CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-13

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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 · two levels

12 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

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