Draft lesson.

Cardinal Ind presentations and their realization

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

A formal colimit is a presheaf, even when its diagram has no colimit in the original category. Its elements remember enough arrows to recognize a cardinal-filtered presentation. We use that recognition to assemble an outer diagram of formal objects, construct realization using only the colimits actually available in the target, and test the resulting comparison with a detecting family.

1. Conventions and complete prerequisites

Fix ZF, set-indexed Choice, the explicit universe-enlargement axiom, and a Grothendieck universe \(\mathcal U\). Let \(\pi\in\mathcal U\) be an arbitrary infinite cardinal. No regularity assumption is made on this parameter. Categories are locally \(\mathcal U\)-small and have ambient sets of object and arrow data. A larger universe contains the functor categories when needed; small indices continue to mean \(\mathcal U\)-small indices. The universe transport proofs and cardinal companion supply all set encodings, finite-union bounds, square bijections and set-indexed choices used below. Exercises 3–4 specialize to \(\pi=\aleph_0\); its regularity is the finite-union calculation in the cardinal companion.

We retain the complete whole-arrow filteredness criterion and cofinal transfer, its small full receiving-witness construction, and its bounded Set comparison. The tests involve fewer than \(\pi\) total arrows, including identities and composites. The index of a small \(\pi\)-filtered colimit may itself have at least \(\pi\) arrows. The complete criterion implies ordinary filteredness and includes the empty diagram.

We use the cardinal-foundation, bounded-exchange, accessibility and bounded-colimit results in the companion lessons linked below.

Write \(h_X(T)=\operatorname{Hom}_C(T,X)\), with the complete Hom encodings and Yoneda maps understood. For a presheaf \(B\), its element category \(E_B\) has objects \((X,b)\), with \(b\in B(X)\), and arrows

Its identity, composition and size properties are proved in the element-category construction. Its Homs are subsets of those in \(C\). We use the ordinary Ind recognition and presentation-to-elements cofinality, together with the density cocone and universe bridge. These constructions include every element arrow and every original density leg.

An ordinary ind-object is a small filtered presheaf colimit of representables. We use the full subcategory

Morphisms are all presheaf natural transformations between the selected objects. The complete pointwise colimit factors and formal Hom formula supply the actual class maps, both variances and both naturalities. No colimits in \(C\) are presumed by (1.2).

2. A represented object and a broader recognition theorem

Represented-object statement. Let \(A\in C\), regarded as its represented presheaf \(h_A\). Then \(h_A\) belongs to \(\operatorname{Ind}^{\pi}(C)\) and has a small \(\pi\)-filtered presentation by objects of \(C\). In particular these two assertions are equivalent in this represented-object scope.

Proof. The element category of \(h_A\) is the slice \(C/A\). Its terminal object is \((A,1_A)\): from \((X,f:X\to A)\) the unique element arrow to it is \(f\). The retained universal-element proof identifies this terminal object with the represented presheaf. A terminal object receives every diagram by its unique structural arrows, whose composites agree by uniqueness. Thus this element category is \(\pi\)-filtered for every infinite \(\pi\), including the empty test. The one-object identity category is small and \(\pi\)-filtered by the same argument. The constant diagram at \(A\) on it has formal colimit \(h_A\), with its identity coprojection. This supplies the presentation and ordinary Ind membership, proving both assertions. \(\square\)

We now prove a genuinely broader statement, with a presheaf as input.

Theorem 2.1 (broader recognition). For \(B\in\operatorname{Ind}(C)\), the following are equivalent:

1. \(B\in\operatorname{Ind}^{\pi}(C)\).
2. There is a small \(\pi\)-filtered \(I\), a functor \(\alpha:I\to C\), and a specified isomorphism \(B\simeq\operatorname{colim}_{i\in I}h_{\alpha(i)}\) in presheaves.

Proof. For the first implication, the complete ordinary recognition theorem gives cofinal smallness of \(E_B\). Apply the retained general-cardinal full receiving-witness construction to its local smallness and \(\pi\)-filteredness. It gives a small full \(\pi\)-filtered \(W\subseteq E_B\). Its entire arrow set is the tagged union of the small element Homs over \(\operatorname{Ob}(W)^2\). Let \(\alpha\) forget each label and send every arrow to its actual \(C\)-arrow. The retained density theorem, followed by its complete cofinal cocone factor, gives

This is the specified natural density comparison, with all coprojections, rather than an abstract bijection of values.

