Draft lesson.

Accessible objects from a single generator

For the adjoint criteria without the single-generator hypotheses, with completeness and cardinal bounds proved, use Presentable probes and accessible adjoints. A single detecting object can control the size of a category's accessible pieces without preserving epimorphisms. Two cardinal bounds first make an object accessible. A uniform quotient bound then turns accessibility into a bound on its generator maps. We use that characterization to choose a small family of pieces, construct every labelled presentation and recover the whole category through cardinal Ind objects.

1. Hypotheses, comparisons and cardinal parameters

Use set-indexed Choice, a fixed Grothendieck universe and the explicit universe and Hom encodings. Small means bijective to a universe member. Categories have ambient sets of object and arrow data and small Hom sets. An enlargement contains the functor categories used below; it does not change which indices are small. All cardinal parameters are small cardinals. A bound on a diagram counts its entire morphism set, including identities and composites.

Let \(C\) satisfy the following hypotheses, denoted \(H\): all small colimits and finite limits exist; small ordinary filtered colimits preserve finite limits; there is a detecting object \(G\); and every epimorphism is strict. Detection means that a bijection \(\operatorname{Hom}_C(G,v)\) forces the actual arrow \(v\) to be invertible. Strictness means the specified coimage-to-image comparison is invertible, equivalently, for an epimorphism, that it is the coequalizer of its kernel pair. These are the generator and strict quotient conventions.

Write

The component of \(\varepsilon_X\) labelled by \(t\) is \(t\). It is epic, including when \(T_X\) is empty, by the complete evaluation argument. The coproduct functor on actual functions sends \(a: E\to F\) to the unique map satisfying \(P(a)j_e=j_{a(e)}\); its identity and composition laws follow on every component.

For an infinite cardinal \(\pi\), an object \(X\) is \(\pi\)-accessible if, for every small \(\pi\)-filtered \(D: I\to C\) with colimit legs \(c_i\), the specified function

is bijective. The comparison proof and both naturalities use the actual arrows and chosen colimit legs. Let \(C_\pi\) be the full ambient subcategory on these objects. It retains all \(C\)-arrows between them, not only representatives of arrows.

For Sections 2–3 and 6–8, fix parameters satisfying

Condition (d) includes the empty coproduct. The exponent in (c) is \(\pi_0\). Applying it to \(\rho=2\), and using Cantor's theorem, gives \(\pi_0<\pi\). Every \(\pi\)-filtered index is therefore \(\pi_0\)-filtered: a diagram with fewer than \(\pi_0\) arrows also has fewer than \(\pi\) arrows. The very same comparison (1.2) shows that \(G\) is \(\pi\)-accessible.

We will use three complete proofs from Coproduct covers and generator bounds: the stage epimorphism lemma, with both alternatives, the two-Hom-bound accessibility theorem, and the quotient-probe bound. The latter says, under all of (1.3),

Its proof uses the whole small-subcover diagram, not surjectivity of the probe of an epimorphism. Section 5 below explains why parameters (1.3) exist beyond any prescribed small cardinal under \(H\).

2. Accessibility is exactly a generator-probe bound

Theorem 2.1. Under \(H\) and (1.3), for every object \(X\),

Proof of the right-to-left implication. Put \(A=T_X\). Its cardinal is below \(\pi\). Apply (1.4) to the identity epimorphism of \(P_A\). This gives \(|\operatorname{Hom}(G,P_A)|<\pi\). Together with \(|\operatorname{Hom}(G,X)|<\pi\), these are exactly the two bounds of the linked accessibility theorem. That theorem proves bijectivity of (1.2) for every small \(\pi\)-filtered diagram, so \(X\in C_\pi\). In particular the second bound is on \(P_A\), not inferred by applying a probe to \(\varepsilon_X\).

Proof of the left-to-right implication. Suppose \(X\in C_\pi\). Let

with all inclusion arrows. It is small: its object set is a subset of \(\mathcal P(T_X)\), and its arrow set is the corresponding subset of its square. Its identities and composites are the ordinary inclusions. The full small-subset proof shows that it is \(\pi\)-filtered. Explicitly, regularity bounds a union of fewer than \(\pi\) subsets of size below \(\pi\); that union receives the entire family. The empty family is received by \(\varnothing\). Thinness makes every parallel-arrow family equalized by the target identity.

There is a specified coproduct colimit

Here is its whole factor property. A compatible family \(h_A: P_A\to Z\) gives, for each \(t\in T_X\), the component \(h_{\{t\}}j_t: G\to Z\). Singletons are stages because \(\pi\) is infinite. They define one map \(h: P_{T_X}\to Z\). For every \(t\in A\), compatibility with \(\{t\}\subseteq A\) makes the restriction of \(h\) equal to \(h_A\) on that component, hence on \(P_A\). The empty stage has initial source and the equality holds there as well. Conversely restriction gives the compatible family, and both operations are inverse and commute with postcomposition. If \(T_X\) is empty, \(J_X\) has its one empty stage and (2.3) is the identity colimit of the initial coproduct.

Compose (2.3) with \(\varepsilon_X\). The resulting map from the colimit to \(X\) is epic. Apply the accessible-target alternative of the stage epimorphism lemma. Some actual stage map

is epic, with \(|A|<\pi\). Formula (1.4) now gives \(|T_X|<\pi\). This proves the other implication. No claim that \(|J_X|<\pi\) was made or needed. \(\square\)

3. A small family of representatives and closure properties

Theorem 3.1. Under the same hypotheses, \(C_\pi\) is essentially small and is closed under actual epimorphic quotients, subobjects and finite limits in \(C\).

Essential smallness means that there is a small full category of representatives meeting every isomorphism class. It does not mean that all actual labels of isomorphic objects form a small set.

