\) into the total arrow set. For \(\pi=\aleph_0\), this category is the finite-subset category of \(A\); it is filtered regardless of how large the small set \(A\) is.

4. The canonical comparison for maps into a colimit

Return to an arbitrary infinite \(\pi\). Let \(C\) be locally \(\mathcal U\)-small and admit colimits of all small \(\pi\)-filtered diagrams. This existence condition does not ask for any other colimits or finite operations in \(C\). Let \(I\) be a small \(\pi\)-filtered category and \(\alpha:I\to C\). Choose a colimit \(L\) with its coprojections \(c_i:\alpha(i)\to L\). The complete cocone universal property and structural maps supply this convention and the maps induced by natural transformations.

For \(X\in C\), the covariant set diagram is

We retain the complete Hom functor and encoding proof. In particular its identity and composite laws, and its natural bijections to universe-member Hom values, apply to (4.1). Thus the Set colimit below exists in the same universe. The complete filtered quotient construction supplies its classes and actual transition relations.

The cocone equations \(c_j\alpha(s)=c_i\) define the canonical function

Indeed a transition sends \(f\) to \(\alpha(s)f\), and \(c_j\alpha(s)f=c_i f\). The retained quotient factor property therefore gives one well-defined function on classes. The symbols \(f\) and \(c_i\) are arrows of \(C\); no underlying elements of its objects are assumed. In an encoded Hom set, decode \(f\), form \(c_i f\), then encode the result. Cancellation of the retained bijections verifies precisely the same formula and equations.

For a natural transformation \(\eta:\alpha\to\beta\), write \(c_i'\) for the latter colimit cocone and \(\ell:L\to L'\) for its induced map, so \(\ell c_i=c_i'\eta_i\). Naturality of \(\eta\) gives the induced Set-colimit map \([i,f]\mapsto[i,\eta_i f]\). The comparison commutes with these maps because

For an arrow \(u:X'\to X\), precomposition gives \([i,f]\mapsto[i,fu]\) on the source and \(g\mapsto gu\) on the target. These maps commute with (4.2), since \(c_i(fu)=(c_i f)u\). Identities and composites act as identities and composites by the retained Hom laws and quotient formulas. Thus the comparison is natural in \(\alpha\) and contravariant in \(X\).

Choosing a different colimit cocone changes (4.2) by the unique isomorphism \(v:L\to L'\) with \(vc_i=c_i'\), provided by that same universal-property interface. Explicitly \(\chi'_{X,\alpha}=\operatorname{Hom}_C(X,v)\chi_{X,\alpha}\). Its bijectivity is therefore independent of the choice.

Definition 4.1. An object \(X\) is \(\pi\)-accessible if (4.2) is bijective for every small \(\pi\)-filtered \(I\) and every \(\alpha:I\to C\). Write \(C_\pi\) for the full subcategory on these objects: for any such \(X,Y\), its Hom set is exactly \(\operatorname{Hom}_C(X,Y)\), with the original identities and composition. No skeleton or smallness of its entire object set is part of this definition.

The two parts of bijectivity have concrete meanings. Surjectivity says every \(g:X\to L\) can be written \(c_i f\) for a stage map \(f:X\to\alpha(i)\). Injectivity says that if \(c_i f=c_j g\), then there are actual arrows \(p:i\to k\), \(q:j\to k\) with

The equivalence of equality of quotient classes and this common-stage witness is the complete retained filtered equality theorem. Empty Hom values do not introduce any choice: a representative is selected only for an existing class or an existing target map.

5. Discrete categories and changing the cardinal

Proposition 5.1. Every object of a discrete category is \(\pi\)-accessible for every infinite \(\pi\).

Proof. A \(\pi\)-filtered index is nonempty and ordinarily filtered. If \(\alpha:I\to C\) has discrete target, every arrow \(s:i\to j\) is sent to an identity, so \(\alpha(i)=\alpha(j)\) literally. Given any two objects of \(I\), arrows from them to a common receiver make their values equal. Hence \(\alpha\) is literally constant at one object \(Y\), and every arrow is \(1_Y\).

A filtered category is connected: each pair has a two-leg zigzag through a receiver. The complete connected constant-diagram argument gives its colimit as \(Y\), with all coprojections \(1_Y\), in any ambient category. Thus a discrete \(C\) has all the colimits required here, without needing other colimits.

For fixed \(X\), the Hom diagram is constant at \(\operatorname{Hom}_C(X,Y)\), a singleton when \(X=Y\) and empty otherwise. The same connected constant-diagram proof in sets gives its colimit as that set. Formula (4.2) is the identity under this identification, since every \(c_i=1_Y\). It is bijective in both cases. If \(C\) is empty, there are no objects \(X\) and no functors from nonempty filtered indices to \(C\); the statement and the required existence condition are vacuous. In all cases \(C_\pi=C\) as full categories. \(\square\)

Proposition 5.2. Let \(\pi'\leq\pi\) be infinite cardinals. If \(C\) admits all small \(\pi'\)-filtered colimits, then it admits all small \(\pi\)-filtered colimits and \(C_{\pi'}\) is a full subcategory of \(C_\pi\).

