Presentable probes and accessible adjoint functors

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

Small presentable objects can control a category without being projective or arising from one generator. We first recover the category from its presentable probes and construct all its limits. We then obtain the two adjoint criteria: preservation of colimits gives a right adjoint, while preservation of limits together with accessibility gives a left adjoint. The converse accessibility assertion for right adjoints requires a cardinal bound, not preservation of ordinary filtered colimits.

Use fixed universes, set-indexed choice, locally small categories and small diagrams. An enlargement contains their object and arrow data, so chosen universal objects can be selected there. Cardinals below are small, infinite and regular. A diagram is \(\kappa\)-small when its entire set of arrows has cardinality below \(\kappa\); it is \(\kappa\)-filtered when every such diagram into it has a cocone. In particular a filtered index is nonempty. An object \(a\) is \(\kappa\)-presentable if the canonical map

is bijective for every small \(\kappa\)-filtered diagram with colimit legs \(c_i\). This is the property called \(\kappa\)-accessibility of an object in Cardinal Ind presentations. A category is locally \(\kappa\)-presentable if it has all small colimits and a small family of \(\kappa\)-presentable objects such that every object is a small \(\kappa\)-filtered colimit of that family. A functor is accessible if it preserves small \(\tau\)-filtered colimits for some regular \(\tau\).

We use the elementary category and universe conventions of Universes and small categories. The single-generator accessibility theorem supplies additional concrete bounds under its stronger hypotheses; those hypotheses are not imposed here. The solution-set construction and extension from representables give related universal constructions. All cardinal, density and adjoint steps needed below are proved here. Basic reference: [Riehl].

1. The set comparison and small colimits of probes

Lemma 1.1. In sets, \(\kappa\)-filtered colimits commute with \(\kappa\)-small limits, through the canonical comparison.

Proof. A filtered colimit of sets consists of represented elements modulo eventual equality: two elements have the same class exactly when their images agree at some common receiving stage. Indeed the usual colimit quotient is generated by the diagram arrows. A finite zigzag of such generating identifications can be put over one stage, and its parallel maps can be equalized there. This proves the eventual-equality description.

A \(\kappa\)-filtered category also admits a cocone on any graph with fewer than \(\kappa\) vertices and edges. First receive all its objects using the discrete diagram. At the common receiver, compare, for every edge, its two induced arrows from the edge source. A formal star with two parallel arrows for each such pair is a \(\kappa\)-small category: it has no nonidentity composites. A cocone on that star equalizes all the pairs at once. The resulting maps give the required graph cocone. This argument includes finite graphs when \(\kappa=\omega\).

For a diagram with directions \(I\), \(\kappa\)-filtered, and \(J\), \(\kappa\)-small, consider a compatible \(J\)-tuple of classes in the \(I\)-colimits. Choose one representative per object of \(J\). There are fewer than \(\kappa\) representatives; a cocone puts them all at one stage. Each \(J\)-arrow equation holds eventually. Choose a witness for each equation and put all these witnesses, their witnessing maps and the original representative stage in a graph of fewer than \(\kappa\) edges. The preceding argument gives a cocone at which the tuple satisfies every equation. It represents an element of the limit at that stage, proving surjectivity.

For injectivity, two represented compatible tuples whose coordinates have equal colimit classes have fewer than \(\kappa\) eventual-equality witnesses. A common cocone for those witnesses equates the tuples at a single stage. Thus their classes in the colimit of limits coincide. These constructions concern exactly the map that takes a tuple to its coordinate classes. If \(J\) is empty, both limits are singleton sets, and the colimit of singleton sets is a singleton because \(I\) is nonempty and connected. \(\square\)

Corollary 1.2. A colimit of a \(\kappa\)-small diagram of \(\kappa\)-presentable objects is \(\kappa\)-presentable. The same holds for a colimit specified by a directed graph with fewer than \(\kappa\) vertices and edges.