Small representatives. The strict-quotient encoding proof makes the strict quotients of a fixed object a small family up to compatible target isomorphism. Under \(H\), every epimorphism is strict, so it covers all epimorphic quotients here. For each initial cardinal \(\alpha<\pi\), choose a small representative family of quotients

The initial cardinals below \(\pi\) are a small subset of \(\pi\). Universe closure and set-indexed Choice therefore give one small tagged family of these representatives. Every representative target belongs to \(C_\pi\), by (1.4) and (2.1).

If \(X\in C_\pi\), then \(|T_X|<\pi\). Choose a bijection \(a: T_X\to\alpha\) to its initial cardinal. The coproduct functor makes \(P(a)\) an isomorphism, with inverse \(P(a^{-1})\) by its identity and composition laws. Thus

is an epimorphism and is isomorphic, over \(P_\alpha\), to one of (3.1). Its target is therefore isomorphic to a representative. Take the full subcategory on the chosen target set. Its whole arrow set is the tagged union of the small sets \(\operatorname{Hom}_C(Q,Q')\) over the small square of its object set. It is small and contains every identity and composite. Since \(G\in C_\pi\), we may add \(G\) itself to this small object set. Denote the resulting small full category by \(\mathcal A\). It still meets every isomorphism class of \(C_\pi\).

Quotients. If \(e: X\twoheadrightarrow Y\) and \(X\in C_\pi\), the composite \(e\varepsilon_X: P_{T_X}\to Y\) is epic. The index has cardinal below \(\pi\) by (2.1). Apply (1.4), then (2.1), to obtain \(Y\in C_\pi\). This is closure for the original quotient arrow in \(C\), not only for a selected representative quotient.

Subobjects. If \(m: X\hookrightarrow Y\) and \(Y\in C_\pi\), postcomposition with \(m\) injects \(T_X\) into \(T_Y\): cancellation of \(m\) proves injectivity. Hence \(|T_X|<\pi\), and (2.1) puts the actual domain \(X\) in \(C_\pi\).

Finite limits. Let \(D: J\to C_\pi\) be a finite diagram and let \(L\) be its actual \(C\)-limit, which exists by \(H\), with projections \(d_j:L\to D(j)\). The map \(h\mapsto(d_jh)_j\) sends a map \(G\to L\) to a compatible tuple. Conversely, any compatible tuple \(a_j:G\to D(j)\) is a cone from \(G\), so the limit property supplies its unique factor \(h:G\to L\) with \(d_jh=a_j\) for every \(j\). Projection recovers every tuple, and uniqueness recovers every starting map. Thus these are inverse bijections, including the empty diagram, when \(L\) is terminal and both sides are singletons. The compatible-family construction gives the corresponding set of tuples and its product size bound. In particular,

The product is finite and all its factors have cardinal below \(\pi\). Choose an infinite bound \(\mu<\pi\) for the finitely many factors if one is infinite; the infinite-square and finite-product bounds bound this product by \(\mu\). If every factor is finite, their finite product is finite. An empty factor makes the product empty; an empty entire product is a singleton. All these cardinals are below the infinite \(\pi\). Thus \(|T_L|<\pi\) and \(L\in C_\pi\). Fullness retains the original limit factor property and its arrows. This argument is about finite limits, not arbitrary products. \(\square\)

4. One accessibility cardinal for any small family

Theorem 4.1. Assume only \(H\). For any small family of objects \((X_s)_{s\in S}\), there is an infinite regular \(\eta\) for which every \(X_s\) is \(\eta\)-accessible.

For each \(s\), form the two small sets

The family index is small. Choose the universe encodings of these values and form their whole tagged union, keeping the two kinds of value separately. The tagged-union closure makes this one small set. Thus a single small cardinal \(\lambda\) bounds every \(|T_{X_s}|\) and \(|V_s|\). Choose \(\lambda\) infinite, and put \(\eta=\lambda^+\). The regular-successor proof makes \(\eta\) regular. Both bounds are strictly below \(\eta\), so the two-Hom-bound theorem applies to each \(X_s\).

For an empty family we may take \(\eta=\aleph_0\): there are no objects to test. For a nonempty family, every construction above is indexed by its actual small set. There is no choice from a proper class of objects and no prior assumption that \(C\) is accessible. \(\square\)

5. Parameters beyond a prescribed bound and small epimorphic lifts

Theorem 5.1. Under \(H\), given a small cardinal \(\kappa\), one can choose (1.3) with \(\pi>\pi_0>\kappa\). For \(S=C_\pi\), the following statements hold: \(S\) is full and essentially small; it contains the detecting object \(G\); it is closed under quotients, subobjects and finite limits; and for every epimorphism \(f: X\twoheadrightarrow Y\) with \(Y\in S\), there is a monomorphism \(g: Z\hookrightarrow X\), with \(Z\in S\), such that \(fg\) is epic.

Choosing the parameters. Let \(\lambda\) be an infinite small cardinal at least \(\kappa\), \(|T_G|\) and \(|\operatorname{Hom}(G,P_{T_G})|\). Take \(\pi_0=\lambda^+\). Both generator bounds are strictly below this regular cardinal, so the two-Hom-bound theorem makes \(G\) \(\pi_0\)-accessible. This step has not used (2.1).

For each initial cardinal \(\alpha<\pi_0\), collect a small representative family of strict quotients of \(P_\alpha\) and the small \(G\)-Hom sets of all their targets. The same quotient encoding as in Section 3 gives one small tagged family. Choose a cardinal \(\pi_1\geq\pi_0\) strictly larger than the cardinal of that union. Reindexing \(P_E\) by a bijection and transporting across the target isomorphism to its representative shows that \(|T_Q|<\pi_1\) for every quotient in (1.3)(d), including an empty \(E\).

