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Draft lesson.

Relative injectivity and small test families

Written with GPT-6.1 Sol (OpenAI). Original exposition: CC0.

For unique extension tests and the accessible reflector they generate, use Small orthogonality and reflective localizations. An extension problem asks whether a map defined on one object continues across a specified arrow. Ordinary injectivity uses every monomorphism as a test. Other problems use a chosen family of arrows, and their solutions need not live in an abelian category. We prove existence from small tests and a factorization bound, then give a criterion under which those tests control a larger class. The criterion uses objects whose outgoing arrows all split. We prove that categorical maximality statement without an order or a monomorphism assumption.

The cardinal and ordinal facts used below are proved in Testing injectivity on generators and constructing enough injectives, Lemma 7.1. For accessibility and its canonical Hom comparison, see Accessible objects from a single generator. We use the universe conventions in Universes and small categories. Basic references for universal constructions and the ordinary injective application are Emily Riehl's Category Theory in Context and the Stacks Project's Injectives in Grothendieck categories, listed below. Every argument needed here is supplied in this lesson or these linked prerequisites.

All categories are locally small in a fixed universe, and all indexing sets and diagrams are small in that universe. We use Choice. A subcategory need not be full: it may specify which arrows are allowed while keeping the same objects as its ambient category. A filtered category is nonempty; every two objects have a common receiving object, and every pair of parallel arrows becomes equal after composition with some arrow. For an infinite cardinal \(\kappa\), a category is \(\kappa\)-filtered when every diagram with fewer than \(\kappa\) arrows admits a cocone.

1. Extension tests and their variance

Fix a family \(\mathcal T\) of arrows in a category \(\mathcal C\). An object \(E\) is \(\mathcal T\)-injective if, for every \(s: A\to B\) in \(\mathcal T\), restriction is surjective:

\[ \operatorname{Hom}_{\mathcal C}(B,E) \longrightarrow\operatorname{Hom}_{\mathcal C}(A,E), \qquad v\longmapsto vs. \tag{1.1} \]

Thus every map \(a: A\to E\) has an extension \(b: B\to E\) satisfying \(bs=a\). The extension need not be unique, and the testing arrow need not be monic. We allow a family that is larger than a set, provided the individual Hom sets are small.

The dual definition is \(\mathcal T\)-projectivity: the object is injective for the opposite arrows in \(\mathcal C^{\mathrm{op}}\). Concretely, for \(s: A\to B\), every arrow \(E\to B\) must lift to \(E\to A\). Reversing arrows therefore exchanges extension and lifting, and exchanges filtered colimits with cofiltered limits. When \(\mathcal C\) is abelian and \(\mathcal T\) is the family of all monomorphisms, (1.1) is exactly the ordinary injectivity condition. No additional exactness assertion is hidden in that identification.

Let \(\mathcal D\subset\mathcal C\) be a subcategory. We will use two assumptions about its allowed changes:

The second condition says that one extension problem can be solved after an allowed change of target. It does not require a pushout to exist. If the appropriate pushout does exist and its target change belongs to \(\mathcal D\), it supplies such a completion.

2. Completing a whole small family

Lemma 2.1. Under the two assumptions above, a small family of problems

\[ u_i: A_i\longrightarrow B_i\quad\text{in }\mathcal D, \qquad a_i: A_i\longrightarrow X\quad\text{in }\mathcal C \tag{2.1} \]

with \(X\in\mathcal D\) can be solved after one allowed change \(j: X\to X'\): there are ambient maps \(b_i: B_i\to X'\) with \(b_i u_i=ja_i\) for every \(i\).

Proof. For the empty family take \(j=1_X\). Otherwise well-order its index set, with ordinal length \(\mu\). Construct \(X_\alpha\) and maps \(X_\alpha\to X_\beta\) in \(\mathcal D\), for \(\alpha\leq\beta\leq\mu\). Start with \(X_0=X\). At stage \(\alpha+1\), apply the second assumption to \(u_\alpha\) and the composite \(A_\alpha\to X\to X_\alpha\). This gives \(X_\alpha\to X_{\alpha+1}\) and an extension from \(B_\alpha\). Compose every earlier extension with that target change.