Proof. Maps out of a colimit are compatible families of maps out of its objects. Apply Lemma 1.1 to interchange that limit with the filtered colimit in (1.1). The resulting composite is the actual comparison on each colimit leg, so it proves presentability. For a graph, compatibility is the equalizer of two maps between the product indexed by its vertices and the product indexed by its edges. Both products have fewer than \(\kappa\) factors. The same proof applies, even when the free category on the graph has infinitely many paths. \(\square\)

A retract of a \(\kappa\)-presentable object is \(\kappa\)-presentable. To see this, let \(r:a\to b\), \(s:b\to a\) satisfy \(rs=1_b\). Precomposition makes \(\operatorname{Hom}(b,-)\) a retract of \(\operatorname{Hom}(a,-)\), naturally in the target. Its comparison (1.1) is the corresponding retract of the invertible comparison for \(a\). If \(i,p\) are the inclusion and retraction on the source, and \(i',p'\) those on the target, an inverse for this retract is \(p\,\chi_a^{-1}i'\); the commuting squares and \(pi=p'i'=1\) verify both inverse identities.

2. A genuinely small full category of presentable objects

Fix a locally \(\kappa\)-presentable category \(C\) and a small presenting family \(A\).

Lemma 2.1. The \(\kappa\)-presentable objects have a small set of isomorphism representatives. A full subcategory \(B\) on such representatives has all \(\kappa\)-small colimits, and they are its colimits in \(C\).

Proof. If \(b\) is \(\kappa\)-presentable, express \(b=\operatorname{colim}_i a_i\) with \(a_i\in A\) and a \(\kappa\)-filtered index. The identity of \(b\) factors as

Thus \(b\) is a retract of a presenting object. For each \(a\in A\), there is only a small set of idempotents in \(\operatorname{Hom}(a,a)\). Each such idempotent \(e\) splits in \(C\): take the coequalizer \(q:a\to b\) of \(1_a,e\). Since \(e^2=e\), there is \(j:b\to a\) with \(jq=e\). Then \(qjq=qe=q\), and the epimorphism \(q\) gives \(qj=1_b\). A coequalizer is epic because maps agreeing after it agree by its universal uniqueness.

Two splittings of the same idempotent give isomorphic retracts: the maps between them are \(r'j\) and \(rj'\), whose composites are identities using \(jr=j'r'=e\). Consequently a chosen splitting for every idempotent on every member of \(A\) is a small representative family for all \(\kappa\)-presentable objects. Its full category is small because it has small Hom sets and a small object set. Retract presentability was proved in Section 1.

A \(\kappa\)-small colimit of objects of \(B\), taken in \(C\), is \(\kappa\)-presentable by Corollary 1.2. Transport its cocone to a representative in \(B\). Fullness makes that cocone a colimit in \(B\) too, and the transported cone remains the specified colimit in \(C\). This includes the initial object. \(\square\)

Write \(Y:C\to\operatorname{Set}^{B^{op}}\) for restricted Yoneda:

Proposition 2.2. The functor \(Y\) is fully faithful. For each \(c\), the category \(B\downarrow c\), retaining every arrow between its presentations, is small and \(\kappa\)-filtered, and its canonical diagram has colimit \(c\).

Proof. For full faithfulness, choose a \(\kappa\)-filtered presentation \(c=\operatorname{colim}_i a_i\) by presenting objects, replacing each by its representative in \(B\). A natural transformation \(\alpha:Y(c)\to Y(d)\) sends its legs \(c_i:a_i\to c\) to a compatible family \(d_i:a_i\to d\). The colimit gives a unique \(h:c\to d\) with \(hc_i=d_i\). Any \(t:b\to c\), with \(b\in B\), factors as \(c_i u\), because \(b\) is presentable. Naturality gives \(\alpha_b(t)=d_i u=ht\). Thus \(\alpha=Y(h)\), and the colimit legs also prove uniqueness.

Objects of \(B\downarrow c\) form the small tagged union of the sets \(\operatorname{Hom}(b,c)\); its arrows are a small tagged union of subsets of the Hom sets in \(B\). A \(\kappa\)-small diagram in this comma category has an underlying colimit \(b\in B\) by Lemma 2.1. Its compatible maps to \(c\) give \(b\to c\), hence a cocone in the comma category. The empty diagram uses the initial object of \(B\), so this index is nonempty.