Put \(\pi=(2^{\pi_1})^+\). It is regular. If \(\rho<\pi\), then \(\rho\leq2^{\pi_1}\), and the currying and infinite-cardinal product proof gives

For the middle equality, the explicit currying bijection sends \(F: \pi_0\to2^{\pi_1}\) to the function \((a,b)\mapsto F(b)(a)\), and its inverse sends \(h\) to \(b\mapsto(a\mapsto h(a,b))\). The square bound gives \(\pi_1\cdot\pi_0=\pi_1\). Zero and singleton bases are included in the base-injection inequality. The uniform bound \(\pi_1<\pi\) gives (d); the generator step gave (b). Thus all four conditions are obtained, with the prescribed strict lower bounds. These are the same complete parameter constructions as in the coproduct-cover lesson.

Lifting the epimorphism. Write \(T_X=\operatorname{Hom}(G,X)\) and use the full subset diagram (2.2)–(2.3) for this set. Its colimit is \(P_{T_X}\), and \(f\varepsilon_X: P_{T_X}\to Y\) is epic. Since \(Y\) is \(\pi\)-accessible, the accessible-target stage lemma gives a subset \(A\subseteq T_X\), with \(|A|<\pi\), for which

has epic composite. The arrow \(p\) need not be epic. Use its specified coimage factorization

The complete coimage composition proof gives \(q\) strict epic and \(g\) monic under the assumption that all epimorphisms are strict. No identification of this coimage with the image of an arbitrary arrow is required. The quotient bound (1.4) applied to \(q\) gives \(|T_Z|<\pi\), so (2.1) puts \(Z\) in \(S\). If two maps out of \(Y\) agree after \(fg\), they agree after \(fgq=fp\); epiness of \(fp\) makes them equal. Thus \(fg\) is epic, as required.

The other conclusions follow from Sections 1–3. The construction includes an empty \(T_X\) or empty \(A\): their subset indices are nonempty, the coproduct is initial, and the same coimage and cancellation proofs apply. It does not assign these witnesses functorially to all epimorphisms. \(\square\)

6. Closure under bounded colimits

Theorem 6.1. If \(D: J\to C_\pi\) and \(|\operatorname{Mor}J|<\pi\), its actual colimit in \(C\) belongs to \(C_\pi\). With the parameters of Section 5, this includes every diagram with at most \(\kappa\) arrows.

The colimit exists by \(H\). The complete bounded accessible-colimit theorem proves the assertion. To identify exactly its comparison, write \(B=\operatorname{colim}_JD\), with legs \(b_j\), and test a small \(\pi\)-filtered \(X: I\to C\), with colimit \(L\). Hom out of \(B\) is the compatible-family limit over \(J^{\mathrm{op}}\), by restriction along every \(b_j\). The full chain is

The first and last maps are actual cocone restriction and factor maps. The middle exchange is the bounded Set theorem with every arrow counted; its hypotheses are precisely the bound on \(\operatorname{Mor}J\) and the filtering of \(I\). The next maps are the accessibility bijections for each \(D_j\), natural under every arrow of \(J\). Their inverses form a natural family, so they give the asserted bijection on compatible tuples. The route of \([i,h]\) finishes at the unique map whose composite with every \(b_j\) is \(c_i h b_j\), hence it is \(c_i h\). Thus (6.1) is exactly the comparison (1.2) for \(B\), proving the theorem.

For an empty \(J\), \(B\) is initial and its Hom sets are singletons. The nonempty \(I\) has singleton colimit for this constant identity diagram. This is also the empty-limit case of the same exchange theorem. Initial objects therefore belong to \(C_\pi\). Identities inject \(\operatorname{Ob}J\) into \(\operatorname{Mor}J\); no additional object bound or bound only on generators of a graph is substituted. \(\square\)

7. A small category of all representative labels

Keep a small full representative category \(\mathcal A\subseteq C_\pi\), containing \(G\), as in Section 3. For every \(X\in C\), define

Its objects are all pairs \((A,u)\) with \(A\in\mathcal A\) and \(u: A\to X\). An arrow \((A,u)\to(B,v)\) is an actual \(a: A\to B\) with \(va=u\). Its identity is \(1_A\); composing \(a\) with \(b\) gives \(ba\), and the slice equations give \(wba=va=u\). The forgetful functor \(U_X: I_X\to\mathcal A\) keeps these actual maps.

Smallness. Its object set is a tagged union of the small sets \(\operatorname{Hom}_C(A,X)\) over a small index. Its whole arrow set is a subset of the tagged union of \(\operatorname{Hom}_C(A,B)\) over the small square of its object set. Both are small by the universe encodings. The size need not be below \(\pi\).

Filtering. Let \(F: J\to I_X\) have fewer than \(\pi\) total arrows. The underlying \(U_XF\)-diagram has colimit \(B\in C_\pi\) by Section 6. The labels of \(F\) give an actual cocone to \(X\), hence a unique \(b: B\to X\). Choose a representative \(A_0\in\mathcal A\) and an isomorphism \(v: B\to A_0\). The label \((A_0,bv^{-1})\), with receiving legs \(v c_j\), is a cocone to \(F\) in \(I_X\): the equations are

for every actual \(s: j\to j'\). The construction includes the empty \(J\), whose colimit is initial. Thus it supplies a nonempty receiving category and proves full \(\pi\)-filteredness.

Cofinality among all accessible labels. Let \(D_X=(C_\pi\downarrow X)\). The inclusion \(I_X\to D_X\) is full. Given \((Y,t)\in D_X\), choose \(A\in\mathcal A\) and \(s: A\xrightarrow{\sim}Y\). The arrow \(s^{-1}\) from \((Y,t)\) to \((A,ts)\) is an object of its incoming comma category. It is initial there: for any comma object \(a: (Y,t)\to(B,w)\), the unique required arrow is \(as: A\to B\), since \((as)s^{-1}=a\); fullness places it in \(I_X\). Thus every such comma category is nonempty and connected. The cofinal cocone proof with specified legs applies. This proves cofinal smallness without asserting that the complete ambient set of accessible labels is itself small.

