CounterexampleConstruction: Literature-sourcedVerification: 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 in Set need not commute with countably infinite products

Statement refuted

Filtered colimits in $S e t$ commute with arbitrary set-indexed products.

Facts & Assumptions

Given: Positive integers $i, j$ and sets $M_{i, j} = {1, \dots, i}$ , with inclusions as $i$ increases.

[F1]

A category is filtered when it is nonempty, every two objects have a common target, and every parallel pair is equalized at a later stage (Filtered categories and filtered colimits).

[L1]

Filtered colimits commute with finite, not asserted infinite, limits in $S e t$ (Filtered colimits commute with finite limits in Set).

[F2]

Products in a category represent families of coordinate maps (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations).

Counterexample

technique · bounded sequences

1.1

The positive-integer chain is nonempty, two indices have their maximum as a common target, and it has no distinct parallel arrows, so it is filtered by [F1]. For fixed $i$ , the countable product $\prod_{j \geq 1} M_{i, j}$ is the set of positive-integer sequences all of whose entries are at most $i$ .

F1F2

1.2

For each $j$ , the filtered colimit ${colim}_{i} M_{i, j}$ is $N_{> 0}$ . The product of these coordinatewise colimits is the set of all positive-integer sequences, including $(1, 2, 3, \dots)$ , which is unbounded.

2.1

Its filtered colimit over $i$ is therefore the set of bounded positive-integer sequences: a sequence appears at some stage exactly when one integer bounds all its coordinates.

step 1.1

3.1

Hence the canonical map from step 2.1 to the set in step 1.2 is not surjective. The arbitrary-product assertion is false, while [L1] is not contradicted because the product is infinite.

L1step 2.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 10 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

The Stacks Project, Categories, Section 4.19 (standard reference, not scraped)