202510171510

Tags : Category Theory

Free-Forgetful Adjunction from Compact Hausdorff Spaces is Monadic

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A topological space can be defined as a set $X$ endowed with a closure operator $\overline{(-)}:PX\to PX$ such that the following properties hold

$$\begin{array}{ccccccc}\overline{\varnothing }=\varnothing & & A\subseteq \overline{A}& & \overline{\overline{A}}=\overline{A}& & \overline{A\cup B}=\overline{A}\cup \overline{B}\end{array}$$

A function $f$ is continuous if $f(\overline{A})\subseteq \overline{f(A)}$ and is called closed if the equality holds. All continuous functions between Compact Hausdorff Spaces are closed.

Consider an absolute coequalizer diagram in $\text{Set}$.

$$X\underset{g}{\stackrel{f}{\rightrightarrows }}Y\stackrel{h}{\twoheadrightarrow }Z$$

The powerset functor preserves this, giving rise to the following diagram:

Since $f$ and $g$ are maps of hausdorff spaces, these functions make the diagram commute.

The induced function defines a closure operator on $Z$ so that it makes $Z$ a topological space, as $h$ is surjective, continuous and closed. By our categorization of compact hausdorff spaces, we get a unique coequalizer map defined in $\text{cHaus}$.

To see that $h$ has the universal property, we must prove that the right square commutes:

This is direct from the fact that $h_{\ast }$ is an epimorphism.

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References

• Compactness
• Hausdorff Property
• Topological Spaces
• Monads and Comonads