TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13‡ 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.

Filtered colimits commute with finite limits in Set

Statement

For every small filtered category $J$ and finite category $K$ , and every $D : J \times K \to S e t$ , the canonical comparison

${colim}_{j} \lim_{k} D (j, k) ⟶ \lim_{k} {colim}_{j} D (j, k)$

is a bijection. This includes the empty finite limit.

Facts & Assumptions

Given: The categories and diagram in the statement.

[F1]

Filteredness combines finitely many objects at a common later stage and coequalizes finitely many parallel arrows (Filtered categories and filtered colimits).

[L1]

Equality of two elements in a filtered Set-colimit occurs at one common later stage (Two representatives in a filtered colimit of sets are equal exactly when they become equal at one common later stage).

[L2]

Set-limits are compatible tuples and Set-colimits are tagged quotients (Set has all small limits, realized as compatible tuples in a set-indexed product, Set has all small colimits, realized as a quotient of a set-indexed disjoint union).

[L3]

Finite limits can be tested using finite products and equalizers (Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals).

[L4]

A canonical comparison is invertible exactly when the limit is preserved (A functor preserves a chosen limit exactly when its canonical comparison to the chosen target limit is an isomorphism, and dually for colimits).

Proof

technique · representatives

1.1

By [L3], it suffices to prove that a filtered colimit preserves finite products and equalizers. For a finite product, a tuple on the right has finitely many coordinates, each represented at some stage. Repeated use of [F1] moves all representatives to one common stage, producing a tuple there. Thus the comparison is surjective.

F1L2L3

1.2

If two common-stage tuples have the same image, [L1] makes each coordinate equal at some later stage. There are finitely many coordinates, so repeated use of [F1] moves all those equalities to one stage. The tuples then agree, which proves injectivity.

F1L1L2

1.3

For an equalizer, an element on the right is represented by $x$ at some stage and its two images become equal in the filtered colimit. By [L1], after moving $x$ to one later stage its two images are equal there, so it is represented by a stagewise equalizer element. This proves surjectivity.

L1L2

1.4

If two stagewise equalizer elements become equal in the ambient filtered colimit, [L1] makes them equal at a common later stage; functoriality keeps them inside the later equalizer. This proves injectivity.

L1L2

1.5

For the empty product, each stage and the target are singletons. The filtered category is nonempty by [F1], so the colimit of the constant singleton diagram is a singleton, not empty.

F1L2

2.1

Steps 1.1 to 1.5 make every finite-limit comparison bijective. By [L4], filtered colimits preserve finite limits, proving the displayed assertion.

L3L4step 1.1step 1.2step 1.3step 1.4step 1.5∎

Depends on

Used by

Filtered colimits in Set need not commute with countably infinite products Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 47 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, Theorem 3.8.9 (standard reference, not scraped)