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edited Dec 10, 2018 at 13:27

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I agree that Leonid Positselski’s first answer seems probably what the writer had in mingmind: given an adjunction, restricting to the categories of “fixed points” on each side yields an equivalence. Here are two important examples in nature, both involving the category of topological spaces:

There’s an adjunction between the categories of preordered sets and topological spaces, sending a preordered set $(X,\leq)$ to $X$ with the topology of down-closed sets, and sending a topological space $Y$ to $Y$ with its specialisation order. All preorders are fixpoints; on the other side, the fixpoints are exactly the Alexandrov spaces, i.e. spaces where arbitrary intersections of opens are open. Restricting to this subcategory shows that the category of preorders is equivalent (in fact, isomorphic!) to the category of Alexandrov spaces.
The adjunction between the categories of topological spaces and locales, sending a topological space to its frame/locale of opens and sending a locale to its space of points, restricts to the equivalence between spatial locales and sober spaces.

I agree that Leonid Positselski’s first answer seems probably what the writer had in ming: given an adjunction, restricting to the categories of “fixed points” on each side yields an equivalence. Here are two important examples in nature, both involving the category of topological spaces:

There’s an adjunction between the categories of preordered sets and topological spaces, sending a preordered set $(X,\leq)$ to $X$ with the topology of down-closed sets, and sending a topological space $Y$ to $Y$ with its specialisation order. All preorders are fixpoints; on the other side, the fixpoints are exactly the Alexandrov spaces, i.e. spaces where arbitrary intersections of opens are open. Restricting to this subcategory shows that the category of preorders is equivalent (in fact, isomorphic!) to the category of Alexandrov spaces.
The adjunction between the categories of topological spaces and locales, sending a topological space to its frame/locale of opens and sending a locale to its space of points, restricts to the equivalence between spatial locales and sober spaces.

I agree that Leonid Positselski’s first answer seems probably what the writer had in mind: given an adjunction, restricting to the categories of “fixed points” on each side yields an equivalence. Here are two important examples in nature, both involving the category of topological spaces:

There’s an adjunction between the categories of preordered sets and topological spaces, sending a preordered set $(X,\leq)$ to $X$ with the topology of down-closed sets, and sending a topological space $Y$ to $Y$ with its specialisation order. All preorders are fixpoints; on the other side, the fixpoints are exactly the Alexandrov spaces, i.e. spaces where arbitrary intersections of opens are open. Restricting to this subcategory shows that the category of preorders is equivalent (in fact, isomorphic!) to the category of Alexandrov spaces.
The adjunction between the categories of topological spaces and locales, sending a topological space to its frame/locale of opens and sending a locale to its space of points, restricts to the equivalence between spatial locales and sober spaces.

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created Dec 7, 2018 at 18:23

Peter LeFanu Lumsdaine

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There’s an adjunction between the categories of preordered sets and topological spaces, sending a preordered set $(X,\leq)$ to $X$ with the topology of down-closed sets, and sending a topological space $Y$ to $Y$ with its specialisation order. All preorders are fixpoints; on the other side, the fixpoints are exactly the Alexandrov spaces, i.e. spaces where arbitrary intersections of opens are open. Restricting to this subcategory shows that the category of preorders is equivalent (in fact, isomorphic!) to the category of Alexandrov spaces.
The adjunction between the categories of topological spaces and locales, sending a topological space to its frame/locale of opens and sending a locale to its space of points, restricts to the equivalence between spatial locales and sober spaces.