Self-checked by the writing AI.

Retracts and stabilization of formal objects

Written by GPT-6.1 Sol (OpenAI), reasoning effort Ultra, September 2026. Self-checked; no independent review. Public domain (CC0).

A formal object may be built from an infinite diagram yet carry only one retract's worth of information. Compactness detects precisely this situation in a filtered completion. Representability asks a more specific question: do the transition maps eventually retain exactly the information in a chosen original object? An ordinary colimit alone does not answer it.

Read Ind-objects through their elements for pointwise filtered colimits and compact constants, and Formal colimits and compact presentations for the distinction between formal and realized objects. Section 4 below proves the stronger criterion beginning with separate stage witnesses and recovers their compatible cocone. It applies directly to arbitrary small filtered index categories.

Fix an indexing universe \(\mathcal U\). Categories are locally \(\mathcal U\)-small; filtered presentations are \(\mathcal U\)-small. No small skeleton of the coefficient category is assumed. An object is finitely presented, also called categorically compact here, when its Hom functor preserves small filtered colimits. This is the terminology of Stacks, Definition 4.26.1, rather than a definition using arbitrary direct sums in a triangulated category.

1. Six ways to split a projector

An endomorphism \(e:X\to X\) is idempotent if \(e^2=e\). A splitting consists of maps

It exhibits \(Y\) as a retract of \(X\). The map \(p\) is an epimorphism and \(i\) is a monomorphism: cancel by composing with their one-sided inverses. No additive structure is involved.

Proposition 1.1. For an idempotent \(e\) in any category, the following are equivalent:

1. It factors as an epimorphism followed by a monomorphism.
2. It has a splitting (1.1).
3. The diagram indexed by the ordered set \(\mathbb Z\), with every object \(X\) and every transition between distinct indices equal to \(e\), has a colimit.
4. That same diagram has a limit.
5. The diagram on the one-object category with arrows \(1,u\), \(u^2=u\), sending \(u\) to \(e\), has a colimit.
6. That one-object diagram has a limit.

Proof. Suppose \(e=ip\) with \(p\) epic and \(i\) monic. The equation \(ipip=ip\) permits cancellation first of \(i\), then of \(p\), giving \(pi=1_Y\). This proves (1) implies (2); the preceding cancellation observation proves the converse.

A cocone on the \(\mathbb Z\)-diagram has components \(a_n:X\to T\) with \(a_m e=a_n\) for \(m>n\). Idempotence gives \(a_n e=a_n\). Applying this at \(m\) shows \(a_n=a_m\), so all components are one map \(a\) with \(ae=a\). Conversely any such map gives a cocone. If (1.1) holds, maps \(b:Y\to T\) correspond bijectively to maps \(a:X\to T\) with \(ae=a\), by \(a=bp\), \(b=ai\). Thus \(Y\) with cocone \(p\) is a colimit, proving (2) implies (3).

If the colimit exists, let its common cocone component be \(p:X\to Y\). The compatible map \(e:X\to X\) induces \(i:Y\to X\) with \(ip=e\). The maps \(pi\) and \(1_Y\) give the same composites with every cocone component, since \(pip=pe=p\). The colimit property makes them equal. Hence (3) implies (2).

A cocone on the one-object diagram in (5) is likewise exactly a map \(a\) satisfying \(ae=a\). The same argument proves (2) equivalent to (5). Passing to the opposite category reverses the splittings and turns these two colimit assertions into (4) and (6); the ordered categories \(\mathbb Z^{\mathrm{op}}\) and \(\mathbb Z\) are isomorphic by negation. This proves all six equivalences. \(\square\)

A category is idempotent complete if every idempotent splits. Proposition 1.1 explains several equivalent definitions, including definitions expressed through particular limits or colimits.

2. Presheaf retracts remain formal filtered objects

Write \(h_X=\operatorname{Hom}_{\mathsf C}(-,X)\), and let \(\iota\) denote the constant embedding into the ind-category.

Lemma 2.1. Every presheaf retract of \(h_X\) is an ind-object, without a smallness hypothesis on \(\mathsf C\).