For the converse, \(\pi\)-filteredness implies ordinary filteredness, so the given presentation is an ordinary Ind presentation. Under its isomorphism, let \(d_i\in B(\alpha(i))\) be the structural class of \(1_{\alpha(i)}\). The complete presentation-to-elements proof gives the functor

The quotient equation \(B(\alpha(s))d_{i'}=d_i\) makes every image an element arrow. Identity and composition are those of \(\alpha\). Its incoming comma at \((T,t)\) is the category of representatives \(T\to\alpha(i)\) of \(t\); the retained proof establishes nonemptiness and connectedness by all generating zigzags. Hence the functor is cofinal. General-cardinal cofinal transfer makes \(E_B\) \(\pi\)-filtered, including every below-\(\pi\) parallel family with one common target leg. This proves (1). \(\square\)

Neither theorem turns a quoted colimit into an actual \(C\)-colimit. The represented-object statement has an object \(A\in C\) as its input; Theorem 2.1 explicitly has the broader input \(B\in\operatorname{Ind}(C)\).

3. An outer colimit with a controlled inner witness

Theorem 3.1. Let \(I\) be small and \(\pi\)-filtered and let \(i\mapsto B_i\) be a functor into \(\operatorname{Ind}^{\pi}(C)\). Its pointwise presheaf colimit \(B\) belongs to \(\operatorname{Ind}^{\pi}(C)\). Its original presheaf coprojections are a colimit cocone in that full subcategory.

Proof. Use the retained pointwise theorem to construct

The complete theorem supplies all class relations, contravariant arrow laws and the unique natural factor. We prove the general-cardinal membership.

Let \(K\to E_B\) be any diagram with \(|\operatorname{Mor}(K)|<\pi\). Write its objects \((X_k,\xi_k)\) and its arrows \(x_f:X_k\to X_{k'}\), so

The underlying objects and arrows form a functor \(X:K\to C\). Define the whole two-variable Set diagram

Its \(K^{\mathrm{op}}\)-maps are \(B_i(x_f)\); its \(I\)-maps are the components of the outer transformations. Their commutation is precisely the naturality of those transformations. Thus all product arrows and functor laws are supplied by the retained diagram conventions. Since identities bound the object count of \(K\), the whole index is small after the retained transport. All values and map graphs have the same universe-member encodings.

The tuple \((\xi_k)_k\) lies in \(\lim_{K^{\mathrm{op}}}\operatorname{colim}_I D\) by (3.2). The complete bounded Set comparison is surjective. Its explicit class formula therefore gives a single \(i\) and elements \(b_k\in B_i(X_k)\) representing every \(\xi_k\), with all equations

already true at that stage. This is a lift of the entire functor to \(E_{B_i}\), not a collection of unrelated stage representatives. The comparison's construction synchronizes the actual total arrow family and uses no regularity.

Because \(E_{B_i}\) is \(\pi\)-filtered, that lifted diagram has a cocone to some \((Y,b)\), with actual arrows \(t_k:X_k\to Y\) satisfying

The same arrows give a cocone to \((Y,[i,b])\) in \(E_B\), since (3.1) preserves the equations. If \(K\) is empty, choose an outer stage using nonemptiness of \(I\), and then an object of its nonempty \(E_{B_i}\); this gives the empty cocone directly. We have received every whole-arrow bounded diagram. The retained criterion proves \(E_B\) \(\pi\)-filtered.

Here is a small witness, with its full arrow bound. For each outer \(i\), choose the small full receiving category \(W_i\subseteq E_{B_i}\) from Theorem 2.1. Set

The tagged set \(S\) is small by the complete small-family union and transport proof. This uses smallness of \(I\) and of each \(W_i\), without requiring either cardinal to be below \(\pi\). Let \(R\) be its object image, a small quotient of \(S\), and let \(W\subseteq E_B\) be full on \(R\). Choose one preimage in \(S\) for each image object. Then its actual total arrows inject into

Indeed tag an element arrow by the chosen source and target preimages and retain its underlying \(C\)-arrow. This is injective, since the tags recover its endpoints and element Homs are subsets of the indicated Homs. Consequently

The right side is \(\mathcal U\)-small by local smallness and the retained tagged-union proof. This is a bound on all arrows, including identities and composites; it is not a claim that their cardinal is below \(\pi\).

For any \((X,\xi)\in E_B\), choose a representative \(\xi=[i,b]\). The object \((X,b)\) in \(E_{B_i}\) has a receiving arrow to a member of \(W_i\). Its underlying arrow becomes an element arrow from \((X,\xi)\) to the corresponding member of \(R\). Thus \(W\) receives every object of \(E_B\). The complete general-cardinal receiving construction makes it \(\pi\)-filtered and cofinal. Density on this small witness gives a small \(\pi\)-filtered presentation of \(B\). In particular \(B\) is an ordinary ind-object and belongs to (1.2).