Finally a cocone from this whole comma diagram to \(d\) restricts to a cocone on the chosen presentation \(a_i\to c\), giving \(h:c\to d\). Every \(b\to c\) factors through one of those legs, and the corresponding factor is an arrow of the full comma category. Cocone compatibility forces its component to be \(h\) composed with \(b\to c\). This proves the canonical colimit claim, including every arrow of the index. \(\square\)

In particular \(B\) detects isomorphisms: if every \(\operatorname{Hom}(b,f)\) is bijective, \(Y(f)\) has a natural inverse given by the inverse component functions. Full faithfulness lifts that inverse and both its identities to \(C\).

3. The essential image constructs all limits

Call a presheaf \(P:B^{op}\to\operatorname{Set}\) \(\kappa\)-continuous here if it sends every \(\kappa\)-small colimit cone of \(B\) to a limit cone of sets. The direction is contravariant.

Theorem 3.1. The essential image of \(Y\) consists exactly of these presheaves.

Proof. A represented restriction \(Y(c)\) has this property by the universal property of every such colimit in \(C\). Conversely, form the category \(E_P\) whose objects are \((b,x)\) with \(x\in P(b)\), and whose arrows \((b,x)\to(b',x')\) are \(u:b\to b'\) satisfying \(P(u)(x')=x\). This is small for the same tagged-union reason as \(B\downarrow c\).

It is \(\kappa\)-filtered. For a \(\kappa\)-small diagram of these elements, take the colimit \(b\) of its underlying diagram in \(B\). The compatible elements give, by \(\kappa\)-continuity, one \(x\in P(b)\) restricting to all of them. Its colimit legs give a cocone in \(E_P\). For the empty diagram, continuity gives a unique element of \(P(0)\), so \(E_P\) is nonempty.

There is a canonical presheaf map

At \(a\in B\), it sends the representative \(u:a\to b\), at \((b,x)\), to \(P(u)(x)\). It is surjective: \(t\in P(a)\) is represented by \(1_a\) at \((a,t)\). If \(u:a\to b\) and \(v:a\to b'\) give the same \(t\), the object \((a,t)\) has arrows \(u,v\) to their two element objects. Both classes therefore equal the class of \(1_a\) at \((a,t)\). This proves injectivity. Precomposition in \(a\) commutes with the formula, proving that (3.1) is a natural isomorphism.

Now take \(c_P=\operatorname{colim}_{(b,x)\in E_P}b\) in \(C\). Every \(a\in B\) is presentable and \(E_P\) is \(\kappa\)-filtered, so

These are the actual colimit-leg and element-evaluation maps, natural in \(a\). Thus \(Y(c_P)\simeq P\). \(\square\)

Corollary 3.2. Every locally presentable category has all small limits.

Proof. For a small diagram \((c_j)\) in \(C\), take the pointwise presheaf limit \(P=\lim_jY(c_j)\). Limits of sets commute with limits of sets: both iterated limits are the same tuples satisfying both families of arrow equations. Hence \(P\) still sends \(\kappa\)-small colimits in \(B\) to limits. By Theorem 3.1 it is \(Y(c)\) for some \(c\). For any \(z\), full faithfulness gives

The projections and their unique cone factors are exactly the ones supplied by this bijection. Thus \(c\) is the limit. The empty diagram gives the terminal object by the same argument. No assertion about the number of epimorphic quotients is needed. \(\square\)

4. Raising the regular cardinal and bounding individual objects

Theorem 4.1. If \(C\) is locally \(\kappa\)-presentable and \(\lambda\ge\kappa\) is regular, then \(C\) is locally \(\lambda\)-presentable. Every object of \(C\) is \(\mu\)-presentable for some regular \(\mu\).

Proof. A \(\lambda\)-filtered category is \(\kappa\)-filtered, so every \(\kappa\)-presentable object is \(\lambda\)-presentable through the very same comparison (1.1). Take the colimits of all graphs with fewer than \(\lambda\) vertices and edges labelled by the small category \(B\). Up to relabelling their vertices and edges by ordinals below \(\lambda\), these are a small set of graphs. A vertex label is an object of \(B\), and an edge label is an appropriate \(B\)-arrow. Their chosen colimits consequently form a small family \(S_\lambda\). Corollary 1.2 makes every member \(\lambda\)-presentable.

