TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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The compatible-tuple construction satisfies the inverse-limit universal property in groups

Statement

The compatible-tuple construction satisfies the inverse-limit universal property in the category of groups.

Facts & Assumptions

Given: An inverse system $((G_{i}), φ_{i j})$ and a group $H$ with a cone of homomorphisms $f_{i} : H \to G_{i}$ satisfying $φ_{i j} \circ f_{j} = f_{i}$ for all $i \leq j$ .

[L1]

The inverse limit consists exactly of the compatible tuples, and $π_{i} ((g_{j})) = g_{i}$ for the coordinate projections (The inverse limit is the set of compatible tuples in the Cartesian product, The inverse limit has its canonical coordinate projection maps).

[F1]

Group homomorphisms preserve products and identities (Monoid homomorphism and group homomorphism).

Proof

technique · direct

1.1

Define $u : H \to \underset{\leftarrow}{l i m} G_{i}$ by $u (h) : = (f_{i} (h))_{i \in I} .$ The cone identities make this tuple compatible, so [L1] shows that $u$ lands in the inverse limit.

L1givenconstruct

2.1

For $h, h^{'} \in H$ one has $u (h h^{'}) = (f_{i} (h h^{'}))_{i} = (f_{i} (h) f_{i} (h^{'}))_{i} = u (h) u (h^{'}),$ and similarly $u (e_{H}) = e$ . Hence $u$ is a group homomorphism by [F1]. Also $π_{i} \circ u = f_{i}$ for every $i$ .

F1step 1.1algebra

3.1

If $v : H \to \underset{\leftarrow}{l i m} G_{i}$ is another homomorphism with $π_{i} \circ v = f_{i}$ for every $i$ , then every coordinate of $v (h)$ agrees with the corresponding coordinate of $u (h)$ . Tuples are equal exactly when all coordinates are equal, so $v = u$ .

L1step 2.1

4.1

Steps 1.1 through 3.1 give existence and uniqueness of the mediating homomorphism.

step 1.1step 2.1step 3.1∎

Depends on

Used by

Dependency tree · two levels

9 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

Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)