The actual colimit. The labels give a comparison

It is invertible. The accessibility of \(G\) identifies \(\operatorname{Hom}(G,\lambda_X)\) with

Surjectivity uses the actual label \((G,t)\) for each \(t\in T_X\), with representative \(1_G\). For injectivity, take two representatives with the same image. Receive their indices at one \((B,v)\), giving maps \(h',k': G\to B\) with \(vh'=vk'\). Form the categorical coequalizer \(q: B\to Q\) of these two maps. Its walking parallel-pair category has four total arrows: two identities and two parallel arrows. Section 6 therefore puts \(Q\) in \(C_\pi\). Its factor property gives \(w: Q\to X\), with \(wq=v\). Choose \(s: Q\xrightarrow{\sim}A_0\), with \(A_0\in\mathcal A\). The map \(sq\) is an actual transition in \(I_X\) from \((B,v)\) to \((A_0,ws^{-1})\), and

This is a later-stage equality witness for the two original classes, proving injectivity. No subtraction or additive structure was used. Hence (7.4) is bijective; detection by \(G\) makes the particular comparison \(\lambda_X\) invertible. Empty Hom sets require no representative choices, and \(I_X\) remains nonempty. \(\square\)

8. Realization and a natural equivalence

For a category \(D\), \(\operatorname{Ind}^{\pi}(D)\) denotes the full subcategory of ordinary Ind objects whose element category is \(\pi\)-filtered. The complete cardinal recognition theorem identifies these with presheaves admitting a small \(\pi\)-filtered presentation by representables. Its morphisms are all presheaf natural transformations between the selected objects.

Theorem 8.1. Under \(H\) and (1.3), realization is an equivalence

It may equally be constructed from the small full representative category \(\mathcal A\).

The complete realization construction applies: \(H\) supplies every required small \(\pi\)-filtered colimit, and the pieces are accessible by definition. For a presentation \(B\cong\operatorname{colim}_Ih_{A_i}\), it chooses \(\sigma_\pi B=\operatorname{colim}_IA_i\). A transformation is sent to the unique map preserving the prescribed stage legs. Its identity, composition, presentation independence and Hom bijections are proved there.

For clarity, full faithfulness uses the chain

The inner comparison is bijective for each accessible \(A_i\). Its natural inverse gives an isomorphism of the full outer diagrams before taking the limit. No exchange of an unbounded limit with a colimit is claimed. On a stage representative \([j,a: A_i\to B_j]\), the actual map sends the \(i\)-th leg to \(c_j^F a\), which is precisely the specified realization map. The linked proof therefore proves full faithfulness for that functor, not an unrelated bijection of Hom sets.

Section 7 supplies a small \(\pi\)-filtered presentation of every \(X\) by pieces of \(C_\pi\), with its compatible isomorphism \(\lambda_X\). The realized formal object is therefore isomorphic to \(X\). This proves essential surjectivity and (8.1).

Here are explicit natural comparisons. In the following formulas \(\sigma_\pi\) denotes realization on \(\operatorname{Ind}^{\pi}(\mathcal A)\); the final paragraph transports this equivalence to full \(C_\pi\). Using the fixed \(\mathcal A\), define the presheaf

It is a cardinal Ind object by the recognition theorem. For \(f: X\to Y\), the index functor \(F_f: I_X\to I_Y\) sends \((A,u)\) to \((A,fu)\) and retains every underlying arrow. The slice equation remains true after postcomposition with \(f\); identities and composites are unchanged. Define \(\tau(f)\) by the pointwise class formula

Here \(T\) is an object of \(\mathcal A\). The formula respects every generating identification, since the same transition arrow is retained. Precomposition in \(T\) commutes with the formula, giving a natural transformation. Identities and composition follow on representatives. Its realization has the specified legs \(f u\); colimit uniqueness gives

If realization chose different presentations, transport each \(\lambda_X\) by the unique presentation comparison from the linked construction; (8.5) is preserved by its naturality.

Full faithfulness defines the unique unit \(\eta_B: B\to\tau(\sigma_\pi B)\) by

This arrow is invertible: full faithfulness lifts the inverse \(\lambda_{\sigma_\pi B}\), and faithfulness makes both composites identities. Equation (8.5) proves naturality of \(\eta\) after applying \(\sigma_\pi\), hence by faithfulness. The counit is \(\lambda: \sigma_\pi\tau\to1_C\), already natural by (8.5). Its first triangle is the defining equality \(\lambda_{\sigma_\pi B}\sigma_\pi(\eta_B)=1\). For the second, (8.5), applied to \(\lambda_X\), gives \(\sigma_\pi(\tau(\lambda_X))=\lambda_{\sigma_\pi\tau X}\). Multiplying by (8.6) at \(B=\tau X\) gives the identity under \(\sigma_\pi\); faithfulness gives the second triangle in the Ind category.

Finally, using \(\mathcal A\) does not change the full-subcategory assertion (8.1). Its inclusion in \(C_\pi\) is fully faithful and essentially surjective. Choose representative isomorphisms and the checked quasi-inverse; the choices range over the ambient object set in the stated enlargement. Precomposition with these inverse equivalences gives inverse equivalences of presheaf categories. Pointwise colimits are preserved, and each representable is sent to the corresponding representable by the explicit Hom isomorphism. Hence a small \(\pi\)-filtered presentation retains its entire index, every arrow and each leg. The presheaf equivalences restrict to the cardinal Ind subcategories. On represented pieces both realizations are the original object; their unique colimit factors give the natural comparison on every formal object. Thus the construction on \(\mathcal A\) yields precisely the realization equivalence for full \(C_\pi\). \(\square\)

9. Two worked examples

Sets with a singleton probe. In small sets take \(G=1\). Evaluation at the unique point identifies \(T_X\) with \(X\), and bijectivity of that probe is invertibility of the Set map. Small colimits are the tagged-quotient construction; finite limits are compatible tuples. Their filtered exactness is the complete finite-limit calculation. A Set epimorphism is onto: if \(y\) is outside its image, two functions to \(2\), one constantly zero and the other supported at \(y\), contradict epiness. An onto map is the coequalizer of its kernel pair: a function constant on each fibre has one well-defined value at every target point, and these values give its unique factor. Hence it is strict. These facts verify \(H\).