For \(c=\operatorname{colim}_{i\in I}a_i\), use a small presenting diagram valued in \(B\). Let \(J\) run over subgraphs of its underlying object-arrow graph having fewer than \(\lambda\) vertices and edges and containing both endpoints of every selected edge. Empty subgraphs are allowed. Inclusion makes this a small \(\lambda\)-filtered poset: a union of fewer than \(\lambda\) such graphs is another one, by regularity. It need not be a subcategory closed under composites.

For each graph \(J\), form its graph colimit \(c_J\). Inclusions give canonical maps \(c_J\to c_{J'}\); uniqueness on their vertex legs proves identities and composition. A compatible cocone from these \(c_J\) is exactly one map from each \(a_i\) satisfying every original \(I\)-arrow equation. To see both directions, use a one-vertex subgraph to recover each component and a one-edge subgraph to recover each equation; conversely the components give the unique compatible factor from each graph colimit. Therefore

Every \(c_J\) is isomorphic to a member of \(S_\lambda\). Transporting the entire diagram by chosen such isomorphisms retains its colimit and its \(\lambda\)-filtered index. This proves local \(\lambda\)-presentability, including \(\lambda=\kappa=\omega\); finite graphs suffice even though their freely generated categories can have infinitely many paths.

For the final assertion, keep one small presenting category \(I\) for \(c\). Choose a small regular \(\mu\ge\kappa\) larger than its entire arrow set. Its objects \(a_i\) are \(\mu\)-presentable, and Corollary 1.2 applied to that \(\mu\)-small diagram makes \(c\) \(\mu\)-presentable. Such a regular cardinal exists inside the fixed universe by taking a sufficiently large successor cardinal. \(\square\)

Thus a small family of objects has a common presentability bound: take a regular cardinal above the union of the small set of their bounds. Increasing a bound preserves presentability, since a more highly filtered diagram satisfies the earlier filtering condition.

The graph argument also works if the initial small presentable family expresses each object as an arbitrary small colimit rather than a filtered one. Taking \(\lambda=\kappa\) and replacing that family by its \(\kappa\)-small graph colimits supplies the filtered presentations in our definition. Thus the two common colimit-generation formulations give the same locally presentable categories.

5. From a real solution set to a left adjoint

We give the needed initial-object argument explicitly. A family in a category is weakly initial if every object receives at least one map from a member of the family.

Lemma 5.1 (Freyd). A complete locally small category with a small weakly initial family has an initial object.

Proof. Let \(D\) be the full small subcategory on the family, and \(i=\lim_{d\in D}d\), with projections \(p_d\). For an object \(x\), choose \(h:d\to x\) and set \(\ell_x=hp_d\). This does not depend on the choice. Given \(h:d\to x\) and \(h':d'\to x\), their pullback receives a map from a family member \(d''\). The resulting arrows \(d''\to d,d''\to d'\) lie in the full \(D\); the limit equations and the pullback equation equate the two composites from \(i\).

For \(f:x\to y\), the choice \(fh:d\to y\) shows \(f\ell_x=\ell_y\). At a family object \(d\), using \(1_d\) shows \(\ell_d=p_d\). Naturality at every projection \(p_d:i\to d\) now gives \(p_d\ell_i=p_d\). Limit projections jointly determine a map, hence \(\ell_i=1_i\). For any \(v:i\to x\), naturality gives \(v=v\ell_i=\ell_x\). Thus there is exactly one such map and \(i\) is initial. Completeness includes a terminal object, so its weakly initial family cannot be empty. \(\square\)

Theorem 5.2. Let \(G:C\to D\) be a functor between locally presentable categories. If \(G\) is accessible and preserves all small limits, it has a left adjoint.