At a nonzero limit stage \(\beta\), take the colimit of the earlier ordinal diagram in \(\mathcal D\). The index is filtered, so the first assumption applies. Its cocone maps belong to \(\mathcal D\); all earlier extensions continue to this colimit as ambient maps. Define all other transition maps by composition with these cocone maps. Their identities and composition equations follow from the colimit property.

The map \(X\to X_\mu\) is an allowed arrow, and each extension made at stage \(i+1\) composes to the required \(b_i: B_i\to X_\mu\). Every square remains commutative under these compositions. \(\square\)

The assertion is simultaneous existence. Arbitrary choices of completions do not make this construction a functor. Functoriality requires further structure, such as chosen universal pushouts whose induced maps can be checked.

3. Saturation through a long ordinal

For this section, \(\mathcal T\) is a small set of arrows of \(\mathcal D\). Assume there is one small infinite cardinal \(\kappa\) such that, for every domain \(A\) of these arrows and every small \(\kappa\)-filtered diagram \((X_i)\) in \(\mathcal D\), the canonical comparison

\[ \operatorname{colim}_i\operatorname{Hom}_{\mathcal C}(A,X_i) \longrightarrow \operatorname{Hom}_{\mathcal C}\!\left(A,\operatorname{colim}_iX_i\right) \tag{3.1} \]

is surjective. Ordinary \(\kappa\)-accessibility of each domain is a sufficient stronger assumption: it makes this comparison bijective for every such existing colimit in \(\mathcal C\). Only factorization, hence surjectivity, is used here. If necessary enlarge \(\kappa\) to an infinite regular cardinal; a diagram filtered for the larger cardinal is filtered for the original one.

Theorem 3.1. Every \(X\in\mathcal D\) has an arrow \(X\to E\) in \(\mathcal D\) with \(E\) \(\mathcal T\)-injective in \(\mathcal C\).

Proof. Choose a small limit ordinal \(\theta\) of cofinality greater than \(\kappa\), using the linked cardinal lemma. We construct \(X_\alpha\) for \(\alpha\leq\theta\), with coherent transition maps in \(\mathcal D\). Start at \(X_0=X\). At each successor stage, form the set

\[ \Gamma_\alpha= \{(s,a): s: A_s\to B_s\text{ belongs to }\mathcal T, \ a: A_s\to X_\alpha\}. \tag{3.2} \]

It is small: the testing arrows form a set and every Hom set is small. Lemma 2.1 completes all these problems after one arrow \(X_\alpha\to X_{\alpha+1}\) in \(\mathcal D\). At every nonzero limit stage use the colimit, with all its cocone and transition maps, exactly as in that lemma. Set \(E=X_\theta\).

The ordinal category of indices \(\alpha<\theta\) is \(\kappa\)-filtered. The indices appearing in a diagram with fewer than \(\kappa\) arrows are bounded below \(\theta\), by the cofinality condition, and an upper bound supplies a cocone. The empty diagram has a cocone because \(0<\theta\). Therefore (3.1) applies to its colimit \(E\).

For a test \(s: A_s\to B_s\) and any \(a: A_s\to E\), surjectivity in (3.1) supplies a stage \(\alpha<\theta\) and \(a_\alpha: A_s\to X_\alpha\) whose composite is \(a\). The labelled problem \((s,a_\alpha)\) was completed at the next stage. Since \(\theta\) is a limit, \(\alpha+1<\theta\). The resulting map \(B_s\to X_{\alpha+1}\to E\) extends \(a\). Hence \(E\) is \(\mathcal T\)-injective. The initial cocone map \(X\to E\) belongs to \(\mathcal D\), as required. \(\square\)

The hypotheses allow the chosen class of target changes to consist of monomorphisms, epimorphisms or other arrows. They do not assert that all arrows of \(\mathcal C\) lie in \(\mathcal D\), or that the result is a universal arrow to the injective objects. Exercise 4 distinguishes an embedding from a reflection.