Proof. A \(\pi\)-filtered index is \(\pi'\)-filtered by the retained monotonicity criterion. Its colimit therefore exists by the stated hypothesis. For a \(\pi'\)-accessible \(X\), the canonical comparison for any such index is already bijective. It is the very same function (4.2), not a replacement comparison. This proves \(\pi\)-accessibility. Fullness of both subcategories retains all ambient arrows between the selected objects, yielding the full inclusion. No regularity is used. \(\square\)

More generally the implication applies whenever the two definitions and their requisite comparisons are defined; it does not manufacture missing colimits from the object property alone.

At \(\pi=\aleph_0\), the retained criterion identifies the indices with ordinary filtered categories. Formula (4.2) is exactly the definition of finite presentation. Hence \(C_{\aleph_0}\) is the full category of objects of finite presentation, under the stated filtered-colimit existence hypothesis. No finite limits in \(C\) have been inserted.

One may ask for an additional condition ensuring that every object belongs to some \(C_\pi\). That is a further property of \(C\), to be established under further hypotheses. The definition and propositions here do not assert an unconditional union theorem or assign an accessibility cardinal to every object.

6. Four exercises with complete solutions

1. (introductory) A well-order with a visible upper tail

Let \(J\) consist of symbols \(a_n\), for \(n\in\omega\), followed by three symbols \(t_0<t_1<t_2\). Order the \(a_n\) by their indices and put each below every \(t_j\). Take \(\pi=\aleph_0\). Compute \(I\) from (2.1), its receivers for finite families, and its supremum in \(J\). Is \(I\to J\) cofinal in the incoming-comma convention?

Solution. The predecessors of \(a_n\) are exactly \(\{a_0,\ldots,a_{n-1}\}\), of cardinal \(n\). The predecessors of each \(t_j\) contain every \(a_n\), so have infinite cardinal. Therefore \(I=\{a_n:n\in\omega\}\).

A nonempty finite family of these objects has the receiver \(a_m\), where \(m\) is the largest occurring index; its legs are the unique order arrows. For the empty family take \(a_0\). All parallel arrows agree, so their target identity supplies every required equalizer. No \(a_n\) is greatest, since \(a_{n+1}\) is larger. In \(J\), every upper bound of \(I\) is one of \(t_0,t_1,t_2\); their least is \(t_0\). Thus \(\sup_J I=t_0\notin I\).

In the incoming-comma convention, cofinality requires every incoming comma to be nonempty and connected. An object of the incoming comma \((t_0\downarrow I)\) would be an arrow \(t_0\to a_n\) in \(J\). None exists, so that comma category is empty and the inclusion fails the incoming-comma nonemptiness test. This conclusion needs no unproved cofinality theorem. The tagged set \((\omega\times\{0\})\sqcup(3\times\{1\})\) encodes \(J\) in \(\mathcal U\), and all order arrows are a subset of its square.

2. (intermediate) Prescribing elements of every small subset

Let \(\pi\) be regular, let \(A\) be small and let \(T\subseteq A\) be arbitrary. Consider the full inclusion category

Determine when it is empty. When nonempty, prove it is small and \(\pi\)-filtered, give a least receiver for a below-\(\pi\) family, and determine when it has a terminal object. Include all empty-set cases.

Solution. If \(|T|\geq\pi\), every superset has cardinal at least \(\pi\), so \(P_T\) is empty and is not \(\pi\)-filtered. If \(|T|<\pi\), \(T\) itself is an object, so the category is nonempty. Its objects are a subset of those in Section 3; all arrows are the corresponding inclusion pairs. The same subset and product bounds, transported by a bijection of \(A\) to a universe member if necessary, prove smallness of the whole category.

Assume \(|T|<\pi\). For \(S\subseteq P_T\) with \(|S|<\pi\), put

Regularity bounds the union of the \(B\)'s below \(\pi\), and the finite-union bound includes \(T\). Thus \(W\in P_T\). Every member includes into \(W\). Any receiver must contain \(T\) and all these members, hence must contain \(W\); it is the least receiver. For \(S=\varnothing\), it is \(T\). The identity of the common target equalizes any parallel family by thinness. This proves \(\pi\)-filteredness.

If \(|A|<\pi\), then \(A\) is an object and is terminal, since every object includes into it. Conversely suppose \(B_*\) terminal. For every \(a\in A\), the set \(T\cup\{a\}\) has cardinal below \(\pi\), by the finite-union bound, so it is an object. Its inclusion into \(B_*\) shows \(a\in B_*\). Hence \(B_*=A\), forcing \(|A|<\pi\). When \(A=\varnothing\), necessarily \(T=\varnothing\), and the sole object is both least and terminal. When \(T=\varnothing\) and \(A\) is arbitrary, the empty family is received by \(\varnothing\); the same terminal classification still applies.