Finally the pointwise presheaf universal factor remains a unique factor among \(\operatorname{Ind}^{\pi}(C)\) objects, because that subcategory is full. This proves the colimit assertion, retaining every coprojection and its induced-map identity and composition laws. Empty values are allowed throughout: a tuple or class is lifted only after it exists. If \(C\) is empty, its Ind category is empty, so there is no diagram from a nonempty \(\pi\)-filtered \(I\) into it; the theorem is then vacuous. \(\square\)

4. Labels over a bounded colimit

Assume in this section that \(C\) admits colimits of every diagram whose total arrow cardinal is below \(\pi\). Fix \(B\in\operatorname{Ind}(C)\). For \(\phi:J\to C\) of that size, write \(L_\phi\) for its colimit and \(c_j:\phi(j)\to L_\phi\) for its coprojections. The compatible-tuple and cocone proofs define

For \(f:j\to j'\), its compatibility is \(B(\phi(f))B(c_{j'})b=B(c_{j'}\phi(f))b=B(c_j)b\). For \(\theta:B\to B'\), naturality at every \(c_j\) makes (4.1) commute with its components. For a transformation \(v:\phi\to\psi\), the induced colimit arrow \(\ell:L_\phi\to L_\psi\) satisfies \(\ell c_j=c'_jv_j\). Contravariance gives

This is naturality in the diagram, with its precise reversed direction. Colimit choices are related by the retained unique compatible isomorphism.

Theorem 4.1. The following are equivalent:

1. \(B\in\operatorname{Ind}^{\pi}(C)\).
2. Every comparison (4.1) is surjective.
3. Every comparison (4.1) is bijective.

Proof that (2) implies (1). Take a whole-arrow bounded diagram \(J\to E_B\). Forget its labels to obtain \(\phi:J\to C\). Its labels \(b_j\in B(\phi(j))\) are a compatible tuple, because each actual element arrow satisfies (1.1). Surjectivity supplies \(b\in B(L_\phi)\) with \(B(c_j)b=b_j\). Therefore every coprojection is an element arrow

Their whole cocone equations are the original \(C\)-colimit equations. This proves the bounded cocone criterion in \(E_B\). For empty \(J\), the colimit is the initial object \(0\), and the target tuple set is the singleton of empty tuples. Surjectivity forces \(B(0)\ne\varnothing\) and supplies a receiving object for this empty test. Thus \(E_B\) is \(\pi\)-filtered and (1) follows. The lifted element \(b\) is a label; no coprojection is asserted invertible.

Proof that (1) implies (3). Take a small \(\pi\)-filtered presentation \(B\simeq\operatorname{colim}_{i\in I}h_{\alpha(i)}\) from Theorem 2.1. The retained Hom-out-of-a-colimit factor and the complete bounded Set comparison give the following chain, in the direction of the canonical map:

The middle Set diagram is \((j,i)\mapsto\operatorname{Hom}_C(\phi(j),\alpha(i))\); its first variable is contravariant and its second covariant, with the retained commuting Hom actions. Its whole \(J\)-arrow count is below \(\pi\), so the companion theorem applies to arbitrary infinite \(\pi\). On a representative \([i,f:L_\phi\to\alpha(i)]\), the chain sends

Thus the chain is precisely \(\rho_{B,\phi}\), and its inverse is the inverse of that canonical map. The original comparison's naturality just proved makes this inverse natural too, by composing its squares with the two inverse functions. The empty \(J\) case is included in the bounded comparison. Statement (3) implies (2) for the same map simply because a bijection is surjective. \(\square\)

For a concrete distinction, let \(C=\mathsf{Set}_{\mathcal U}\), let \(J\) be discrete on two objects, and let both values of \(\phi\) be singletons. Take \(B=h_{\{0,1\}}\). The colimit is a two-element set, and (4.1) takes a function from it to \(\{0,1\}\) to its two chosen values. This is a bijection. Each coprojection from a singleton into the two-element colimit nevertheless fails to be an isomorphism. Surjectivity of the label map is about extending compatible functions, not inverting these arrows.

5. Realization when only the required colimits exist

Return to the hypothesis that \(C\) admits all small \(\pi\)-filtered colimits. Let \(D=C_\pi\) be the full subcategory of \(\pi\)-accessible objects, using the complete accessibility comparison and its naturalities. Thus \(D\) keeps every ambient \(C\)-arrow between its objects.