Corollary 3.2 (a common rank for small test families). Suppose the ambient category \(\mathcal C\) has all small colimits and finite limits, its small filtered colimits preserve finite limits, it has an object \(G\) whose Hom functor detects isomorphisms, and every epimorphism is the coequalizer of its kernel pair. Let \(\mathcal D\subseteq\mathcal C\) satisfy the two target-change assumptions of Section 1. For every small set \(\mathcal T\) of arrows of \(\mathcal D\), each \(X\in\mathcal D\) has an allowed arrow \(X\to E\) with \(E\) \(\mathcal T\)-injective in \(\mathcal C\). In particular, a Grothendieck abelian category satisfies these ambient hypotheses.

Proof. The domains \(A_s\), indexed by the small set of tests \(s\in\mathcal T\), form a small family of objects. The common-cardinal theorem, Theorem 4.1, applies under precisely these ambient hypotheses: its strict epimorphisms are the stated kernel-pair coequalizers. It supplies an infinite regular \(\kappa\) for which every \(A_s\) is \(\kappa\)-accessible in \(\mathcal C\). A small \(\kappa\)-filtered diagram in \(\mathcal D\) is ordinarily filtered, and its chosen colimit is also its colimit in \(\mathcal C\) by the first target-change assumption. The accessibility comparison therefore gives exactly the bijection, and hence the surjectivity, required in (3.1). Theorem 3.1 now gives the asserted arrow and extensions. For an empty test set, take \(E=X\) and the identity arrow.

To verify the Grothendieck case, small colimits and finite limits exist, and AB5 makes each small filtered-colimit functor exact. It preserves finite biproducts, since they are finite coproducts, and it preserves kernels by exactness. The finite-limit difference construction then proves preservation of all finite limits. A generator detects isomorphisms by the abelian detection theorem.

Finally, an epimorphism \(q:B\to A\) in an abelian category is the cokernel of its kernel \(k:K\to B\). For its kernel pair \(p_1,p_2:B\times_A B\rightrightarrows B\), the difference \(p_1-p_2\) factors through \(k\). Thus every map killing \(k\) equalizes \(p_1,p_2\). Conversely, the pair \((k,0)\) factors through this kernel pair, so a map equalizing \(p_1,p_2\) kills \(k\). The cokernel factor property of \(q\) is therefore exactly its kernel-pair coequalizer property. This verifies the last ambient hypothesis and completes the specialization. \(\square\)

4. Objects whose outgoing arrows split

Call \(M\) split maximal when every arrow \(u: M\to N\) has a left inverse \(r: N\to M\). This condition is also called quasi-terminality. An endomorphism of \(M\) is then an automorphism. Indeed, if \(vu=1_M\), choose \(w\) with \(wv=1_M\). Then

\[ w=w(vu)=(wv)u=u, \qquad uv=wv=1_M. \tag{4.1} \]

It need not be a terminal object: neither existence of arrows from other objects nor uniqueness of those arrows is part of the definition.

Lemma 4.1 (returning after an outgoing arrow). A nonempty essentially small category with all small filtered colimits has an object \(L\) such that an arrow \(L\to N\) always implies the existence of an arrow \(N\to L\).

Proof. We may work in an equivalent small category. Equivalences preserve and reflect colimit universal properties: applying their full and faithful Hom bijections transports the cocone factorization and its uniqueness. Existence of the stated return arrows is also invariant under these equivalences.

Consider the set of filtered subcategories of this small category, ordered by inclusion; they need not be full. It is nonempty because the subcategory consisting of one object and its identity is filtered. The union of an inclusion chain is a subcategory: any finite collection of identities or compositions belongs to one member of the chain. It is filtered for the same reason. Any two of its objects and any pair of its parallel arrows occur together in one filtered member, which supplies the needed receiving object or equalization. Thus every nonempty chain has an upper bound, and a chosen one-object subcategory bounds the empty chain. Zorn's lemma supplies a maximal filtered subcategory \(\mathcal H\).