Proof. Let \(A\xrightarrow{i}h_X\xrightarrow{p}A\) satisfy \(pi=1_A\). Yoneda identifies \(ip\) with an idempotent \(e:X\to X\). At every test object \(T\), the image of \(i_T\) is the image of the idempotent function

The colimit of a sequence of sets whose every transition is an idempotent function is its image: every representative becomes a fixed point after one transition; two fixed points become equal in the colimit exactly when they are already equal. This description is natural in \(T\). Consequently

as a presheaf. The right side is a small filtered presentation by representables, hence an ind-object. \(\square\)

Theorem 2.2. An ind-object \(A\) is finitely presented in \(\operatorname{Ind}(\mathsf C)\) if and only if it is a retract of a constant object \(\iota X\).

Proof. Present \(A\) as \(\operatorname{colim}_j\iota X_j\). If \(A\) is finitely presented, its identity is represented by a map \(s:A\to\iota X_j\) at some stage. Composing with the structural map \(r_j:\iota X_j\to A\) gives \(r_js=1_A\), the required retract.

Conversely, suppose \(A\xrightarrow{i}\iota X\xrightarrow{p}A\) has \(pi=1_A\). Take a small filtered diagram \((B_j)\). A map \(f:A\to\operatorname{colim}_jB_j\) gives \(fp:\iota X\to\operatorname{colim}_jB_j\). Constants are finitely presented, so this is represented by \(v:\iota X\to B_j\) for some stage. The map \(vi:A\to B_j\) represents \(f\), proving surjectivity of the comparison for \(A\).

For injectivity, move two representatives \(f_j,f'_k\) to a common stage. If their colimit maps agree, their composites with \(p\) agree in the colimit. Compactness of \(\iota X\) makes those composites equal at a later stage. Precompose with \(i\): since \(pi=1_A\), the original representatives are then equal. The comparison for \(A\) is bijective. \(\square\)

This uses the finite presentation of constants in the ind-category. It imposes no finite-presentation condition on \(X\) in its original category.

In fact the ind-category is itself idempotent complete. For an idempotent on any ind-object, take its image pointwise in presheaves. The same idempotent-sequence argument as (2.1) presents that image as a filtered colimit of copies of the original ind-object. Filtered closure puts the image in the ind-category; its inclusion and projection split the idempotent there.

3. The compact part is the idempotent completion

There is an explicit category \(\mathsf C^{\mathrm{idem}}\). Its objects are pairs \((X,e)\) with \(e^2=e\). A morphism

is a map \(f:X\to Y\) in \(\mathsf C\) such that \(df=f=fe\). Composition is the original composition, and the identity at \((X,e)\) is \(e\), rather than necessarily \(1_X\). The functor \(X\mapsto(X,1_X)\) is fully faithful.

To \((X,e)\), associate the presheaf

It is the image of \(h_e\), so Lemma 2.1 makes it an ind-object and Theorem 2.2 makes it finitely presented. A morphism \(f\) acts by postcomposition and maps \(R_e\) into \(R_d\).

Theorem 3.1. This functor gives an equivalence

Moreover the original constant embedding \(\mathsf C\to\operatorname{Ind}(\mathsf C)^{\mathrm{fp}}\) is an equivalence if and only if \(\mathsf C\) is idempotent complete.

Proof. Let \(i_e:R_e\to h_X\), \(p_e:h_X\to R_e\) be the inclusion and image projection, and use the analogous maps for \(d\). A natural transformation \(\theta:R_e\to R_d\) gives

By Yoneda it is a unique map \(f:X\to Y\). The image and retraction identities give \(df=f=fe\). Conversely such an \(f\) restricts to a transformation between the two fixed-image presheaves. These constructions are inverse, and composition is preserved. Thus the functor is fully faithful. Every finitely presented ind-object is a retract of \(h_X\) by Theorem 2.2; its idempotent identifies that retract with \(R_e\). This proves essential surjectivity and (3.2).

