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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

1.1F1F2F3F4F5

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 $J$ .

1.2F1construct

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.

2.1step 1.2F1

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

3.1step 1.1step 2.1L1F2F3∎

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.

Depends on

Used by

With the SAFT initial comma objects supplied for all spaces, they assemble into the compact-Hausdorff reflection and agree on Tychonoff spaces with the constructed Stone-Cech adjunction Theorem

Dependency tree · two levels

25 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

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)