Let \(L\) be the colimit of its inclusion diagram, with cocone maps \(a_H: H\to L\). If \(u: L\to N\) and \(N\) were outside \(\mathcal H\), adjoin that object, its identity and, for each old object \(H\), the one arrow \(ua_H: H\to N\). These arrows are closed under precomposition with the old arrows, by the cocone equations. There are no new arrows from \(N\) to an old object. The enlarged subcategory therefore has terminal object \(N\), is filtered, and strictly contains \(\mathcal H\), a contradiction. Hence \(N\) already belongs to \(\mathcal H\), and its cocone map \(a_N: N\to L\) is the required return. \(\square\)

Apply this lemma to the category of arrows out of a fixed object \(A\). It is essentially small when the original category is: choose object representatives, then use their small Hom sets to choose representatives of the arrows out of \(A\). It has all small filtered colimits. Such a diagram consists of a filtered diagram of targets and compatible arrows from \(A\); the nonempty filtered index makes these a map from \(A\) to the target colimit. Factorization of a cocone in the arrow category is exactly its underlying colimit factorization together with that one compatibility equation.

Consequently, for every \(A\) there is \(f: A\to B\) such that

\[ \text{for every }u: B\to N\text{ there is }v: N\to B \text{ with }vuf=f. \tag{4.2} \]

Lemma 4.2 (one cocone map is epic). Let \(\mathcal E\) be small, let \(\rho>|\operatorname{Mor}(\mathcal E)|\) be a small infinite cardinal, and let \((Y_i)\) be a small \(\rho\)-filtered diagram with colimit \(Y\) in \(\mathcal E\). One of its cocone maps \(q_i: Y_i\to Y\) is an epimorphism.

Proof. Among the actual cocone arrows \(q_i\), choose one index for each distinct arrow of \(\mathcal E\). There are fewer than \(\rho\) choices. A cocone over their discrete index diagram gives one receiving index \(i_0\) and arrows from all the chosen indices to \(i_0\). Every original \(q_i\) equals one chosen arrow, with the same domain object, and hence factors through \(q_{i_0}\). If two maps out of \(Y\) agree after \(q_{i_0}\), they agree after every \(q_i\). The colimit property says they are equal. Thus \(q_{i_0}\) is epic. This uses only the total number of arrows and the joint epimorphic property of a colimit cocone. \(\square\)

Theorem 4.3 (categorical maximality). A nonempty essentially small category admitting all small filtered colimits has a split maximal object.

Proof. Again replace the category by an equivalent small one, and choose \(\rho\) greater than its arrow cardinal. Choose a small limit ordinal \(\theta\) with cofinality greater than \(\rho\). Beginning at any \(Y_0\), construct a coherent ordinal diagram. At every successor choose \(f_\alpha: Y_\alpha\to Y_{\alpha+1}\) with property (4.2); at every nonzero limit take the colimit of all earlier stages. Let \(M\) be the colimit over \(\alpha<\theta\), with maps \(q_\alpha: Y_\alpha\to M\).

This ordinal index is \(\rho\)-filtered, so Lemma 4.2 gives an epic \(q_\alpha\). For any \(u: M\to N\), property (4.2) applied to \(u q_{\alpha+1}\) gives \(v: N\to Y_{\alpha+1}\) satisfying

\[ v u q_{\alpha+1} f_\alpha=f_\alpha. \tag{4.3} \]

Put \(r=q_{\alpha+1}v\). The transition equation \(q_{\alpha+1}f_\alpha=q_\alpha\) gives \(ruq_\alpha=q_\alpha\). Since \(q_\alpha\) is epic, \(ru=1_M\). This supplies a left inverse for every outgoing \(u\), which proves split maximality. Equivalence transports these left inverses back to the original category. \(\square\)

The epimorphism in the proof is categorical right cancellation. The category need not be abelian, its arrows need not be monomorphisms, and no Hom functor is assumed to preserve its filtered colimits.

5. Recognizing the larger extension class

We return to \(\mathcal D\subset\mathcal C\) and the two assumptions in Section 1. Now \(\mathcal T\subset\operatorname{Mor}(\mathcal D)\) can be an arbitrary family. For an arrow \(f: X\to Z\) of \(\mathcal D\), consider the following test property: every pullback square with top arrow in \(\mathcal T\) admits a diagonal \(\ell\) making both triangles commute.

\[ \begin{array}{ccc} A&\xrightarrow{s}&B\\ {a}\downarrow&&\downarrow{b}\\ X&\xrightarrow{f}&Z \end{array} \qquad \ell: B\longrightarrow X, \quad \ell s=a,\quad f\ell=b. \tag{5.1} \]

