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Limits and colimits 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 filtered completion supplies filtered colimits by construction. Other limits and colimits require a separate argument. Some can be computed in the surrounding presheaf category. Others exist inside the completion even though their presheaf counterparts give the wrong answer. Keeping track of the category in which a construction occurs is part of its universal property.

Read Ind-objects through their elements for recognition and pointwise filtered colimits. Use Finite diagrams and comma objects, Section 2 for the complete simultaneous stage construction on arbitrary small filtered categories. Filtered stages and finite limits, Sections 1–2 prove the finite graph and equality witnesses, and Section 4 proves the full finite-set interchange. We also retain the complete coproduct/coequalizer construction of colimits.

Fix an indexing universe \(\mathcal U\). All indexing sets and categories are small in this universe, and \(\mathsf C\) is locally \(\mathcal U\)-small. Write \(h_X=\operatorname{Hom}_{\mathsf C}(-,X)\), \(\iota X=h_X\), and \(\widehat{\mathsf C}=\operatorname{Fun}(\mathsf C^{\mathrm{op}},\mathsf{Set}_{\mathcal U})\), interpreted in a larger ambient universe when necessary. The inclusion of the ind-category into this presheaf category is fully faithful.

1. Finite colimits are tested contravariantly

Suppose \(D:K\to\mathsf C\) is a finite diagram and has colimit \(Q\). Every representable presheaf sends it to a limit:

For an ind-object \(L=\operatorname{colim}_i h_{X_i}\), the same assertion holds. Evaluate its presentation at \(Q\), use (1.1) at each stage, and commute the filtered colimit with the finite limit of sets. Thus

This also proves that \(\iota Q\), with its specified structural maps, is a colimit of \(\iota D\) inside \(\operatorname{Ind}(\mathsf C)\): Yoneda identifies the two sides of (1.2) with the required Hom sets into any \(L\). Hence the constant embedding preserves every existing finite colimit, whether or not \(\mathsf C\) has all finite colimits. In particular, an initial object of \(\mathsf C\) remains initial in its ind-category.

When \(\mathsf C\) does have all finite colimits, there is a useful converse.

Proposition 1.1. A presheaf \(L\) is an ind-object if and only if it sends finite colimits in \(\mathsf C\) to limits of sets and its category of elements is cofinally small.

Proof. Necessity follows from (1.2) and the recognition theorem. Conversely, use the initial object \(0\): preservation of its empty colimit says \(L(0)\) is a singleton, so the category of elements is nonempty. For two elements \(x\in L(X)\), \(y\in L(Y)\), the bijection

supplies an element receiving both of them. If \(f,g:(X,x)\rightrightarrows(Y,y)\) are parallel element morphisms, then \(L(f)(y)=L(g)(y)\). Let \(q:Y\to Q\) be their coequalizer in \(\mathsf C\). The finite-colimit hypothesis on \(L\) says exactly that \(y=L(q)(z)\) for a unique \(z\in L(Q)\). The element morphism \(q:(Y,y)\to(Q,z)\) equalizes \(f,g\). Its category of elements is therefore filtered. Cofinal smallness and the recognition theorem imply that \(L\) is an ind-object. \(\square\)

The cofinal-smallness condition remains necessary for a large coefficient category. Finite equations alone do not supply a small presentation. For a small \(\mathsf C\), its element category is small, so this extra condition is automatic.

There is also a comma-category formulation requiring no finite colimits in \(\mathsf C\). For every ind-object \(L\), the category \((\iota\downarrow L)\) is its category of elements. The recognition theorem makes it filtered and cofinally small. These are, respectively, the comma-category conditions called right exact and right small for the functor \(\iota\). When finite colimits exist, the right-exact condition also has the finite-colimit preservation meaning proved above. The comma formulation retains information even when such colimits are unavailable.

2. Independent product indices

There is a set-theoretic fact about products that differs from commuting an infinite product with one common filtered colimit. Suppose \(J\) is small, each \(I_j\) is a small filtered category, and \(S_j:I_j\to\mathsf{Set}_{\mathcal U}\). Put \(I=\prod_{j\in J}I_j\). This category is small and filtered: choose objects, common targets, and equalizers in each coordinate. These choices use the usual axiom of choice for a small family. For empty \(J\), \(I\) is the terminal category.

The canonical map is a bijection:

Indeed, choose a representative for each coordinate of an element on the right. Their indices form one object of \(I\), proving surjectivity. If two product representatives have the same image, equality in each coordinate is witnessed at a common later coordinate stage. Choose those stages and witnessing arrows together. They form an object and arrows in \(I\) at which the whole product representatives agree. This proves injectivity.