We first construct realization on \(\operatorname{Ind}^{\pi}(D)\). For each \(A\) choose an admissible small \(\pi\)-filtered presentation \(A\simeq\operatorname{colim}_{i\in I}h_{\alpha(i)}\), with \(\alpha:I\to D\), and put \(L_A=\operatorname{colim}_I\alpha\) in \(C\). Theorem 2.1 supplies such presentations. All choices are indexed by ambient sets, not by a global choice principle: small category encodings range over a set in a larger universe containing \(\mathcal U\), and their functors, presentation isomorphisms and colimit cocones also range over ambient sets. No arbitrary ordinary-filtered presentation is realized unless it meets this \(\pi\)-filtered requirement.

The complete intrinsic representation construction in Ind-objects, Section 3 supplies its Hom-factor mechanism. We apply that mechanism only to these admissible presentations. For \(Y\in C\), write \(h_Y|_D\) for the restricted Hom presheaf. The formal factor property, Yoneda and the actual \(C\)-colimit property give

Every arrow \(g:L_A\to Y\) corresponds to the transformation whose restriction to the \(i\)-th representable is postcomposition by \(gc_i^A\). This is natural in \(Y\). The right side is intrinsic to \(A\); its bijection to a small compatible-family limit also supplies its \(\mathcal U\)-smallness, even when the full object set of \(D\) is larger.

Let \(j_A:A\to h_{L_A}|_D\) be the transformation corresponding to \(1_{L_A}\) in (5.1). Its actual value is

A natural transformation \(\theta:A\to B\) defines a unique arrow

by the bijection (5.1) at \(Y=L_B\). The identity arrow satisfies this equation for \(\theta=1_A\). For \(\theta:A\to B\) and \(\zeta:B\to F\), the composite of their arrows satisfies it because

Uniqueness gives the identity and composition laws. Thus \(A\mapsto L_A\) is an actual functor \(\sigma_\pi:\operatorname{Ind}^{\pi}(D)\to C\).

A different admissible presentation gives \(L_A'\) and \(j_A'\), representing the same intrinsic right side of (5.1). The complete compatible representation-uniqueness theorem gives a unique isomorphism \(v_A:L_A\to L_A'\) with \(h_{v_A}j_A=j_A'\). For \(\theta:A\to B\), both \(v_B\sigma_\pi(\theta)\) and \(\sigma_\pi'(\theta)v_A\) have transformation \(j_B'\theta\) after \(j_A\); (5.1) makes them equal. Hence these comparisons are natural on every arrow. This constructs presentation-independent realization with only the target colimits in the hypothesis.

Theorem 5.1. This realization functor is fully faithful.

Proof. Let \(A\simeq\operatorname{colim}_Ih_{\alpha(i)}\) and \(B\simeq\operatorname{colim}_Jh_{\beta(j)}\) be admissible presentations. The exact Hom chain is

The first bijection is the complete formal Hom formula. The equality uses fullness of \(D\). For each \(i\), \(\alpha(i)\) is \(\pi\)-accessible and \(J\) is small and \(\pi\)-filtered, so its canonical comparison sends \([j,f]\) to \(c_j^Bf\) bijectively. Its contravariant naturality in \(\alpha(i)\) makes these bijections an isomorphism of the entire \(I^{\mathrm{op}}\)-diagram. Their componentwise inverse is natural by the retained natural-inverse theorem, so they induce the stated compatible-family bijection. The last bijection is Hom out of the actual colimit \(L_A\).

To identify the actual map, restrict \(\theta\) to its \(i\)-th stage. Yoneda gives an element \(b_i\in B(\alpha(i))\). If it is represented by \([j,f:\alpha(i)\to\beta(j)]\), the chain prescribes the leg \(c_j^Bf\) on \(\alpha(i)\). Equations (5.2)–(5.3) prescribe exactly the same leg for \(\sigma_\pi(\theta)c_i^A\). The unique actual colimit factor therefore is \(\sigma_\pi(\theta)\). This is a bijection on every Hom set, natural in both whole formal arguments through their actual composition laws. It proves full faithfulness.

No limit and colimit have been exchanged in (5.5): each inner comparison was replaced by a bijection before taking the limit. In particular the bounded interchange theorem is not a premise of this Hom chain. If \(C\) is empty, \(D\) and its Ind category are empty and the same fully faithful assertion is vacuous. \(\square\)

6. Detecting families and small costructured presentations

Now assume that \(C\) admits all small colimits. Let \((G_\nu)_{\nu\in N}\) be a small detecting family, each \(G_\nu\) \(\pi\)-accessible. Detection means that a \(C\)-arrow is invertible whenever each separate function \(\operatorname{Hom}_C(G_\nu,-)\) sends it to a bijection. We use exactly the complete separate-probe convention. We do not replace the outputs by their product.