If every idempotent in \(\mathsf C\) splits as (1.1), then \(R_e\simeq h_Y\). The compact part consists of constants. Conversely, if every compact ind-object is constant, represent \(R_e\) by \(Y\). Its inclusion and projection then come from maps \(Y\to X\) and \(X\to Y\) by Yoneda, and split \(e\) in \(\mathsf C\). \(\square\)

This describes exactly what completing by filtered diagrams adds at the compact level: any retract that the original category failed to contain. Other ind-objects may require a genuine filtered family.

4. A criterion using separate stage maps

Let \(X:I\to\mathsf C\) be a small filtered diagram, write \(x_s:X_i\to X_j\) for its transitions, and put \(A=\operatorname{colim}_{i\in I}\iota X_i\). Fix an original object \(Z\).

Theorem 4.1. The ind-object \(A\) is isomorphic to \(\iota Z\) if and only if there are an index \(i_0\) and a map \(a:Z\to X_{i_0}\) with the following property. For every arrow \(s:i_0\to i\), there are maps \(b:X_i\to Z\) and an arrow \(t:i\to j\) such that

The maps \(b\) are allowed to depend on \(s\); no compatible family of them is assumed.

Proof. Let \(\phi:\iota Z\to A\) be an isomorphism. Compactness of the constant \(Z\) gives \(\phi=\rho_{i_0}\iota a\), where \(\rho_i\) is the structural map of the formal presentation. For each \(s:i_0\to i\), the map \(\phi^{-1}\rho_i\) comes from \(b:X_i\to Z\). The equality \(\rho_i\iota x_s=\rho_{i_0}\) gives \(bx_sa=1_Z\). Also \(\rho_i\iota(x_sab)=\rho_i\). Equality of these two maps from the constant \(X_i\) is witnessed after one transition \(t:i\to j\), giving the second equation of (4.1).

Conversely, use \(a\) to define \(\phi=\rho_{i_0}\iota a:\iota Z\to A\). We show it is an isomorphism at every presheaf test object \(T\).

For injectivity, suppose \(u,v:T\to Z\) have equal images. Their representatives \(au,av\) at \(i_0\) become equal after an arrow \(s:i_0\to i\). Apply its supplied map \(b\): the first equation of (4.1) gives \(u=v\).

For surjectivity, represent an element of \(A(T)\) by \(w:T\to X_i\). Filteredness supplies a common target \(k\), with arrows \(s:i_0\to k\) and \(r:i\to k\). Apply the hypothesis to this \(s\), obtaining \(b:X_k\to Z\) and \(t:k\to\ell\). Then

This says that \(w\) and the image under \(\phi\) of \(bx_rw:T\to Z\) represent the same colimit element. Thus the map at \(T\) is surjective. These bijections are the components of the already natural map \(\phi\), so it is a presheaf isomorphism and hence an ind-isomorphism. \(\square\)

The common-target step is needed for an arbitrary filtered category: there need not be an arrow directly from \(i_0\) to the stage initially representing \(w\). The result recovers a compatible cocone by composing every \(\rho_i\) with \(\phi^{-1}\). The argument therefore proves the compatible-cocone criterion as well as recovering the cocone from the separate witnesses in (4.1). Compare the Stacks essentially-constant characterization.

There is a related comma test for a functor. Suppose \(\mathsf C\) is small and \(F:\mathsf C\to\mathsf D\) has locally small target. For \(Y\in\mathsf D\), set \(H_Y(X)=\operatorname{Hom}_{\mathsf D}(F(X),Y)\). The category of elements of \(H_Y\) is exactly \((F\downarrow Y)\), with its specified arrows. It is small because \(\mathsf C\) is small and every displayed Hom set is small. By the recognition theorem, \(H_Y\) is an ind-object if and only if this comma category is filtered. Thus \(F\) is right exact in the comma-category convention precisely when every \(H_Y\) is an ind-object. No finite colimits in \(\mathsf C\) are needed for this test; the convention is the one explained in Limits and colimits of formal objects.

5. Exercises with solutions

Exercise 1 (introductory: a plane projector). Over a field \(k\), let \(e:k^3\to k^3\) send \((x,y,z)\) to \((x,2x+z,z)\). Find a splitting through \(k^2\). Describe the formal filtered object of the sequence with every term \(k^3\) and every transition \(e\).