Theorem 2.1. Fix a small set \(J\). If every \(J\)-indexed product of representable presheaves belongs to \(\operatorname{Ind}(\mathsf C)\), then the presheaf product of every \(J\)-indexed family of ind-objects belongs to it. This presheaf product is their product in the ind-category.

Proof. Choose presentations \(A_j=\operatorname{colim}_{i_j\in I_j}h_{X_{j,i_j}}\). Apply (2.1) at every test object of \(\mathsf C\). The resulting bijections are natural in that test object and give

Each stage on the right is an ind-object by hypothesis. Closure under small filtered colimits makes the result an ind-object. Since the subcategory is full, the presheaf product cone has its universal property there too. \(\square\)

Each coordinate in (2.1) has its own freely chosen index. Exercise 2 shows why replacing these tuples by one common scalar index can fail for an infinite product.

Theorem 2.2. If the presheaf equalizer of every parallel pair of maps between representables belongs to the ind-category, then the same is true for every parallel pair between ind-objects.

Proof. By simultaneous strictification, Section 2, write the pair as stage maps \(f_i,g_i:X_i\rightrightarrows Y_i\) on one small filtered index category. Let \(E_i\) be the presheaf equalizer of \(h_{f_i},h_{g_i}\). These equalizers form a diagram, since transition squares commute. At each test object, filtered colimits of sets commute with equalizers. Thus the presheaf equalizer of the original formal pair is \(\operatorname{colim}_i E_i\). Every \(E_i\) belongs to the ind-category by hypothesis, and filtered closure proves the assertion. Fullness supplies the equalizer universal property. \(\square\)

If \(T\) is terminal in \(\mathsf C\), then \(h_T\) is the terminal presheaf, so it is terminal in the full ind-subcategory. This assertion needs no other limits in \(\mathsf C\).

Corollary 2.3. If \(\mathsf C\) has finite limits, then \(\operatorname{Ind}(\mathsf C)\) has finite limits, and both the constant embedding and the presheaf inclusion preserve them. If \(\mathsf C\) has all small limits, the corresponding assertions hold for all small limits.

Proof. Yoneda preserves every existing limit, since its value at a test object is a Hom functor in the second variable. Products and equalizers of representables are therefore representable under the stated hypotheses. Apply Theorems 2.1 and 2.2, then express a limit as an equalizer between products indexed by the objects and arrows of its diagram. These are presheaf constructions, so the inclusion preserves them. Yoneda gives preservation by the constant embedding. The empty product is included: a terminal object of \(\mathsf C\) gives the terminal presheaf. \(\square\)

3. Colimits formed inside the completion

For colimits, evaluating a presheaf at the stage colimit is more useful than taking a colimit of presheaves.

Theorem 3.1. The following assertions hold separately.

1. If \(\mathsf C\) has coequalizers, then \(\operatorname{Ind}(\mathsf C)\) has coequalizers.
2. If \(\mathsf C\) has finite coproducts, including the empty one, then \(\operatorname{Ind}(\mathsf C)\) has all small coproducts.
3. If \(\mathsf C\) has finite colimits, then \(\operatorname{Ind}(\mathsf C)\) has all small colimits.

The constant embedding preserves the existing finite constructions in all three assertions.

Proof. For (1), use the same simultaneous construction to strictify the given parallel pair as \(f_i,g_i:X_i\rightrightarrows Y_i\). Set \(Q_i=\operatorname{coeq}(f_i,g_i)\) in \(\mathsf C\). The transition squares induce maps between the \(Q_i\), forming a filtered diagram. We claim \(Q=\operatorname{colim}_i\iota Q_i\) is the desired coequalizer in the ind-category. For any ind-object \(L\), equation (1.2) for a coequalizer gives

Here \(A=\operatorname{colim}_i\iota X_i\), \(B=\operatorname{colim}_i\iota Y_i\). The last step commutes limits with limits. All bijections identify the specified maps, proving the universal property.

For (2), first form a finite coproduct of presented objects on a common filtered product of their index categories. At a stage take the coproduct in \(\mathsf C\). Equation (1.2), with a finite product of sets on its right, and the Hom formula show that the filtered formal colimit of these stages is the coproduct of the original ind-objects. The empty case is \(\iota0\), whose maps into every ind-object form the singleton \(L(0)\).

For a small family \((A_j)_{j\in J}\), use its finite subsets \(T\subset J\), including the empty subset. They form a small filtered poset, with union giving a common upper bound. Finite coproducts and their canonical inclusions form a diagram. Then

To verify this, maps from the right-hand object to \(L\) are compatible families of maps from its finite partial coproducts. Such a family is exactly one map \(A_j\to L\) for each \(j\in J\), with no further equations. This is the coproduct universal property.