Take \(\pi_0=\aleph_0\) and \(\pi=(2^{\aleph_0})^+\). The singleton is \(\aleph_0\)-accessible by the complete small-set comparison. Quotients of finitely many singleton copies are finite, so (d) holds. For \(\rho<\pi\), currying gives \(\rho^{\aleph_0}\leq(2^{\aleph_0})^{\aleph_0}=2^{\aleph_0}<\pi\). Thus (1.3) holds. Theorem 2.1 identifies \(C_\pi\) with sets of cardinal below \(\pi\). Initial-cardinal sets give a small representative category. Its labelled index over \(X\) retains all functions from these small domains to \(X\), including noninjective functions. Formula (7.4) is exactly evaluation of a labelled point.

A small subobject over a quotient. In the same Set category, let \(X=\mathbb Z\), \(Y=\{0,1\}\), and let \(f(n)\) be its parity. The subset \(Z=\{0,1\}\subseteq\mathbb Z\) gives a monic \(g\) whose composite \(fg\) is bijective, hence epic. A stage in (5.2) can be the two labels picking \(0\) and \(1\); its coimage is their two-element image, with the displayed inclusion \(g\). More generally, for any onto \(f: X\to Y\) with \(|Y|<\pi\), set-indexed Choice selects one \(x_y\) in each fibre. Then \(Z=\{x_y:y\in Y\}\) has cardinal \(|Y|<\pi\), and \(f|_Z\) is bijective. For empty \(Y\), a map \(X\to Y\) forces empty \(X\), and the empty subobject suffices. This stronger Set witness is an example; the general categorical proof uses coimages and only concludes epiness of \(fg\).

10. Four graded exercises with complete solutions

Exercise 1. Beginner: why a regular cardinal alone is insufficient

In small sets let \(G=\mathbb N\), \(X=2\), and \(\pi=\aleph_0\). Prove that \(G\) detects isomorphisms and \(X\) is \(\pi\)-accessible, but \(|\operatorname{Hom}(G,X)|\not<\pi\). Explain why this does not contradict Theorem 2.1.

Solution. Suppose postcomposition with \(v: A\to B\) gives a bijection from functions \(\mathbb N\to A\) to functions \(\mathbb N\to B\). If \(v(a)=v(a')\), the two constant functions have equal images; injectivity makes them equal, and evaluation at zero gives \(a=a'\). For each \(b\in B\), surjectivity lifts the constant function with value \(b\). Evaluating a lift at zero gives a preimage of \(b\). If \(A\) is empty and \(B\) nonempty, the constant function has no lift, so that case cannot satisfy the probe condition. Thus \(v\) is bijective in every case and its inverse is a Set map. Detection follows.

The two-element set is \(\aleph_0\)-accessible by the proved finite-set direction of the small-set comparison. Its \(G\)-Hom set is the set of all binary sequences. Already the sequences having a single one at position \(n\) give an injection of \(\mathbb N\) into it, so it is not finite. Theorem 2.1 requires the entire package (1.3), not just regularity of \(\pi\). In particular an infinite \(\pi_0\) satisfying (c) would have \(2^{\pi_0}<\aleph_0\), which is impossible. No such package exists at this \(\pi\); the characterization is not asserted there. \(\square\)

Exercise 2. Intermediate: every small subset and its colimit

For an infinite regular \(\pi\) and a small set \(X\), let \(J\) be the full inclusion category of subsets of \(X\) of cardinal below \(\pi\). Verify its \(\pi\)-filteredness and prove that its inclusion diagram has the specified colimit \(X\), including empty \(X\). Show that postcomposition by a function \(f: X\to Y\) gives the correct target function on that colimit.

Solution. The object set is a subset of \(\mathcal P(X)\); all arrows are inclusion pairs, and identities and composites are included. Fewer than \(\pi\) objects have union of cardinal below \(\pi\) by regularity, and this union receives every inclusion in the family. The empty family is received by the empty subset. Every parallel family in a thin category consists of the same arrow, so its target identity equalizes it. The full criterion therefore makes \(J\) \(\pi\)-filtered.

For a compatible family \(a_A: A\to T\), define \(a(x)=a_{\{x\}}(x)\). The singleton stage exists because \(\pi\) is infinite. If \(x\in A\), compatibility with \(\{x\}\subseteq A\) gives \(a(x)=a_A(x)\). These equations prove every required restriction and force uniqueness. Conversely restricting any function \(X\to T\) gives the compatible family. Both constructions commute with postcomposition. For empty \(X\), the sole stage is empty and there is the unique empty factor.

The functions \(f|_A: A\to Y\) form the target cocone. Its unique factor is \(f\), because every singleton restriction has value \(f(x)\). One may also retain a diagram of images: \(A\mapsto f(A)\) preserves inclusions and has restricted maps \(A\to f(A)\) natural on every arrow. The cardinal of \(f(A)\) is at most \(|A|\); however, this image construction does not discard the original labelled functions of \(I_X\), nor does it identify two of its objects solely because their images coincide. \(\square\)

Exercise 3. Advanced: representative quotients and the size of the code

Explain the complete kernel-pair code behind (3.1). Prove that two strict quotients with the same subobject of \(P_\alpha\times P_\alpha\) are uniquely isomorphic over \(P_\alpha\). Distinguish the resulting essential smallness from a bound below \(\pi\) on the number of representatives or from smallness of all actual object labels.