For every \(X\in C\), assume additionally that the costructured category

is cofinally small. Its objects are labels \(u:Y\to X\), with \(Y\in D\); an arrow \((Y,u)\to(Z,v)\) is a \(D\)-arrow \(r:Y\to Z\) with \(vr=u\). The complete slice construction retains identities, composites, its forgetful functor and its locally small Hom subsets.

We retain the complete bounded accessible-colimit proof and costructured-filtering proof in the bounded-accessibility lesson. Colimits with fewer than \(\pi\) total diagram arrows of \(\pi\)-accessible objects remain \(\pi\)-accessible, and these colimits make \(D_X\) \(\pi\)-filtered. Both retained bodies include every arrow and the empty diagram. These are the two results used here. Cofinal smallness of \(D_X\) is the separate hypothesis above, not a consequence of detection or local smallness.

Theorem 6.1. Under these assumptions, \(\sigma_\pi:\operatorname{Ind}^{\pi}(C_\pi)\to C\) is an equivalence.

Proof. For \(X\), apply the complete general-cardinal receiving construction to \(D_X\). It gives a small full \(\pi\)-filtered \(W_X\subseteq D_X\). Write its objects as \((Y_w,u_w:Y_w\to X)\) and its arrows as their actual slice arrows. Its total arrow set is

Every summand is a subset of a small \(C\)-Hom, and the pair index is small. Hence all arrows are small, with the explicit bound by the corresponding tagged union of ambient Homs. Identities and composites are included. The underlying functor \(W_X\to D\) has a formal colimit \(A_X\) in \(\operatorname{Ind}^{\pi}(D)\), by Theorem 2.1. Realizing this particular presentation gives \(L_X^W=\operatorname{colim}_{w\in W_X}Y_w\). The slice equations make the \(u_w\) a cocone and give

The complete cofinal cocone theorem extends every original structural leg to the full \(D_X\); it supplies the same full-index universal property without demanding arbitrary large colimits in \(C\).

The functor \(\sigma_\pi\) may have chosen another admissible presentation of \(A_X\). Section 5 gives its unique presentation comparison \(v_X:\sigma_\pi(A_X)\xrightarrow{\sim}L_X^W\), with the corresponding formal legs preserved. Define \(\lambda_X=\lambda_X^Wv_X:\sigma_\pi(A_X)\to X\). We first prove that \(\lambda_X^W\) is invertible, and then use this transported comparison for the chosen functor.

Fix a generator \(G_\nu\). Its accessibility identifies the actual probe of \(\lambda_X^W\) with

For surjectivity, a given \(t:G_\nu\to X\) is an object \((G_\nu,t)\) of \(D_X\). The receiving property supplies an actual slice arrow \(a:(G_\nu,t)\to(Y_w,u_w)\), so \(u_wa=t\). Thus \([w,a]\) maps to \(t\). It is unnecessary for the original labelled generator object itself to belong to \(W_X\).

For injectivity, take representatives \([w,f]\), \([v,g]\) with equal images in (6.4). First receive their stages at one \(z\in W_X\), with slice arrows \(a:w\to z\), \(b:v\to z\). Put \(f'=af\), \(g'=bg\). Their composites with \(u_z\) agree. Form the categorical coequalizer of the pair

The walking parallel-pair category has exactly four total arrows: its two identities and its two parallel arrows. Thus its cardinal is below every infinite \(\pi\). The retained accessibility-closure theorem puts \(Q\) in \(D\), since both input objects are in \(D\). The full coequalizer factor gives \(\bar u:Q\to X\) with \(\bar u q=u_z\), so \(q\) is an arrow in \(D_X\). This uses no subtraction or additive structure.

Move \((Q,\bar u)\) to a member \((Y_t,u_t)\) of \(W_X\) by a receiving slice arrow \(r:Q\to Y_t\). The composite \(rq:Y_z\to Y_t\) satisfies \(u_t rq=u_z\). Fullness makes it an actual arrow of \(W_X\) from the old stage \(z\) to \(t\), even when \(Q\) is not an object of \(W_X\). Since \(rqf'=rqg'\), the two representatives agree after this transition and hence have the same class. This proves injectivity of (6.4).

Every separate probe of \(\lambda_X^W\) is therefore bijective. Detection makes it an isomorphism, so \(\lambda_X=\lambda_X^Wv_X\) is also an isomorphism. No inference from being monic and epic, no finite limits and no regularity were used. Full faithfulness is Theorem 5.1, and these isomorphisms for the chosen functor establish essential surjectivity.