Proof: the solution set. Fix \(d\in D\). Choose a regular \(\mu\) at least the local-presentability bounds for \(C,D\), the bound for accessibility of \(G\), and an individual presentability bound for \(d\). Theorem 4.1 makes \(C\) locally \(\mu\)-presentable, and \(G\) preserves \(\mu\)-filtered colimits. Let \(B_\mu\) be its small full presentable category. The family

is small in \(d\downarrow G\), including all such arrows \(u\).

For \(v:d\to G(c)\), write \(c=\operatorname{colim}_j b_j\) as a \(\mu\)-filtered presentation. Since \(G\) preserves that colimit and \(d\) is \(\mu\)-presentable, there are \(j\) and \(u:d\to G(b_j)\) with

Thus \(c_j\) is an actual comma arrow from a member of (5.1) to \((c,v)\). This is a weakly initial family, not merely a bound on possible object names.

The comma category is locally small. It is complete: take the limit of its underlying \(C\)-diagram. Limit preservation by \(G\) turns the compatible maps from \(d\) into a unique map \(d\to G(\lim c_j)\). The same cone uniqueness proves the comma limit property, including the empty diagram. Lemma 5.1 gives an initial object \((L(d),\eta_d)\).

Proof: arrows and the natural bijection. For \(a:d\to d'\), define \(L(a)\) as the unique map satisfying

Initiality gives it by viewing the right side as a comma object. Identity and composite maps satisfy exactly the same equation, so \(L\) is a functor. For every \(c\), initiality supplies the inverse of

Postcomposition proves naturality in \(c\), and (5.3) proves naturality in \(d\). This is the adjunction \(L\dashv G\).

Its counit \(\epsilon_c:L(G(c))\to c\) is defined by \(G(\epsilon_c)\eta_{G(c)}=1_{G(c)}\). This is one triangle identity; naturality follows from uniqueness in (5.4). For the other, naturality of \(\eta\) at \(\eta_d:d\to G(L(d))\) gives

The injectivity of (5.4) makes \(\epsilon_{L(d)}L(\eta_d)=1_{L(d)}\). Hence both actual unit and counit maps satisfy their triangle identities. \(\square\)

6. Every right adjoint has an accessibility bound

Theorem 6.1. A functor \(G:C\to D\) between locally presentable categories has a left adjoint exactly when it preserves all small limits and is accessible.

Proof. Sufficiency is Theorem 5.2. Suppose \(L\dashv G\). For a limit cone in \(C\), the natural adjunction and the Hom universal property give

These maps compose with the actual cone legs. For all \(d\), this is exactly the limit universal property, so \(G\) preserves limits.

Choose a small full presenting category \(B\) of \(\kappa\)-presentable probes in \(D\). The objects \(L(b)\), for \(b\in B\), have a common presentability bound \(\mu\) in \(C\), by Theorem 4.1. Increase it to at least \(\kappa\). Each \(b\) is also \(\mu\)-presentable. For a small \(\mu\)-filtered diagram \((c_i)\), let \(\gamma\) be the canonical comparison below. On applying \(\operatorname{Hom}_D(b,-)\), it becomes the composite

Every step is natural and uses the original colimit legs. Hence it is precisely \(\operatorname{Hom}_D(b,\gamma)\), where

All these tested maps are bijective. The conservative presentable probes from Proposition 2.2 make \(\gamma\) invertible. Thus \(G\) is accessible. The bound depends on the objects \(L(b)\); it need not be the original local-presentability bound of either category. \(\square\)

7. Colimit preservation gives the other adjoint

Theorem 7.1. Let \(C\) be locally presentable and \(D\) locally small. A functor \(F:C\to D\) has a right adjoint exactly when it preserves every small colimit. No local-presentability hypothesis on \(D\) is required.

Proof: construction. Use the small full category \(B\) of presentable objects from Section 2. For \(d\in D\), define

Its restriction maps are precomposition with \(F\), and maps \(d\to d'\) act by postcomposition. Local smallness makes it a small-valued presheaf. If \(F\) preserves colimits, \(P_d\) sends every \(\kappa\)-small colimit in \(B\) to a limit of sets. Theorem 3.1 gives an object \(R(d)\) and natural identifications

Postcomposition \(P_d\to P_{d'}\) lifts by full faithfulness to \(R(d)\to R(d')\). Identities and compositions lift to their identities and compositions, so this defines a functor \(R\).