We impose three additional assumptions, stating the ambient categories explicitly:

\[ \begin{array}{ccc} A&\xrightarrow{s}&B\\ {a}\downarrow&&\downarrow{j}\\ X&\xrightarrow{g}&P \end{array} \qquad h: P\longrightarrow Z, \quad hg=f,\quad hj=b, \quad g,h\in\operatorname{Mor}(\mathcal D). \tag{5.2} \]

The factorization assumption concerns pullback squares. It does not claim that every square is a pullback, or that an arbitrary ambient pushout has its structure maps in \(\mathcal D\). The last assumption is a genuine isomorphism-detection hypothesis, not a consequence of injectivity.

Theorem 5.1. Under these five assumptions, every \(\mathcal T\)-injective object of \(\mathcal C\) is injective for the entire family \(\operatorname{Mor}(\mathcal D)\).

Proof. Let \(E\) be \(\mathcal T\)-injective, and fix \(f: X\to Z\) in \(\mathcal D\) and \(a: X\to E\) in \(\mathcal C\). Form a category \(\mathcal P\) of partial extensions. Its objects are diagrams

\[ X\xrightarrow{u}P\xrightarrow{v}Z, \qquad vu=f, \qquad w: P\longrightarrow E, \quad wu=a, \quad u,v\in\operatorname{Mor}(\mathcal D). \tag{5.3} \]

An arrow to a second such object is \(t: P\to P'\) in \(\mathcal D\) with \(tu=u'\), \(v't=v\) and \(w't=w\). The object \((1_X,f,a)\) makes \(\mathcal P\) nonempty.

This category is essentially small. Choose a small representative family of arrows \(u: X\to P\) using the first additional assumption. For each representative target, the possible \(v\) and \(w\) lie in small Hom sets. Keep just the pairs satisfying (5.3). Isomorphisms in the allowed arrow category transport these pairs and give representatives of every partial extension. The resulting representative category has small Hom sets, hence a small set of arrows.

It also has all small filtered colimits. For a filtered diagram of partial extensions, take the target colimit \(P\) in \(\mathcal D\). Its cocone maps yield \(u: X\to P\) in \(\mathcal D\), because the maps from the fixed \(X\) agree after transitions and the index is nonempty and filtered. The compatible \(v_i\) give \(v: P\to Z\) in \(\mathcal D\) by the colimit property there. Preservation of this colimit in \(\mathcal C\) gives \(w: P\to E\). Their required equations follow after every target cocone map; the equation \(wu=a\) follows using any index. The same universal properties give the colimit factorization in \(\mathcal P\) and its uniqueness.

Theorem 4.3 now supplies a split maximal partial extension \((u_0,v_0,w_0)\) with intermediate object \(P_0\). We prove that \(v_0\) has property (5.1). Take any such pullback square for \(v_0\) and \(s: A\to B\) in \(\mathcal T\). Assumption (5.2) supplies

\[ P_0\xrightarrow{g}Q\xrightarrow{h}Z, \qquad hg=v_0, \qquad Q=P_0\amalg_A B, \tag{5.4} \]

with \(g,h\) allowed and with the specified pushout map \(j: B\to Q\). Injectivity of \(E\) for \(s\) extends \(w_0 a: A\to E\) to \(b_E: B\to E\). The pushout property glues \(w_0\) and \(b_E\) into \(w_1: Q\to E\). Therefore \((g u_0,h,w_1)\) is another partial extension, and \(g\) gives an arrow to it from the split maximal one.

Its left inverse in \(\mathcal P\) is an allowed map \(r: Q\to P_0\) satisfying

\[ rg=1_{P_0},\qquad v_0r=h,\qquad w_0r=w_1. \tag{5.5} \]

The diagonal \(\ell=rj: B\to P_0\) then satisfies

\[ \ell s=rga=a, \qquad v_0\ell=hj=b. \tag{5.6} \]

Thus every pullback test for \(v_0\) has a diagonal. The isomorphism-detection assumption makes \(v_0\) invertible in \(\mathcal C\). Finally \(w_0v_0^{-1}: Z\to E\) extends the original \(a\), since \(v_0u_0=f\) and \(w_0u_0=a\). Both the extension and the testing arrow were arbitrary, proving the theorem. \(\square\)

This theorem has no smallness or accessibility assumption on the testing family. Its size control comes from the allowed partial-extension categories. The existence theorem in Section 3 instead uses a small testing set and factorization bounds, without the isomorphism-detection assumptions. These are different uses of smallness.