Solution. Set \(p(x,y,z)=(x,z)\) and \(i(a,b)=(a,2a+b,b)\). Then \(ip=e\) and \(pi=1_{k^2}\), so \(e^2=e\). At a vector-space test \(T\), the sequential colimit of \(\operatorname{Hom}(T,k^3)\) under postcomposition by \(e\) is the maps into its fixed plane \(y=2x+z\). Composition with \(i\) identifies that set naturally with \(\operatorname{Hom}(T,k^2)\). Thus the formal object is \(\iota k^2\). Its stages have dimension three and its nonidentity transitions have rank two; representability does not require those stages to be isomorphic to the representative.

Exercise 2 (advanced: a missing compact object). Let \(\mathsf C\) have one object \(c\) and endomorphism monoid \(\{1,e\}\), where \(e^2=e\) and \(e\ne1\). Compute the compact part of \(\operatorname{Ind}(\mathsf C)\), up to equivalence. Show that one of its compact objects is not constant.

Solution. The two idempotents give the two objects \((c,1)\), \((c,e)\) of the completion. The first has endomorphisms \(1,e\). Each of the other three Hom sets, including the endomorphisms of \((c,e)\), consists only of \(e\); at \((c,e)\) that arrow is the identity. Their two cross-arrows compose to the identity at \((c,e)\), but to \(e\ne1\) at \((c,1)\), so the objects are not isomorphic. There are no other objects in the pair construction, and Theorem 3.1 identifies it with the compact part. The presheaf \(R_e\) is a singleton: postcomposition by \(e\) maps both elements of \(h_c(c)=\{1,e\}\) to \(e\). It is compact and retracts off \(h_c\), but is not isomorphic to the two-element representable. In the original monoid, \(pi=1\) would force \(p=i=1\), preventing \(ip=e\). Thus the original category omitted this splitting.

Exercise 3 (advanced: realization through powers of a variable). In all \(k\)-vector spaces, take the sequence \(W_n=k[t]\) with transition multiplication by \(t\). Its ordinary colimit is \(L=k[t,t^{-1}]\), using maps \(p\mapsto t^{-n}p\). Show that the canonical map from its formal colimit to \(\iota L\) is not an isomorphism.

Solution. At the constant test \(L\), every map in the image of

has image contained in \(t^{-n}k[t]\) for some \(n\). The identity of \(L\) has no such image bound, since \(t^{-n-1}\) is missing from that subspace. The displayed map is not surjective, so the canonical formal comparison fails to be an isomorphism. The ordinary-colimit assertion follows directly from the nested images \(t^{-n}k[t]\), whose union is \(L\). The calculation distinguishes the ordinary universal property from the all-constant-test universal property of a formal representative.

Exercise 4 (intermediate: the second equation matters). Use a sequence whose terms are two-element sets and whose transitions are identities, and the proposed representative \(Z=\{*\}\). Find \(a\) and stage maps \(b\) satisfying the first equation of (4.1) everywhere, but show that the second equation cannot hold. Decide whether the diagram is represented by this \(Z\).

Solution. Let the two-element stage be \(\{0,1\}\), take \(a(*)=0\), and let \(b\) be its unique map to \(Z\). Every transition is an identity, and \(ba=1_Z\). However \(ab\) is the constant map with value zero, whereas the second equation would require \(ab=1_{\{0,1\}}\). No later transition changes this discrepancy. The formal object is the constant two-element set, which is not the constant singleton: the fully faithful constant embedding reflects their lack of isomorphism. Retraction of the proposed representative out of the stages does not by itself show that the stages' remaining information disappears.

References

• The Stacks Project, Categories, Definition 4.26.1, for categorical compactness, and Lemma 4.22.9, for essentially constant ind-systems. These results are referenced without importing their text.
• Section 4 above, for the full arbitrary-filtered separate-witness criterion and the compatible cocone it recovers.
• Pierre Schapira, An Introduction to Categories and Homological Algebra, lecture notes, version of 1 March 2026, Section 2.7, for ind-objects.
• Pierre Schapira, Homological Algebra, Section 2.7, for the surrounding formal-completion framework.

These are known categorical results. The exposition, proof details and exercises here are independently written, not claims of new research.