For arbitrary \(c\), use its full small density presentation \(B\downarrow c\) from Proposition 2.2. Colimit uniqueness and (7.2) give

Here the last comparison uses the actual image under \(F\) of that density colimit.

Its forward map has a useful explicit description. The maps \(F(b)\to d\) corresponding to all \(b\to R(d)\) form a compatible family. Since \(F\) preserves their density colimit, they induce

For \(h:c\to R(d)\), the corresponding arrow is \(\epsilon_dF(h)\). Checking this on every leg \(F(b)\to F(c)\) proves it agrees with (7.3). This formula gives naturality in \(c\). The definition of \(R\) by postcomposition gives naturality in \(d\), first on the presentable legs and then by colimit uniqueness. Thus (7.3) is the natural adjunction bijection \(F\dashv R\).

Its unit \(c\to R(Fc)\) corresponds to \(1_{Fc}\). The counit is (7.4). The first inverse-bijection identity gives \(\epsilon_{Fc}F(\eta_c)=1_{Fc}\). Naturality of the counit gives

The injectivity of (7.3), with \(c=R(d)\), yields \(R(\epsilon_d)\eta_{R(d)}=1_{R(d)}\), the second triangle identity.

Proof: necessity. If \(F\dashv R\) and \(c=\operatorname{colim}_i c_i\), then for every \(d\)

The bijection is given by restriction to the image colimit legs. It proves that \(Fc\) with those legs is the colimit of the \(Fc_i\), including the empty diagram. \(\square\)

Corollary 7.2. For a locally presentable \(C\), a covariant functor \(H:C\to\operatorname{Set}\) is corepresentable exactly when it preserves all small limits and is accessible. A contravariant \(K:C^{op}\to\operatorname{Set}\) is representable exactly when it sends all small colimits of \(C\) to limits of sets.

Proof. In the first case Theorem 6.1 gives \(L\dashv H\), and

by applying the adjunction to the singleton set. Conversely a corepresented functor preserves limits by the Hom universal property, and is accessible by the individual object rank in Theorem 4.1.

For the second case, regard \(K^{op}\) as a functor \(C\to\operatorname{Set}^{op}\). Its colimit preservation is the stated hypothesis. The category \(\operatorname{Set}^{op}\) is locally small, so Theorem 7.1 supplies a right adjoint \(R\), even though local presentability of this opposite category was not assumed. Apply the adjunction with target \(1\):

All identifications are natural. The converse is the contravariant Hom universal property. \(\square\)

8. Four graded exercises with complete solutions

Exercise 1 (foundation: the exact bound for a power)

For an infinite regular \(\lambda\), determine which sets \(A\) are \(\lambda\)-presentable in sets. Determine exactly when \(X\mapsto X^A\) preserves \(\lambda\)-filtered colimits, and give its left adjoint with unit and counit.

Solution. If \(|A|<\lambda\), then \(\operatorname{Hom}(A,-)\) is a product of \(|A|\) copies of the identity functor. Lemma 1.1 makes its comparison with every \(\lambda\)-filtered colimit bijective. This includes \(A=\varnothing\), using the empty-limit case.

If \(|A|\ge\lambda\), let \(S\) range over all subsets of \(A\) of size below \(\lambda\), with inclusions. Regularity makes this poset \(\lambda\)-filtered, and its union is \(A\). The identity \(A\to A\) is not in the image of

Every represented function has image in one such \(S\), whereas the identity has image \(A\). Thus exactly the sets of size below \(\lambda\) are \(\lambda\)-presentable, and exactly for these sets does the power functor preserve \(\lambda\)-filtered colimits.

Currying gives \(\operatorname{Hom}(A\times B,X)\simeq\operatorname{Hom}(B,X^A)\): a map \(h\) goes to \(b\mapsto(a\mapsto h(a,b))\), and evaluation reverses this. These formulas are natural. The unit sends \(b\) to \(a\mapsto(a,b)\), and the counit sends \((a,f)\) to \(f(a)\); substitution verifies both triangle identities. An infinite \(A\) therefore gives a right adjoint which is accessible but not finitary. Sets are locally finitely presentable because every set is the filtered union of its finite subsets, whose presentability was just proved.