6. Two concrete computations

Example 6.1: successive tests on an ordinal chain. Let \(\mathcal C=\mathcal D\) be the poset category

\[ 0<1<2<\cdots<\omega, \qquad \mathcal T=\{n\longrightarrow n+1: n<\omega\}. \tag{6.1} \]

Every nonempty filtered diagram has its supremum as colimit. A one-problem completion is the maximum of its target and the current object. A finite \(n\) is accessible for ordinary filtered colimits: if \(n\leq\sup_i x_i\), some \(x_i\) is at least \(n\). For otherwise all values are bounded by \(n-1\); the case \(n=0\) follows from nonemptiness. The Hom sets have at most one element, so the induced comparison is also injective.

Starting at \(0\), the simultaneous completion of all problems at stage \(k\) can be chosen as \(k\to k+1\). Precisely the tests with \(n\leq k\) have maps into that stage, and each \(n+1\) lies below \(k+1\). At the countable limit the target is \(\omega\), and all tests extend. Any later target can remain \(\omega\). No finite object is \(\mathcal T\)-injective, because its own successor test fails. Thus the only injective target is \(\omega\), and the limit stage is essential in this computation.

Example 6.2: splitting does not give uniqueness. Take the one-object category of a nontrivial small group \(H\). Every arrow is invertible, so its object is split maximal; it is not terminal because its endomorphism set has more than one member.

This category does admit every small filtered colimit. To verify that assertion rather than assume it, let \(F: I\to H\) be a filtered diagram and fix \(i_0\). For any \(i\), choose a span \(i\xrightarrow{a}k\xleftarrow{b}i_0\), and define

\[ t_i=F(b)^{-1}F(a). \tag{6.2} \]

This is independent of the span: place two spans over a common receiving object, then equalize the two resulting paths from \(i\) and the two from \(i_0\). Their images under \(F\) are equal; cancellation in the group gives equality of the two ratios. Applying the same argument to a span and to a given arrow \(c: i\to j\) proves \(t_jF(c)=t_i\). We may take \(t_{i_0}=1\), so these maps form a cocone to the one object.

For any other cocone \((z_i)\), its equations on the chosen span give \(z_i=z_{i_0}t_i\). Thus its unique factorization through this cocone is the arrow \(z_{i_0}\). This is the colimit universal property. The calculation explains how a category can have all filtered colimits and a split maximal object while failing terminal uniqueness.

7. Graded exercises with solutions

Exercise 1 (routine). Show that both nonemptiness and essential smallness are necessary in Theorem 4.3.

Solution. The empty category admits colimits of every filtered diagram into it vacuously, because a filtered index is nonempty and there is no such functor. It has no object, hence no split maximal one.

For size, take the category of all ordinals small in the fixed universe, with one arrow \(\alpha\to\beta\) exactly when \(\alpha\leq\beta\). A small nonempty filtered diagram has the supremum of its values as colimit. That supremum is still small: the union of a small family of small ordinal sets belongs to the universe. This category is locally small, but not essentially small in that universe, since distinct ordinals are nonisomorphic and their entire collection is not a small set. If that collection were small, its union would be a small ordinal; the successor of that union would be another member larger than the union, a contradiction. For each \(\alpha\), the arrow \(\alpha\to\alpha+1\) has no left inverse in the poset. There is no split maximal object. The category itself can be organized in an enlarged ambient universe; that does not make it essentially small in the original one.

Exercise 2 (standard). Prove that \(\mathcal T\)-injective objects are closed under retracts and existing small products. State and prove the dual assertion for projective objects.

Solution. If \(E\xrightarrow{i}J\xrightarrow{p}E\) has \(pi=1_E\) and \(J\) is injective for a test \(s: A\to B\), compose \(a: A\to E\) with \(i\), extend to \(b: B\to J\), and use \(pb\). Then \(pbs=pia=a\).