Solution. A quotient \(q: P_\alpha\to Q\) has kernel pair \(R_q\) with its two projections to \(P_\alpha\). The combined map \(R_q\to P_\alpha\times P_\alpha\) is monic, because its projections determine every map into the pullback. The subobject encoding from the linked generator lesson sends a monomorphism to the image subset of its \(G\)-Hom map. Two such subobjects with the same subset have a pullback whose two projections induce Hom bijections: every probe element in either domain has one matching element in the other, and monicity makes it unique. Detection makes both projections invertible. Thus the subset code identifies their subobject classes.

If two kernel pairs define the same subobject of the product, the domain isomorphism preserves both projections. Strictness says that each quotient is the coequalizer of exactly that pair. One quotient therefore factors uniquely through the other, and conversely. The composites have the original quotient as their composite with the epic source map, so epic cancellation makes them identities. The compatible target isomorphism is unique by the same cancellation.

The code lives in \(\mathcal P(\operatorname{Hom}(G,P_\alpha\times P_\alpha))\), a small set. Taking a small family of \(\alpha<\pi\) and all its small representative sets still gives a small family by universe closure. This does not give a strict cardinal bound below \(\pi\) for that power set or tagged family: taking a power set can increase a below-\(\pi\) cardinal beyond that bound, and the index of all initial cardinals below \(\pi\) was not claimed to have cardinal below \(\pi\). Finally different actual target labels can be isomorphic to the same representative. The argument selects a small full representative category, not the collection of every relabelled target. \(\square\)

Exercise 4. Expert: the natural inverse and its two triangles

Using \(I_X\), prove the formula for \(\tau(f)\) on every representative, and construct the unit and counit of the realization equivalence. Verify both triangle identities and prove that changing the chosen admissible realizations changes this equivalence by compatible natural isomorphisms.

Solution. A transition \(a: (A,u)\to(B,v)\) satisfies \(va=u\). Postcomposition gives \((fv)a=fu\), so it is still the same transition in \(I_Y\). Thus replacing a representative by \([(B,v),ah]\) and applying (8.4) gives the same class as applying (8.4) first and then this transition. This verifies every generating relation and hence the full quotient relation. Precomposition in a test object sends \(h\) to \(hr\) on both routes. Identity and composite functions retain the same label and underlying arrow, proving the functor laws on all classes.

The realized transformation has composite \(fu\) on every original stage leg, so its unique colimit factor obeys (8.5). Every \(\lambda_X\) is invertible by Section 7. Full faithfulness gives one \(\eta_B\) whose image is (8.6), and lifts the inverse image \(\lambda_{\sigma_\pi B}\) to its inverse; faithfulness verifies both composites. For \(\theta: B\to F\), (8.5) at \(\sigma_\pi\theta\) states

This is exactly the image of the unit naturality equation, which follows by faithfulness. The counit naturality is (8.5) itself. The first triangle becomes the defining product of \(\lambda_{\sigma_\pi B}\) with its inverse. For the second, apply (8.5) to \(f=\lambda_X\). Canceling \(\lambda_X\) gives

Multiplying this by \(\sigma_\pi(\eta_{\tau X})\) gives the identity. Faithfulness proves \(\tau(\lambda_X)\eta_{\tau X}=1_{\tau X}\).

For another choice of realization, the representation factor property produces \(v_B: \sigma_\pi B\xrightarrow{\sim}\sigma'_\pi B\) preserving all formal stage legs. The defining equation for realization arrows gives \(v_F\sigma_\pi(\theta)=\sigma'_\pi(\theta)v_B\), by colimit uniqueness, so \(v\) is natural. Transport the counit by \(\lambda'_X=\lambda_Xv_{\tau X}^{-1}\). Its naturality follows by this square and (8.5). The unit for the new choice is the unique arrow whose realized image is the inverse of its transported counit; full faithfulness gives it and both triangle calculations repeat with those inverses. All comparisons therefore preserve the specified colimit legs and change the equivalence only by compatible natural isomorphisms. \(\square\)

References and prerequisite proofs

• Coproduct covers and generator bounds, Sections 1–3, 5–6 and 8, for the explicit hypotheses, evaluation maps, both stage alternatives, two Hom bounds, parameter construction and uniform quotient bound.
• Generators and small quotient families, Sections 3–4, for complete evaluation and kernel-pair encoding; Coimages, images and composition of quotients, Section 4, for the monic coimage factor under the stated strictness hypothesis.
• Bounded colimits, accessible sets and cardinal ind-objects, Sections 1–4, and Common arrows and bounded Set comparisons, Sections 3 and 6, for the exact comparisons and all bounded-diagram arrows.
• Cardinal Ind presentations and their realization, Sections 2 and 5–6, for admissible presentations, all realization arrows, natural presentation comparisons and full faithfulness; Equivalences and chosen representatives, Section 1, for the actual chosen quasi-inverse.
• The Stacks Project, Grothendieck's AB conditions gives the abelian setting of filtered exactness and generators. Its finite-presentation Hom characterization describes the ordinary-filtered module comparison. The arguments here retain the stated general categorical hypotheses and the cardinal-filtered comparison.

11. Grothendieck abelian categories and small exact pieces

Let \(C\) be a locally small Grothendieck abelian category in the universe convention of Section 1, and let \(G\) be a generator. Here we use the Stacks definition of AB5 and a generator: small coproducts exist, directed colimits of short exact systems remain short exact, and every proper subobject \(N\hookrightarrow M\) fails to receive at least one map \(G\to M\). The following argument connects those assumptions to every part of \(H\).