3. (intermediate) A retract retains the preservation property

Let \(C\) admit small \(\pi\)-filtered colimits, with arbitrary infinite \(\pi\). Suppose there are arrows \(e:X\to Y\), \(r:Y\to X\) with \(re=1_X\). If \(Y\) is \(\pi\)-accessible, prove that \(X\) is \(\pi\)-accessible by giving an actual inverse to its comparison. Do not assume any finite operations in \(C\).

Solution. Fix any small \(\pi\)-filtered \(I\), any \(\alpha:I\to C\) and its colimit \(L\). Write \(Q_X=\operatorname{colim}_I\operatorname{Hom}(X,\alpha(-))\), and define \(Q_Y\) likewise. Precomposition and its naturality from Section 4 give functions

They are well-defined because precomposition commutes with every stage transition. Their composite satisfies \(ER[i,f]=[i,fre]=[i,f]\), so \(ER=1_{Q_X}\). On the target Hom sets define \(R_L(f)=fr\) and \(E_L(g)=ge\); then \(E_LR_L=1_{\operatorname{Hom}(X,L)}\). Naturality of (4.2) gives

Since \(\chi_Y\) is bijective, define

The two composites are

Thus \(\psi_X\) is the required inverse for this arbitrary diagram. The quantified definition proves \(X\) accessible. Empty Hom sets cause no exception: all maps and inverse equations are functions between the stated sets. Regularity and finite categorical operations were never used.

4. (advanced) Exactly which sets are accessible at a regular cardinal

Let \(\pi\in\mathcal U\) be regular and infinite. Prove that an object \(X\in\mathsf{Set}_{\mathcal U}\) is \(\pi\)-accessible precisely when \(|X|<\pi\). Identify its comparison explicitly. Indicate which direction continues to hold without regularity.

Solution. Suppose first \(|X|<\pi\), and take an arbitrary small \(\pi\)-filtered set diagram \(\alpha\). In the complete bounded Set comparison, take \(K\) to be the discrete category on \(X\). Its total arrows are its identities, so their cardinal is \(|X|<\pi\). Take \(D(x,i)=\alpha(i)\); arrows in the \(I\) direction act by \(\alpha\), and the only \(K\) arrows act identically. Universe closure puts the complete index, value and map data in the stated encoding.

The retained compatible-tuple proof identifies the discrete limit at stage \(i\) with functions \(X\to\alpha(i)\): a tuple \((z_x)_{x\in X}\) is the function \(x\mapsto z_x\), with inverse evaluation at each \(x\). The bounded comparison therefore gives a bijection

In the quotient model the colimit coprojection is \(c_i(z)=[i,z]\). Hence (6.7) is exactly \(\chi_{X,\alpha}\). Other colimit choices give the same result by the compatible isomorphism from Section 4. The complete bounded comparison includes empty \(K\), so \(X=\varnothing\) is included. This direction uses no regularity of \(\pi\).

For the converse use the small \(\pi\)-filtered category \(P=\mathcal P_{<\pi}(X)\) of Section 3 and the diagram taking \(B\) to the actual set \(B\), with every inclusion map. We identify its Set colimit with \(X\) by

This function is well-defined because every transition is inclusion. It is onto: each \(x\in X\) lies in the stage \(\{x\}\). It is injective: representatives \((B,x),(C,x)\) map to the same point in the common stage \(B\cup C\), which has cardinal below \(\pi\) by the finite-union bound. Different points cannot be identified, since \(\theta\) is well-defined and keeps their values. Thus it is a bijection. It intertwines every coprojection with the actual inclusion \(B\hookrightarrow X\), so it identifies their colimit cocones. For \(X=\varnothing\), the sole stage is the empty set and the same map is the unique empty bijection.

Under this identification, the comparison takes a function \(f:X\to B\) at a stage \(B\) to its composite \(X\to B\hookrightarrow X\). If \(X\) were accessible, its identity would be such a composite for some \(B\in P\). The equation forces every \(x\in X\) to belong to \(B\), hence \(B=X\), so \(|X|<\pi\). If \(|X|\geq\pi\), the identity is an explicit target map outside the image and the comparison is not surjective. The use of Section 3 to make \(P\) \(\pi\)-filtered is where this converse uses regularity; we make no converse assertion for singular cardinals.

7. Prerequisites

The cardinal companion supplies well-orders, initial cardinal representatives, the square bijection, finite-union bounds and regularity. The bounded-comparison companion supplies the complete arbitrary-cardinal filteredness criterion and the bounded Set interchange. The universe, Hom, cocone, constant-diagram, tuple-limit and filtered-quotient proofs supply their actual maps and size bounds. The residual arguments above keep those constructions and add only the two poset verifications, the precise preservation comparison and its elementary consequences.