CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

A limit weighted by a $S e t$ -valued weight on a small index category exists in a complete target, and the weighted colimit in a cocomplete one

Statement

Let $J$ be a small category (Small, locally small, and large categories), let $M$ be locally small and let $F : J \to M$ be a diagram.

If $M$ is complete (Finite, small, and large limits and colimits; complete and cocomplete categories), then for every weight $W : J \to S e t$ the weighted limit ${W, F}$ exists (Set-weighted limits and colimits). If $M$ is cocomplete, then for every weight $W : J^{o p} \to S e t$ the weighted colimit $W ⋆ F$ exists.

Local smallness of $M$ is part of the hypothesis and cannot be dropped: the definition of a weighted limit is a representation of a $S e t$ -valued functor built from the hom-sets of $M$ .

These conditions are sufficient and are not asserted to be necessary of the target: a particular weighted limit in a category that is not complete may exist all the same. The definition on this page fixes a small index category, so no large-index weighted limit is asserted here.

Facts & Assumptions

Given: A small category $J$ , a locally small category $M$ that is complete, respectively cocomplete, a diagram $F : J \to M$ , and a weight of the variance named in each clause.

[F3]

A category is small when both $Ob (C)$ and $Mor (C)$ are sets. (Small, locally small, and large categories).

[F4]

The category of elements $\int W$ has objects the pairs $(c, x)$ with $x \in W (c)$ , and a morphism is a morphism of the index category subject to one equation (The category of elements of a covariant functor or a presheaf).

[F5]

A weighted limit ${W, F}$ is an object that represents the functor sending an object to the set of natural transformations from the weight, and a weighted colimit is characterised dually; the construction presupposes $J$ small and $M$ locally small (Set-weighted limits and colimits).

[L1]

For small $J$ the category of elements $\int W$ is small, and a weighted limit is an ordinary limit over the category of elements of the weight: ${W, F}$ exists exactly when $\lim_{\int W} F π$ does, and then they agree (A weighted limit is an ordinary limit over the category of elements of the weight, and a weighted colimit an ordinary colimit over it).

[L2]

Under the same hypotheses, a weighted colimit an ordinary colimit over it: $W ⋆ F$ exists exactly when ${colim}_{\int W} F π$ does, and then they agree (A weighted limit is an ordinary limit over the category of elements of the weight, and a weighted colimit an ordinary colimit over it).

[F6]

A category is complete when it has all small limits and cocomplete when it has all small colimits, a diagram being small when its indexing category is small; Completeness and cocompleteness do not assert the existence of limits or colimits of large diagrams (Finite, small, and large limits and colimits; complete and cocomplete categories).

Proof

technique · direct

1.1

The category of elements $\int W$ is small. Its objects are the pairs $(c, x)$ with $c$ in the set $Ob (J)$ and $x$ in the set $W (c)$ , hence a set; and a morphism of $\int W$ is determined by its domain, its codomain and the underlying morphism of $J$ , so the morphisms form a subclass of a product of three sets and hence a set. The count is carried out rather than asserted, and it uses no choice; the presheaf case counts identically.

F3F4L1given

2.1

For the limit clause, $F π$ is a diagram indexed by $\int W$ , which is small by step 1.1, so completeness of $M$ supplies a limit for it. By [L1] the weighted limit ${W, F}$ then exists and is that limit; $M$ is locally small, so the weighted limit is defined at all.

F5F6L1step 1.1

2.2

For the colimit clause, the same diagram over the small category $\int W$ has a colimit because $M$ is cocomplete, and by [L2] the weighted colimit $W ⋆ F$ exists and is that colimit.

F5F6L2step 1.1

3.1

Both clauses are sufficiency only. A weighted limit is by [F5] an object representing the displayed functor, so completeness or cocompleteness of $M$ is not necessary for a particular weighted object to exist. Smallness of $J$ remains part of the definition in force here, and no converse is claimed.

F5F6step 2.1step 2.2∎

Remarks

The smallness of the category of elements, not of the index category alone, is what the argument needs, and it is the values of the weight that supply the extra objects: a weight taking large values on a small index category would not be $S e t$ -valued, which is why the hypothesis is stated on the weight and not only on $J$ .

The corresponding statement for ends is Ends exist over a small index category in a complete target, and coends in a cocomplete one, and the two are proved the same way: an existence hypothesis on the target, applied to an ordinary limit over a small index category that a comparison theorem has identified with the object in question.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Riehl, Categorical Homotopy Theory, (7.1.8) and (7.2.4) (standard reference, not scraped)
G. M. Kelly, Basic Concepts of Enriched Category Theory (TAC Reprints 10), (3.33)-(3.34) (standard reference, not scraped)

A limit weighted by a SetSet-valued weight on a small index category exists in a complete target, and the weighted colimit in a cocomplete one