For the actual equivalence, retain the complete chosen quasi-inverse construction and all functor checks. Here its chosen objects are \(A_X\) and its comparison is \(\lambda_X\). It defines the unique \(\tau(f):A_X\to A_Y\) by

The retained construction checks identities, composition, naturality of \(\lambda:\sigma_\pi\tau\to1_C\), and the lifted second natural isomorphism \(\tau\sigma_\pi\to1\). All choices are indexed by ambient sets as above. If the detecting family is empty, detection already says every \(C\)-arrow is invertible, so the conclusions about \(\lambda_X^W\) and \(\lambda_X\) hold vacuously; the nonempty witness still comes from \(D_X\)'s filteredness. \(\square\)

There is a useful concrete presentation attached to this proof. Apply the complete small-diagram coproduct/coequalizer construction to \(W_X\to C\). Its two maps are specified on every arrow summand:

The map \(P_0\to X\) with components \(u_w\) is their coequalizer, because the construction gives the colimit \(L_X^W\) and (6.3) identifies it with \(X\). Its factor property and naturality are exactly the retained construction's complete ones. Both indexing sets are small in total arrows. They need not have cardinal below \(\pi\), so this presentation makes no extra claim that the two large coproducts are \(\pi\)-accessible.

7. Four original exercises with complete solutions

Exercise 1. (introductory) Two isolated objects

Let \(C\) have exactly two objects \(c_0,c_1\) and only their identities. For an arbitrary infinite \(\pi\), classify \(\operatorname{Ind}^{\pi}(C)\). Explain why the presheaf coproduct \(h_{c_0}\coprod h_{c_1}\) is excluded, and describe a \(\pi\)-filtered outer diagram in the resulting category.

Solution. A presheaf on \(C\) is a pair of small sets \((S_0,S_1)\). Its element category is discrete on the tagged set

An empty discrete category fails the empty test for filteredness. A discrete category with two distinct objects fails the test consisting of their two identities: there is no common receiver. This test has two total arrows, fewer than every infinite \(\pi\). A discrete category with one object is \(\pi\)-filtered, because that object receives every diagram. Thus \(\pi\)-filteredness of the element category is equivalent to having exactly one element in (7.1). Such a presheaf is isomorphic to exactly one of \(h_{c_0}=(1,\varnothing)\) and \(h_{c_1}=(\varnothing,1)\), and its singleton presentation proves ordinary Ind membership. These are precisely the objects of \(\operatorname{Ind}^{\pi}(C)\), up to isomorphism.

There is no natural transformation from \(h_{c_0}\) to \(h_{c_1}\), since its \(c_0\)-component would be a map \(1\to\varnothing\). The reverse direction fails at \(c_1\). Each representable has only its identity endomorphism. More generally the full subcategory on all isomorphic copies has one arrow between any two copies of the same representable and no arrows across the two classes; the unique arrows within a class are invertible. This records the entire category, not merely its list of isomorphism classes.

The coproduct has values \((1,1)\), so its two element objects have no receiver. It is excluded. The presheaf with both values empty is excluded by the empty test.

If \(I\) is small and \(\pi\)-filtered, every two \(I\)-objects have a receiver. A functor from \(I\) to our full Ind subcategory must therefore take every object into the same one of the two classes. After the unique identifications with the corresponding \(h_{c_r}\), it is a constant diagram whose transition maps are identities. The retained quotient construction gives its pointwise colimit \(h_{c_r}\): nonempty connected \(I\) has a singleton colimit of singleton sets, and the other value remains empty. This explicitly illustrates Theorem 3.1. All element arrows in the classification are the identities of (7.1); the tag encoding bounds the total arrow set as well as the objects.

Exercise 2. (intermediate) One generator can conceal infinitely many arrows

Let \(J\) have one object and endomorphism monoid \((\mathbb N,+)\), and take \(\pi=\aleph_0\). Define \(\phi:J\to\mathsf{Set}_{\mathcal U}\) by \(\phi(*)=\mathbb N\), with \(m\) acting by \(n\mapsto n+m\). Compute its colimit and the comparison \(h_{\{0,1\}}(\operatorname{colim}\phi)\to\lim_{J^{\mathrm{op}}}h_{\{0,1\}}(\phi(*))\). Explain both why its coprojection need not be invertible and why this is not a finite-total-arrow instance of Theorem 4.1.

