TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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.

Under the ultrafilter lemma and dependent choice, Stone-Cech compactification is left adjoint to the compact-Hausdorff inclusion

Statement

Assume the ultrafilter lemma and dependent choice. On the category of Tychonoff spaces, chosen Stone–Čech compactifications define a functor $β$ left adjoint to the full inclusion

$J : C o m p H a u s ↪ T y c h .$

For each Tychonoff space $X$ , the unit is its compactification map $η_{X} : X \to J β X$ .

Facts & Assumptions

Given: The ultrafilter lemma and dependent choice, and a chosen Stone–Čech compactification $(β X, η_{X})$ for every Tychonoff space $X$ .

[F1]

A Stone–Čech compactification $(B, i)$ of $X$ has the property that every continuous map $X \to K$ to a compact Hausdorff space extends uniquely to a continuous map $B \to K$ (The Stone–Čech compactification by its compact-Hausdorff extension property).

[F2]

Under the ultrafilter lemma and dependent choice, the evaluation-closure construction is a Stone–Čech compactification of every Tychonoff space (Under the ultrafilter lemma and dependent choice, the closure of the full evaluation image is the Stone–Čech compactification).

[F3]

Under dependent choice, every compact Hausdorff space embeds in a cube $[0, 1]^{J}$ for some set $J$ (Under dependent choice, every compact Hausdorff space embeds in a unit cube).

[F5]

Every compact Hausdorff space is Tychonoff (Under dependent choice a compact Hausdorff space is Tychonoff, and its disjoint closed sets are separated by continuous functions).

[F4]

A full subcategory contains all ambient morphisms between its objects (Subcategory and full subcategory).

[L1]

Chosen objectwise universal arrows assemble uniquely into a left adjoint (Chosen objectwise universal arrows assemble uniquely into a left adjoint).

Proof

technique · direct

1.1

By [F5] every compact Hausdorff space is Tychonoff, so [F4] makes $J : C o m p H a u s ↪ T y c h$ a well-defined full inclusion; [F3] is what supplies the embedding used inside [F2]. The hypotheses in [F2] supply $(β X, η_{X})$ , and [F1] says precisely that it is a universal arrow from $X$ to $J$ .

F1F2F3F4F5

1.2

For a continuous map $a : X \to Y$ , apply [F1] to $η_{Y} a : X \to β Y$ and define $β a : β X \to β Y$ as its unique extension.

F1construct

2.1

Extension uniqueness gives $β (1_{X}) = 1_{β X}$ and $β (b a) = β (b) β (a)$ , and the defining equations make $η$ natural.

step 1.2F1

3.1

Thus the chosen universal arrows assemble by [L1] into $β ⊣ J$ . The assumptions are exactly those used in [F2] and [F3]; the assembly step adds no choice principle.

step 1.1step 2.1L1F2F3∎

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: 80 results over 18 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

Emily Riehl, Category Theory in Context, 2nd ed., Example 4.6.13 (standard reference, not scraped)
Tom Leinster, Basic Category Theory, Example 6.3.14 (standard reference, not scraped)