Why the abelian assumptions imply H

An abelian category has zero, finite biproducts, kernels and cokernels. Its equalizer of \(f,g:A\rightrightarrows B\) is the kernel of \(f-g\). More generally, for a finite diagram \(D:J\to C\), take the kernel of

The sums in (11.1) are finite biproducts. A map to the first sum lies in the kernel precisely when all its coordinates form a cone. For every starting object, the biproduct factor and then the kernel factor give the unique map with the required coordinates. This is the full finite-limit property, including an empty diagram, for which the kernel is zero and is terminal.

Small coproducts also give all small colimits. For a small diagram \(D:J\to C\), take the cokernel of

Given any target \(T\), a map from the second coproduct is exactly a family \(a_j:D_j\to T\). It kills \(d\) if and only if \(a_kD(s)=a_j\) for every actual arrow \(s\). The cokernel therefore supplies the unique factor for every cocone. Its legs are the cokernel map composed with the \(\iota_j\). Empty sums give the initial zero object; identities and composites in the whole arrow set cause no additional assumption.

We next pass from directed systems to arbitrary small filtered categories. The complete forward proof of Stacks, Lemma 4.21.5 gives a cofinal functor \(u:P\to I\) from a directed poset. Its construction uses finitely generated subcategories of \(I\times\mathbb N\) with a unique final object; inclusions give the actual transition arrows. It preserves smallness: finite generators come from a small total arrow set, so their family, the selected subcategories and their order arrows are small. No assertion that \(I\) itself is a poset is needed. The directed-replacement proof in Universal maps, finite probes and countable indexing, Theorem 6.2 proves the same existence statement by a finite-face construction, including smallness, both functor laws and every incoming-comma equation. Using that functor, the cofinal cocone factor proof gives an isomorphism

For a natural transformation, both routes through (11.3) agree on every \(c_p\), so uniqueness gives its naturality. Restrict an objectwise short exact system over \(I\) along \(u\). AB5 makes its \(P\)-colimit short exact. The three natural isomorphisms (11.3) identify its maps with the original \(I\)-colimit maps, so that sequence is short exact too.

To obtain finite-limit preservation, we must also keep the additive and universal maps. The pointwise kernel and image construction makes \(C^I\) abelian. Kernel transition maps are uniquely determined by \(k_{i'}K(s)=A(s)k_i\); cokernel transition maps satisfy the dual equation. Those equations prove all identities, compositions and natural factors. Images are kernels of the resulting cokernels, and their transition maps come from the same construction. Thus the two short exact systems associated with a transformation \(f:A\to B\) are actual diagrams:

Write \(L=\operatorname{colim}_I\). This functor preserves binary biproducts with their specified maps. Indeed, for every \(T\), cocones from \(A_i\oplus B_i\) to \(T\) are exactly pairs of cocones from \(A\) and \(B\); this gives the colimit object \(LA\oplus LB\). The comparison has restrictions equal to the two original colimit legs followed by the biproduct injections. Conversely, the cocones obtained from the stage injections give its inverse; both inverse equations follow on all stage legs. The corresponding projections are then determined by the same injection equations. Also \(L0=0\), by the zero-cocone factor property. A functor preserving these maps preserves the diagonal and fold maps. The identity

therefore shows that \(L\) is additive.

Apply the short exactness just proved to both systems (11.4). The map \(Lm\) is monic, and \(Lk\) is the kernel of \(Lr\). For any \(t:T\to LA\) with \((Lf)t=0\), cancellation of \(Lm\) gives \((Lr)t=0\). Hence \(t\) factors uniquely through \(Lk\). This proves that \(Lk\) is the actual kernel of \(Lf\), with its universal map. Its composite with each original \(K_i\)-leg is the \(A_i\)-leg followed by \(k_i\), so the resulting kernel comparison is the specified one. Naturality follows by testing a square against the kernel inclusion and these colimit legs.

Because \(L\) is additive, it sends the pointwise difference \(f-g\) to \(Lf-Lg\). It consequently preserves equalizers as well as finite biproducts and zero. Apply these comparisons to the kernel presentation (11.1) of an arbitrary finite diagram in \(C^I\). At every diagram object the resulting map is the original colimit map followed by the original limit projection. The unique cone factor is therefore the canonical finite-limit comparison. It is invertible. This proves the finite-limit clause of \(H\) for every small ordinary filtered \(I\); it does not assume that \(\operatorname{Hom}(G,-)\) preserves those colimits.

The generator is detecting. First, \(\operatorname{Hom}(G,-)\) is faithful: if \(a,b:X\rightrightarrows Y\) agree on all maps from \(G\), all such maps factor through their equalizer. A proper equalizer subobject would contradict the generator condition. The equalizer is thus invertible and \(a=b\). Suppose now that \(\operatorname{Hom}(G,v)\) is bijective. Injectivity and this faithfulness show that \(v\) is monic: equality after \(v\) can be tested after every map from \(G\) to the starting object. Surjectivity shows that \(v\) is epic: for maps \(a,b\) out of its codomain with \(av=bv\), every map \(h\) from \(G\) to that codomain is \(vh'\), and hence \(ah=bh\). Faithfulness gives \(a=b\). A monic epic map in an abelian category has zero kernel and cokernel; its coimage-to-image comparison is that map after the canonical identifications, so it is invertible. Thus \(v\) is invertible.

Finally every epimorphism \(e:A\twoheadrightarrow B\) is strict in the convention of Section 1. The abelian coimage-to-image axiom makes \(e\) a cokernel of its kernel \(k:K\to A\): its cokernel is zero, its image is \(B\), and the coimage comparison identifies \(\operatorname{coker}k\) with \(B\). Form the kernel pair \(r_0,r_1:A\times_B A\rightrightarrows A\). The pair \((k,0)\) factors through this pullback. A map \(t:A\to T\) equalizing \(r_0,r_1\) therefore kills \(k\), and factors uniquely through \(e\). Conversely every factor through \(e\) equalizes the pair. Hence \(e\) is its kernel-pair coequalizer. All four clauses of \(H\) are now established.