Solution. Identity \(0\) acts as the identity, and shift by \(m+m'\) is the composite of the shifts by \(m\) and \(m'\). Thus \(\phi\) is an actual functor. For a cocone \(t:\mathbb N\to T\), its equation at shift \(1\) is \(t(n+1)=t(n)\). Induction makes \(t\) constant, and every constant map satisfies all shift equations. The one-point set \(1\), with its constant coprojection \(c:\mathbb N\to1\), therefore has the unique factor property: a compatible \(t\) factors through the value \(t(0)\). This also computes the tagged quotient; any \(n,n'\) map to their common larger value through shifts, so they have one class.

The target limit is the set of functions \(f:\mathbb N\to\{0,1\}\) satisfying \(f(n+m)=f(n)\) for every \(m,n\). Each is constant, by \(m=1\); conversely the two constant functions satisfy all equations. The comparison sends \(a:1\to\{0,1\}\) to \(a c\). Its inverse sends an invariant \(f\) to the function on \(1\) with value \(f(0)\). These are inverse functions by the equations just proved. Postcomposition with any \(\{0,1\}\to T\) commutes with the formulas.

The coprojection \(c\) is not injective, since \(c(0)=c(1)\), and hence is not an isomorphism of sets. Bijection of the two label sets supplies no inverse to \(c\). Moreover \(\operatorname{Mor}(J)=\mathbb N\) has cardinal \(\aleph_0\), including all powers of the one generator and its identity. Its cardinal is not below \(\pi=\aleph_0\). This particular Hom comparison is bijective by the actual colimit property of a representable; it is not an application of the bounded-shape hypothesis. Counting one generator would give the wrong bound.

Exercise 3. (advanced) Realize shift and parity, including their composite

Take \(C=\mathsf{Set}_{\mathcal U}\), \(\pi=\aleph_0\), and \(D=C_\pi\). Use the complete finite-set accessibility calculation, which places all finite sets in \(D\). Let \(I\) be the poset of finite subsets of \(\mathbb N\), with inclusion arrows, and put \(A=\operatorname{colim}_{F\in I}h_F\) on \(D\). Construct the formal transformations given by shift \(s(n)=n+1\) and parity \(p(n)=n\bmod2\), and compute their realizations and their composite.

Solution. The complete finite-subset receiving construction makes \(I\) filtered: \(F\cup G\) receives \(F,G\), parallel arrows are equal, and \(\varnothing\) receives the empty test. Its objects form a subset of \(\mathcal P(\mathbb N)\); its total arrows are a subset of \(\mathcal P(\mathbb N)^2\). They are \(\mathcal U\)-small. The identities on \(\{n\}\) give infinitely many distinct arrows, so this small filtered presentation does not have fewer than \(\aleph_0\) total arrows.

For every finite \(T\), define

where \(\iota_F:F\hookrightarrow\mathbb N\). Inclusion transition maps respect the formula. Given \(g:T\to\mathbb N\), its finite image gives a representing stage \(F=g(T)\), proving surjectivity. If two representatives induce the same \(g\), move both to \(F\cup G\); their functions into that set are equal pointwise, proving injectivity by the same-stage quotient test. Precomposition by \(v:T'\to T\) sends either route to \(g v\), so this is a natural isomorphism on \(D\). For \(T=\varnothing\) every stage Hom is a singleton and its filtered colimit is a singleton; the same construction works with \(F=\varnothing\).

The inclusion cocone \(F\to\mathbb N\) is an actual Set colimit. Given compatible maps \(t_F:F\to Z\), put \(t(n)=t_{\{n\}}(n)\). Compatibility through \(F\cup\{n\}\) gives \(t|_F=t_F\), and these singleton values force uniqueness. Thus \(L_A=\mathbb N\) can be chosen, and \(j_A\) of (5.2) is exactly (7.2).

Write \(sF=\{n+1:n\in F\}\). The shift transformation is

For an inclusion \(F\subseteq G\), its image is the inclusion \(sF\subseteq sG\), and their composites with the two shift functions agree on every \(n\in F\). Hence (7.3) respects all relations. Precomposition with \(v\) gives the same representative \([sF,s|_F f v]\), proving naturality. Under (7.2), its value is \(s g\).

The parity transformation \(\zeta:A\to h_2\), with \(2=\{0,1\}\in D\), has components \(\zeta_T([F,f])=p|_F f:T\to2\). The inclusion relations and precomposition equations prove that it is well-defined and natural. It becomes \(g\mapsto p g\) under (7.2). Give \(h_2\) its singleton presentation, realized by \(2\). Then the defining equation (5.3), evaluated on every stage leg, says

Uniqueness of the just-proved colimit factor gives \(\sigma_\pi(\theta)=s\) and \(\sigma_\pi(\zeta)=p\). The composite has stage maps \(p s\,\iota_F\), so its realization is \(p s\), in agreement with the whole-arrow composition law. Its value is \(1\) on even \(n\) and \(0\) on odd \(n\). All these functions are maps of the specified small sets. Shift is injective but not surjective; the formal construction and the functor law never required it to be invertible.