For (3), both (1) and (2) apply. The referenced colimit construction gives every small diagram's colimit as a coequalizer between two small coproducts, one indexed by its arrows and one by its objects. This construction also covers the empty diagram. Preservation of existing finite colimits by the constant embedding was proved in (1.2). \(\square\)

The presheaf inclusion need not preserve these coproducts or coequalizers. For instance, if \(0\) is initial in \(\mathsf C\), then \(h_0\) is initial in the ind-category, whereas the initial presheaf is the everywhere-empty functor. The value \(h_0(0)\) contains its identity. Exercise 3 gives a failure for a nonempty coproduct.

The constant embedding has a different limitation. For \(\mathsf C=\mathsf{Set}\), its countable coproduct of copies of \(1\) in the ind-category is the formal filtered union of finite sets. The comparison to \(\iota\mathbb N\) fails at the constant test \(\mathbb N\): its image contains only functions with finite image, missing the identity. This is the finite-image example in Formal colimits and compact presentations. It shows failure to preserve this small coproduct and its filtered presentation, despite preservation of finite colimits. Conversely the presheaf inclusion preserves filtered colimits but can fail on finite colimits. A formal colimit over a nonfiltered diagram, when taken in presheaves, must therefore be labeled as such; it cannot automatically be identified with a colimit inside the ind-category.

4. Exactness and a change of universe

Theorem 4.1. If \(\mathsf C\) has finite limits, then small filtered colimits commute with finite limits in \(\operatorname{Ind}(\mathsf C)\). If \(\mathsf C\) also has finite colimits, these filtered-colimit functors are exact: they preserve both finite limits and finite colimits.

Proof. The presheaf inclusion computes filtered colimits pointwise by the element lesson, and computes finite limits pointwise by Corollary 2.3. At every test object the required interchange is the filtered-colimit/finite-limit bijection for sets. Its naturality makes these pointwise bijections an isomorphism of presheaves and hence of ind-objects.

Under the additional hypothesis, Theorem 3.1 supplies the needed finite colimits in the ind-category. For a fixed small filtered index \(I\), its colimit functor is left adjoint to the constant-diagram functor. A left adjoint preserves existing colimits, so it preserves finite colimits as well. This and the first assertion give exactness. \(\square\)

Exactness here concerns finite categorical limits and colimits. It requires no abelian structure and adds no theorem about derived colimits.

Now let \(\mathcal U\subset\mathcal V\) be indexing universes, with \(\mathsf C\) still locally \(\mathcal U\)-small. A \(\mathcal U\)-small presentation is also a \(\mathcal V\)-small presentation, giving a functor

Theorem 4.2. This functor is fully faithful and preserves \(\mathcal U\)-small filtered colimits. If \(\mathsf C\) has finite colimits, it preserves all \(\mathcal U\)-small colimits. If \(\mathsf C\) has finite limits, it preserves finite limits; if \(\mathsf C\) has all \(\mathcal U\)-small limits, it preserves those limits.

Proof. Between two \(\mathcal U\)-small presentations, the Hom formula takes only \(\mathcal U\)-small limits and colimits of \(\mathcal U\)-small sets. The same operations on these sets in \(\mathcal V\) produce the same set and the same composition maps. This proves full faithfulness. The pointwise description of a \(\mathcal U\)-small filtered colimit is likewise unchanged when sets are regarded in \(\mathcal V\), proving its preservation.

If finite colimits exist in \(\mathsf C\), the stage constructions for coequalizers and finite coproducts in Theorem 3.1 use only \(\mathcal U\)-small presentations and the same objects of \(\mathsf C\). Their universal-property proofs apply to every \(\mathcal V\)-ind-object too: equation (1.2) holds for a \(\mathcal V\)-small filtered presentation. Thus \(j\) preserves these finite constructions. Formula (3.2) for a \(\mathcal U\)-small family uses a \(\mathcal U\)-small filtered poset, so it also preserves that coproduct. The coequalizer construction of arbitrary colimits then proves preservation of all \(\mathcal U\)-small colimits.

For limits of the specified sizes, Corollary 2.3 computes them in presheaves in both universes. Products and equalizers of the given \(\mathcal U\)-valued presheaves have the same pointwise sets in either universe. Hence the comparison preserves these limits. \(\square\)

Full faithfulness does not assert that every larger-universe formal object already has a smaller-universe presentation. The theorem compares the specified objects and constructions.

5. Exercises with solutions

Exercise 1 (introductory: identifying parity). In finite sets let \(B_n=\{0,\ldots,2n+1\}\) and \(A_n=\{0,\ldots,2n-1\}\), taking \(A_0=\varnothing\). Use inclusion transitions and maps \(f_n(r)=r\), \(g_n(r)=r+2\) from \(A_n\) to \(B_n\). Find their stage coequalizers and the coequalizer of the resulting formal parallel maps.