One regular cardinal and a small abelian category

Theorem 11.1. For any prescribed small family \((X_s)_{s\in E}\) in \(C\), there is a small regular cardinal \(\pi>\aleph_0\) such that the full category \(S=C_\pi\) contains that family and \(G\), is essentially small, and is closed under ambient subobjects, quotients, finite limits and colimits with fewer than \(\pi\) total arrows. It is abelian with exact inclusion into \(C\), and it has the ambient countable coproducts. There is a small full abelian representative category \(\mathcal A\subseteq S\) containing the prescribed objects and \(G\), with exact inclusion and countable coproducts preserved up to their specified universal isomorphisms. For any epimorphism \(f:X\twoheadrightarrow Y\) with \(Y\in S\), some actual monomorphism \(g:Z\hookrightarrow X\), with \(Z\in S\), has epic composite \(fg\). Every selected object has the same \(\pi\)-accessibility comparison (1.2).

Proof. Apply Section 4 to the prescribed family with \(G\) adjoined, obtaining one infinite regular \(\eta\). Choose a small \(\kappa\geq\eta,\aleph_0\), and apply Section 5 to obtain its four parameters with \(\pi>\pi_0>\kappa\). Every small \(\pi\)-filtered category is \(\eta\)-filtered, since each diagram with fewer than \(\eta\) arrows also has fewer than \(\pi\) arrows. Thus the original \(\eta\)-accessibility bijections of the prescribed objects apply to every required \(\pi\)-filtered diagram and remain exactly (1.2). The same monotonicity puts \(G\) in \(C_\pi\). Sections 2–3, 5 and 6 give essential smallness, subobjects, quotients, finite limits, all the asserted bounded colimits and the epimorphic lift, under the \(H\) just proved.

To prove the abelian conclusion, use the inherited abelian Hom groups and bilinear composition on the full subcategory \(S\). It contains zero and every ambient finite biproduct: these are finite coproducts, so Section 6 applies. The biproduct injections and projections lie in \(S\) by fullness and retain their ambient universal properties. A kernel of a map in \(S\) is an ambient subobject, so belongs to \(S\); its original kernel map and factors give the internal kernel. A cokernel is an ambient quotient, so likewise belongs to \(S\) with its original cokernel property. Consequently the coimage and image of a map in \(S\) are the ambient ones. Their canonical comparison and its inverse belong to \(S\), by fullness. This is the abelian axiom. It is also the full-subcategory construction.

The inclusion preserves the actual kernels and cokernels, and is additive. A short exact sequence is a kernel followed by its cokernel, so its image is short exact in \(C\). Conversely an ambient short exact sequence all of whose terms lie in \(S\) has the same universal factors inside \(S\), and is short exact there. This describes the inherited exact structure explicitly. For a countable family in \(S\), its discrete indexing category has at most \(\aleph_0<\pi\) total arrows. Section 6 therefore puts its ambient coproduct in \(S\), with the original injections and all-target factor property. No closure under countable products is asserted.

Choose a small set of representatives for \(S\), and adjoin the actual objects in the prescribed small family, \(G\), and an ambient zero object. Their union is small. Take the full category \(\mathcal A\) on that set: its total arrow set is the small tagged union of the small Hom sets over the square of its object set. It remains a representative category for \(S\). For a map in \(\mathcal A\), its ambient kernel and cokernel lie in \(S\). Choose isomorphisms from these objects to representatives and transport their structure maps. Each transported structure map retains its full ambient universal property, because its factors are composed with the chosen isomorphism or its inverse. Fullness then gives the property inside \(\mathcal A\). The same transport applies to finite biproducts and countable coproducts.

Thus \(\mathcal A\) inherits the additive structure, kernels and cokernels; its coimage-to-image comparison is the transported ambient isomorphism, whose inverse is an actual arrow of \(\mathcal A\). It is abelian and its inclusion is exact by the kernel–cokernel argument above. Its countable coproduct injections also form ambient coproducts. The representative category is closed under these constructions up to the specified isomorphisms; it need not contain every relabelled ambient object literally.

For the asserted epimorphic lift, retain the actual \(Z\), \(g\) and \(fg\) supplied by (5.2)–(5.3). The domain \(X\) was arbitrary. If a representative domain is desired, choose \(v:Z\xrightarrow{\sim}A_0\) with \(A_0\in\mathcal A\). Then \(gv^{-1}:A_0\hookrightarrow X\) is monic and \(fgv^{-1}\) is epic. The codomain \(Y\) can likewise be replaced by an isomorphic representative without changing epiness. Accessibility transports through the domain isomorphism: precomposition identifies both sides of (1.2), and the square commutes since \((c_i h)v=c_i(hv)\). Thus all selected pieces retain the same cardinal and the actual comparison, including the prescribed objects, zero and the chosen representatives. \(\square\)

Example 11.2. In \(\mathsf{Ab}\), take \(G=\mathbb Z\) and prescribe \(\mathbb Z/2\) and a chosen countable coproduct \(\bigoplus_n\mathbb Z\). Small coproducts are finite-support sums, and the filtered-module exactness proof, specialized to \(R=\mathbb Z\), supplies AB5. A proper subgroup \(N\subset M\) misses some \(m\in M\); the homomorphism \(n\mapsto nm\) from \(\mathbb Z\) does not factor through \(N\). Thus this is the required generator. The theorem supplies one \(\pi>\aleph_0\) containing these particular objects, their ambient kernels and cokernels, and countable sums of the selected pieces. For the reduction map \(\mathbb Z\to\mathbb Z/2\), the kernel is the original subobject \(2\mathbb Z\hookrightarrow\mathbb Z\), and the cokernel is zero; the representative category transports these maps if different models were selected. More generally the epimorphic lift applies even when its original domain lies outside \(S\). This example asserts the existence of a common cardinal from the proved construction; it does not identify the least possible cardinal.