For a product \(E=\prod_jE_j\), take the components of \(a: A\to E\). Each extends to \(b_j: B\to E_j\); small-indexed Choice selects these extensions together. The product universal property gives \(b: B\to E\), and equality after every projection proves \(bs=a\). For the empty product, the terminal object's unique map from \(B\) restricts to its unique map from \(A\), so the statement includes that case.

In the opposite category these are precisely retracts and coproducts of \(\mathcal T\)-projective objects. Concretely, a map \(\coprod_jP_j\to B\) is a family of maps from the \(P_j\), lift each across a test \(A\to B\), and assemble the lifts by the coproduct property. The empty coproduct is initial and lifts by its unique arrow. For a retract, lift after its retraction from the larger projective object and precompose with its section. The equation for that section and retraction verifies the desired lift.

Exercise 3 (challenging). In abelian groups, take the single test \(0\to\mathbf Z\). Show that every group passes its extension test, but the inclusion \(2\mathbf Z\hookrightarrow\mathbf Z\) satisfies the pullback property (5.1) without being invertible. Explain which extra assumption of Theorem 5.1 this disproves as an automatic assertion.

Solution. There is one map from \(0\) to any group, and it extends to \(\mathbf Z\) by the zero homomorphism. Thus every group is injective for this testing family. Nevertheless \(\mathbf Z\) is not ordinarily injective: the homomorphism \(2n\mapsto n\) on \(2\mathbf Z\) cannot extend to \(\mathbf Z\), since an extension would have \(2f(1)=1\).

Let \(f: 2\mathbf Z\hookrightarrow\mathbf Z\). A square (5.1) with top arrow \(0\to\mathbf Z\) would have bottom-right vertical map \(b: \mathbf Z\to\mathbf Z\), say multiplication by \(m\). Its pullback along \(f\) identifies with \(b^{-1}(2\mathbf Z)\), which contains the nonzero subgroup \(2\mathbf Z\), since \(b(2n)=2mn\). Therefore this pullback cannot be \(0\). There is no pullback square of the required shape, and the condition that every such square admits a diagonal holds vacuously. The inclusion \(f\) is not an isomorphism. Thus the final isomorphism-detection assumption in Section 5 is a substantive restriction on the testing family; it cannot be inferred from the words 'extension test' or 'pullback'.

Exercise 4 (advanced). For sets and the test \(\varnothing\to\{*\}\), give a functorial embedding into test-injective objects. Prove that the full subcategory of those objects is nevertheless not reflective in sets.

Solution. A set \(Y\) is test-injective exactly when it is nonempty: the restriction \(\operatorname{Hom}(\{*\},Y)\to\operatorname{Hom}(\varnothing,Y)\) is the map \(Y\to\{*\}\), which is surjective precisely in that case.

Put \(I(X)=X\amalg\{*\}\), send a function \(u: X\to X'\) to \(u\amalg1\), and use the first summand inclusion \(q_X: X\hookrightarrow I(X)\). These assignments preserve identities and composition on both summands, so define a functor. The equation \(I(u)q_X=q_{X'}u\) proves naturality, and each \(q_X\) is monic. Every target is nonempty and therefore test-injective.

If nonempty sets formed a reflective subcategory, its left adjoint would send \(\varnothing\) to a nonempty set \(L\) with

\[ \operatorname{Hom}(L,Y) \simeq\operatorname{Hom}(\varnothing,Y)=\{*\} \quad\text{for every nonempty }Y. \tag{7.1} \]

For the two-element set \(Y\), the two constant functions from nonempty \(L\) are distinct, contradicting (7.1). No reflection exists. The functorial embedding solves extension existence; it does not give the uniqueness needed for an adjunction. Already \(I(\varnothing)\) has one element and \(I^2(\varnothing)\) has two, so the natural second embedding is not an isomorphism.

8. References

Relative injectivity changes the family of extension problems, while the choice of allowed target changes governs the construction. Passing all the small tests, recognizing every allowed arrow and obtaining a universal adjunction are separate assertions; their proofs above state the hypotheses each requires.