Solution. At stage \(n\), the imposed relations identify each \(r\) with \(r+2\). Their equivalence classes are the even and odd elements of \(B_n\); both classes are nonempty, including at \(n=0\). Thus each coequalizer is the two-element parity set, and every induced transition on it is the identity. Theorem 3.1 gives the constant two-element ind-object as the formal coequalizer. Under \(\operatorname{Ind}(\mathsf{FinSet})\simeq\mathsf{Set}\), the source and target realize to \(\mathbb N\), with maps \(r\mapsto r\) and \(r\mapsto r+2\). Their ordinary coequalizer also has the two parity classes. The stage calculation identifies the entire equivalence relation, including its transitive closure.

Exercise 2 (intermediate: a shared bound is too restrictive). Let \(S_n=\{0,\ldots,n\}\). Describe the image of

Compare it with the independent-index formula (2.1).

Solution. The stages are nested sets of sequences. Their union consists of the bounded sequences of nonnegative integers. The target is every such sequence, so the map is injective but not surjective: the sequence \(j\mapsto j\) is missing. For independent indices, use the filtered poset \(\mathbb N^{\mathbb N}\) ordered coordinatewise. Its two objects have the coordinatewise maximum as an upper bound. At the index \((n_j)\), the product stage consists of sequences with \(j\)-th term at most \(n_j\). Every sequence \((a_j)\) belongs to the stage \((n_j)=(a_j)\). The independent-index colimit is therefore all of \(\mathbb N^{\mathbb N}\), as (2.1) predicts. There need not be a single finite bound for all coordinates.

Exercise 3 (intermediate: two coproducts with different tests). Let \(\mathsf C=\mathsf{FinSet}\), and take a countable family of copies of its one-element set \(1\). Compare their coproduct in the presheaf category with their coproduct in the ind-category. Evaluate the canonical comparison at \(\varnothing\) and at a two-element set.

Solution. The presheaf coproduct \(P=\coprod_{n\in\mathbb N}h_1\) has value \(\mathbb N\) at every finite test set: every map to \(1\) is unique. The ind-coproduct \(Q\) is the filtered formal colimit of its finite partial sums. At a finite test set \(X\), a function \(X\to\mathbb N\) has finite image, so

The presheaf comparison \(P\to Q\) sends the label \(n\) to the constant function with value \(n\). At \(X=\varnothing\), it is the map from \(\mathbb N\) to a singleton, hence fails injectivity. At a two-element test it is the diagonal \(\mathbb N\to\mathbb N^2\), hence fails surjectivity. At a one-element test it is bijective. Testing only at that object would miss both obstructions. Consequently the presheaf inclusion does not preserve this coproduct.

Exercise 4 (advanced: reverse the universal properties). Suppose \(\mathsf C\) has finite limits and finite colimits. Deduce that \(\operatorname{Pro}(\mathsf C)\) has all small limits, finite colimits, and that small cofiltered limits commute with finite colimits. Describe what “pointwise” means for its small cofiltered limits in the covariant functor model.

Solution. Use \(\operatorname{Pro}(\mathsf C)=\operatorname{Ind}(\mathsf C^{\mathrm{op}})^{\mathrm{op}}\). Finite limits in \(\mathsf C\) are finite colimits in \(\mathsf C^{\mathrm{op}}\), so Theorem 3.1 makes its ind-category cocomplete; taking opposites makes the pro-category complete. Finite colimits in \(\mathsf C\) become finite limits in its opposite, so Corollary 2.3 supplies finite colimits in the pro-category. Theorem 4.1, applied to \(\mathsf C^{\mathrm{op}}\), reverses to the asserted interchange of cofiltered limits and finite colimits. The constant embedding preserves both sorts of finite constructions by the corresponding ind assertions.

A pro-object \((X_i)\), with cofiltered index, is represented contravariantly by the covariant functor \(T\mapsto\operatorname{colim}_{i\in I^{\mathrm{op}}}\operatorname{Hom}(X_i,T)\). Thus the fully faithful target is the opposite of the covariant-functor category. A cofiltered limit of pro-objects becomes a pointwise filtered colimit of these covariant functors. It is not computed by a pointwise limit of their sets. The opposite category reverses the universal property and all structural arrows.

References

• The Stacks Project, Categories, Section 4.19, for filtered colimits and finite limits of sets, and Lemma 4.14.12, for the coproduct/coequalizer construction of colimits. The statements are used by reference; their text is not imported.
• Finite diagrams and comma objects, Section 2, for the complete simultaneous strictification, cofinal pointwise comparison and pro dual used here. Filtered stages and finite limits, Sections 1–2 and 4, for the complete graph, equality and finite-limit proofs.
• 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 broader role of formal filtered completions in homological algebra.

These are known categorical results. The proofs, organization, examples and exercise wording here are independently written expository material, not claims of new research.