Exercise 2 (standard: rational Hom is not finitary)

Prove that \(\operatorname{Hom}_{\mathbb Z}(\mathbb Q,-):\operatorname{Ab}\to\operatorname{Ab}\) preserves \(\aleph_1\)-filtered colimits but fails to preserve an explicit ordinary filtered colimit. Specify its left adjoint.

Solution. We use the full module tensor–Hom adjunction in Ring actions and tensor–Hom. Its specialization gives the left adjoint \(\mathbb Q\otimes_{\mathbb Z}-\), with the natural correspondence sending \(f(q\otimes b)\) to \(b\mapsto(q\mapsto f(q\otimes b))\). The balanced tensor relations make the reverse map well defined.

Filtered colimits of abelian groups have their filtered colimit sets as underlying sets. Given two representatives, receive them at a common stage to define their sum; eventual equality proves independence of the stage. Inverses and zero are defined at their stages, and a common receiving stage verifies each group identity. Compatible group homomorphisms then give the unique homomorphism out, proving the universal property and the assertion about underlying sets.

Present \(\mathbb Q\) as the sequential colimit of copies of \(\mathbb Z\), whose map from stage \(n\) to \(n+1\) multiplies by \(n+1\), with leg \(k\mapsto k/n!\) for \(n\ge1\). These legs identify the increasing subgroups \((1/n!)\mathbb Z\) with union \(\mathbb Q\), so this is its colimit. Consequently

This is a countable limit of the underlying sets of \(A\), with the actual compatibility equations. Lemma 1.1 interchanges it with any \(\aleph_1\)-filtered colimit. The resulting bijections respect addition, so the group-valued functor preserves these colimits.

For failure at \(\omega\), let \(B_n=\mathbb Z[1/n!]\), with inclusions. Their union is \(\mathbb Q\). Choose a prime \(p\) not dividing \(n!\), for instance a prime divisor of \(n!+1\). A nonzero element of \(B_n\) cannot be divisible by \(p^k\) inside \(B_n\) for every \(k\): its denominator has no factor \(p\), and its numerator has only finitely many such factors. Every element in the image of a map \(\mathbb Q\to B_n\) is divisible by every \(p^k\), since its preimage is. Hence every such map is zero.

But \(\operatorname{Hom}_{\mathbb Z}(\mathbb Q,\mathbb Q)\simeq\mathbb Q\), by multiplication by the value at \(1\). Additivity and unique divisibility give \(h(m/n)=(m/n)h(1)\); every rational value at \(1\) defines such a map. The ordinary filtered comparison is thus \(0\to\mathbb Q\), and is not surjective. This is compatible with accessibility at the larger bound.

Exercise 3 (advanced: a mixed torsion-and-power adjoint)

Fix \(n\ge2\) and define \(F(A)=A[n]\times A^{\mathbb N}\), where \(A[n]=\{a:na=0\}\). Construct its left adjoint, unit and counit. Prove it is \(\aleph_1\)-accessible but not finitary.

Solution. The left adjoint is

A map \(B\to A[n]\) is exactly a map \(B/nB\to A\): it kills \(nB\), and conversely the image of such a quotient map is annihilated by \(n\). A map \(B\to A^{\mathbb N}\) is exactly a family of maps \(B\to A\), equivalently a map from the direct sum of its labelled copies. The coproduct of the quotient summand with these copies therefore gives a natural bijection \(\operatorname{Hom}(L(B),A)\simeq\operatorname{Hom}(B,F(A))\).

Let \(i_0:B/nB\to L(B)\) and \(i_j:B\to L(B)\) denote the summand inclusions. The unit is

Its first component is indeed annihilated by \(n\). For \(z=(a,(x_j))\in F(A)\), the counit on the first summand \(F(A)/nF(A)\) sends \([z]\) to \(a\). This is well defined because the first component of \(nz'\) is zero. On the copy of \(F(A)\) labelled \(j\), it sends \(z\) to \(x_j\). These maps uniquely give a homomorphism from the direct sum to \(A\).

