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

1.1L1givenconstruct

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.

2.1F1step 1.1algebra

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 $i$ .

3.1L1step 2.1

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 $v = u$ .

4.1step 1.1step 2.1step 3.1∎

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

Depends on

Used by

Dependency tree · two levels

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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)