Exercise 4. (advanced) A coequalizer object outside the chosen witness

Continue with sets and \(\pi=\aleph_0\), and fix \(X=\mathbb N\). Let \(V_n=\{0,\ldots,n-1\}\times\{0,1\}\), labelled by \(u_n(k,\varepsilon)=k\). Let \(W\) be the full subcategory of \((D\downarrow\mathbb N)\) on these labelled objects. Prove that it is a small filtered receiving witness. Exhibit a categorical coequalizer object absent from \(W\), and the actual \(W\)-arrow that replaces it in the probe-colimit equality test. Compute the colimit of \(W\to\mathsf{Set}_{\mathcal U}\).

Solution. By the finite-set accessibility proof cited above, the \(V_n\) are in \(D\). That proof also identifies every \(\aleph_0\)-accessible set as finite. For any object \((Y,u)\) of the slice, the image of \(u\) is therefore finite. Choose \(n\) larger than its elements, or \(n=0\) if \(Y=\varnothing\). The function \(a(y)=(u(y),0)\) then has \(u_n a=u\), giving a receiving arrow into \(W\).

An arrow \(V_n\to V_m\) in \(W\) must preserve \(k\). For \(n\leq m\) it can independently choose the second coordinate of the image of each of the \(2n\) elements, so this Hom has \(2^{2n}\) elements, including one element when \(n=0\). For \(n>m\), the element \((m,0)\in V_n\) has no permitted image, so the Hom is empty. Identities preserve both coordinates; compositions preserve \(k\), and fullness includes every such function. The object index is \(\mathbb N\), and the total arrow set is its tagged union of these finite Homs over \(\mathbb N^2\). The cardinal foundation's infinite square and countable-family bound give cardinal at most \(\aleph_0\); the identities at every \(n\) give the reverse inequality. Hence all arrows form a small set of cardinal \(\aleph_0\), not a set of cardinal below \(\pi\).

The object \(V_0\) receives the empty test. A pair \(V_n,V_m\) has receiving arrows to \(V_{\max(n,m)}\), using the same \(k\) and second coordinate \(0\). For any parallel \(a,b:V_n\to V_m\), postcompose with \(r_m:V_m\to V_{m+1}\), \(r_m(k,\varepsilon)=(k,0)\). The maps \(a,b\) preserve \(k\), so \(r_m a=r_m b\) on every element. The retained ordinary filteredness criterion therefore proves that \(W\) is filtered. It is full and receives every slice object, so its inclusion is cofinal by the retained receiving-witness theorem.

Now take \(f,g:1\to V_1\) selecting \((0,0)\) and \((0,1)\). Let \(Q=1\), let \(q:V_1\to Q\) be constant, and label \(Q\to\mathbb N\) by \(0\). A function out of \(V_1\) equalizes \(f,g\) precisely when its two values agree; it then factors uniquely through \(q\). Thus \(q\) is the categorical coequalizer. The singleton \(Q\) is not any \(V_n\), whose cardinal is \(2n\). Nevertheless the receiving slice arrow \(r:Q\to V_1\) selecting \((0,0)\) satisfies \(u_1r=0\). Fullness makes

an actual arrow of \(W\). It satisfies \(r q f=r q g\). Consequently \([V_1,f]\) and \([V_1,g]\) are equal in \(\operatorname{colim}_W\operatorname{Hom}(1,V_n)\), although the coequalizer object itself is absent from \(W\). This is the precise missing-stage mechanism used in Section 6.

Finally define the Set comparison

All \(W\)-arrows preserve \(k\), so the map descends through every relation. It is surjective: represent \(k\) by \((k,0)\in V_{k+1}\). If two representatives have the same \(k\), move both to \(V_{\max(n,m)}\) with second coordinate \(0\); their images agree there, proving equality of classes. Distinct \(k\)'s cannot have equal classes, since the descended function would then give distinct values to one class. The map is therefore bijective, with inverse \(k\mapsto[k+1,(k,0)]\). The quotient factor property makes this bijection the actual colimit comparison \(\lambda_X^W\), retaining every original leg \(u_n\); another chosen realization uses the compatible isomorphism described in Section 6. Detection here uses the singleton probe: \(\operatorname{Hom}(1,t)\) is bijective exactly when the Set map \(t\) is bijective. The proof used a categorical coequalizer and full labelled maps, without an additive or balanced-category hypothesis.