For the triangle on \(L(B)\), evaluate on the first summand and each labelled copy: the first gives \(i_0([b])\), and the \(j\)-th gives \(i_j(b)\). Thus the composite is the identity. For the triangle on \(F(A)\), its first coordinate recovers \(a\), and its \(j\)-th coordinate recovers \(x_j\). This verifies the other identity.

The functor \(A[n]\) is the equalizer of multiplication by \(n\) and zero. Along with the countable power, the expression for \(F\) is a countable limit of underlying sets. The filtered-group construction in Exercise 2 and Lemma 1.1 prove preservation of \(\aleph_1\)-filtered colimits, with the resulting maps homomorphisms.

Let \(A_m=\bigoplus_{j=1}^m\mathbb Z\), with the usual inclusions and union \(A=\bigoplus_{j\ge1}\mathbb Z\). Their \(n\)-torsion is zero. The element \((e_j)_{j\ge1}\in A^{\mathbb N}\), with \(e_j\) its \(j\)-th basis vector, belongs to no \(A_m^{\mathbb N}\). It is therefore absent from the image of the filtered comparison for \(F\). Hence this right adjoint is not finitary.

Exercise 4 (synthesis: an accessible inclusion with the other adjoint)

Let \(\operatorname{Tor}\) be the full category of torsion abelian groups. Prove it is locally finitely presentable, and that its inclusion into abelian groups is accessible and has a right adjoint, but has no left adjoint.

Solution. Arbitrary direct sums of torsion groups are torsion, since each element has finite support and the orders of its finitely many components have a common finite multiple. Quotients of torsion groups are torsion. The usual coproduct-and-relations construction of an abelian-group colimit consequently stays in \(\operatorname{Tor}\), and its ambient universal property gives every small colimit there.

Every torsion group is the filtered union of its finite subgroups. The subgroup generated by finitely many elements of orders \(m_1,\ldots,m_k\) is an image of the finite group \(\prod_i\mathbb Z/m_i\), hence finite. These finite subgroups receive finite families by taking the subgroup they generate. Finite groups have a small set of isomorphism types, since their addition tables are tables on finite sets.

A finite abelian group is finitely presentable with respect to filtered colimits in \(\operatorname{Tor}\). For a map into such a colimit, choose stage representatives for each of its finitely many elements. A common stage, enlarged once more for its finitely many addition and zero equations, makes those representatives a homomorphism. Two such represented homomorphisms which agree in the colimit agree at a common later stage, using finitely many equality witnesses. This proves both parts of its Hom comparison. Thus the stated small finite family gives local finite presentability. Filtered colimits are created by the inclusion, so the inclusion is accessible.

For an abelian group \(A\), its subgroup \(t(A)\) of finite-order elements is functorial: sums and inverses of such elements have finite order, and homomorphisms send them to finite-order elements. Any map from a torsion group into \(A\) has image in \(t(A)\), uniquely. This is the natural adjunction

Thus inclusion has right adjoint \(t\).

The product of torsion groups \(T_j\) inside \(\operatorname{Tor}\) is \(t(\prod_j T_j)\). Indeed a map from a torsion group into the ambient product has torsion image, so uniquely factors through this subgroup. For \(T_j=\mathbb Z/p^j\) and \(j\ge1\), the ambient product has the element \((1\bmod p^j)_j\) of infinite order: any positive annihilating integer would be divisible by every \(p^j\). Hence its torsion subgroup is a proper subgroup. The inclusion fails to preserve this product, and therefore cannot be a right adjoint, or equivalently cannot have a left adjoint. Accessibility alone does not imply the limit-preservation clause of Theorem 6.1.

References

[Riehl] Emily Riehl, Category Theory in Context, Dover Publications, 2016. Author-hosted full text, Section 4.7, especially the locally presentable adjoint criteria. The presentable-probe, cardinal and solution-set constructions above supply the proofs used in this lesson.