Skip to main content
MathOverflow

Return to Question

Became Hot Network Question
deleted 6 characters in body
Source Link
James E Hanson
James E Hanson
  • 12.4k
  • 3
  • 37
  • 67

Recall that given a category $X$, a concrete category over $X$ is a pair $(A,G)$ where $A$ is a category and $G : A \to X$ is a faithful functor (which we often call the forgetful functor). There is a notion of a concrete category being topological (AHS Definition 21.1, nLab) which abstracts the nice property of the category of topological spaces that limits and colimits are computed in a very uniform way relative to the forgetful functor to sets. In particular, limits and colimits are computed in $\mathrm{Set}$ and then lifted to $\mathrm{Top}$ by computing the initial or final topology relative to the diagram in question. In an arbitrary topological category $(A,G)$, one has access to a similar procedure. There are a lot of examples of this kind of category, such as uniform spaces with uniformly continuous maps, (extended) pseudometric spaces with $1$-Lipschitz maps, preorders with monotone maps, and measurable spaces with measurable maps (with this standard forgetful functors). Moreover, there are nice extensions of $\mathrm{Top}$ that are still topological but are also nicer as categories (such as the categories of convergence spaces and pseudotopological spaces, which are quasitoposes).

There is also a notion of a concrete category (or a functor more generally) being algebraic (AHS Def. 23.19, nLab), which abstracts the nice properties of categories of models of algebraic theories (such as groups, rings, vector spaces over a finitefixed field, etc.). There's also several related weaker and stronger conditions (e.g., monadicity and essential algebraicity). One of the nice aspects of the category of locales, $\mathrm{Loc}$, is that its dual (the category of frames) is monadic (over $\mathrm{Set}$) and therefore algebraic in this sense.

Recall that given a category $X$, a concrete category over $X$ is a pair $(A,G)$ where $A$ is a category and $G : A \to X$ is a faithful functor (which we often call the forgetful functor). There is a notion of a concrete category being topological (AHS Definition 21.1, nLab) which abstracts the nice property of the category of topological spaces that limits and colimits are computed in a very uniform way relative to the forgetful functor to sets. In particular, limits and colimits are computed in $\mathrm{Set}$ and then lifted to $\mathrm{Top}$ by computing the initial or final topology relative to the diagram in question. In an arbitrary topological category $(A,G)$, one has access to a similar procedure. There are a lot of examples of this kind of category, such as uniform spaces with uniformly continuous maps, (extended) pseudometric spaces with $1$-Lipschitz maps, preorders with monotone maps, and measurable spaces with measurable maps (with this standard forgetful functors). Moreover, there are nice extensions of $\mathrm{Top}$ that are still topological but are also nicer as categories (such as the categories of convergence spaces and pseudotopological spaces, which are quasitoposes).

There is also a notion of a concrete category (or a functor more generally) being algebraic (AHS Def. 23.19, nLab), which abstracts the nice properties of categories of models of algebraic theories (such as groups, rings, vector spaces over a finite field, etc.). There's also several related weaker and stronger conditions (e.g., monadicity and essential algebraicity). One of the nice aspects of the category of locales, $\mathrm{Loc}$, is that its dual (the category of frames) is monadic (over $\mathrm{Set}$) and therefore algebraic in this sense.

Recall that given a category $X$, a concrete category over $X$ is a pair $(A,G)$ where $A$ is a category and $G : A \to X$ is a faithful functor (which we often call the forgetful functor). There is a notion of a concrete category being topological (AHS Definition 21.1, nLab) which abstracts the nice property of the category of topological spaces that limits and colimits are computed in a very uniform way relative to the forgetful functor to sets. In particular, limits and colimits are computed in $\mathrm{Set}$ and then lifted to $\mathrm{Top}$ by computing the initial or final topology relative to the diagram in question. In an arbitrary topological category $(A,G)$, one has access to a similar procedure. There are a lot of examples of this kind of category, such as uniform spaces with uniformly continuous maps, (extended) pseudometric spaces with $1$-Lipschitz maps, preorders with monotone maps, and measurable spaces with measurable maps (with this standard forgetful functors). Moreover, there are extensions of $\mathrm{Top}$ that are still topological but are also nicer as categories (such as the categories of convergence spaces and pseudotopological spaces, which are quasitoposes).

There is also a notion of a concrete category (or a functor more generally) being algebraic (AHS Def. 23.19, nLab), which abstracts the nice properties of categories of models of algebraic theories (such as groups, rings, vector spaces over a fixed field, etc.). There's also several related weaker and stronger conditions (e.g., monadicity and essential algebraicity). One of the nice aspects of the category of locales, $\mathrm{Loc}$, is that its dual (the category of frames) is monadic (over $\mathrm{Set}$) and therefore algebraic in this sense.

Source Link
James E Hanson
James E Hanson
  • 12.4k
  • 3
  • 37
  • 67

How algebraic can the dual of a topological category be?

(I'm going to try to use definitions from Abstract and Concrete Categories: The Joy of Cats by Adámek, Herrlich, and Strecker, since both of the adjectives in the title of my question seem to have at least three definitions in the context of category theory.)

Recall that given a category $X$, a concrete category over $X$ is a pair $(A,G)$ where $A$ is a category and $G : A \to X$ is a faithful functor (which we often call the forgetful functor). There is a notion of a concrete category being topological (AHS Definition 21.1, nLab) which abstracts the nice property of the category of topological spaces that limits and colimits are computed in a very uniform way relative to the forgetful functor to sets. In particular, limits and colimits are computed in $\mathrm{Set}$ and then lifted to $\mathrm{Top}$ by computing the initial or final topology relative to the diagram in question. In an arbitrary topological category $(A,G)$, one has access to a similar procedure. There are a lot of examples of this kind of category, such as uniform spaces with uniformly continuous maps, (extended) pseudometric spaces with $1$-Lipschitz maps, preorders with monotone maps, and measurable spaces with measurable maps (with this standard forgetful functors). Moreover, there are nice extensions of $\mathrm{Top}$ that are still topological but are also nicer as categories (such as the categories of convergence spaces and pseudotopological spaces, which are quasitoposes).

There is also a notion of a concrete category (or a functor more generally) being algebraic (AHS Def. 23.19, nLab), which abstracts the nice properties of categories of models of algebraic theories (such as groups, rings, vector spaces over a finite field, etc.). There's also several related weaker and stronger conditions (e.g., monadicity and essential algebraicity). One of the nice aspects of the category of locales, $\mathrm{Loc}$, is that its dual (the category of frames) is monadic (over $\mathrm{Set}$) and therefore algebraic in this sense.

My question is about the extent to which you can have a category that resembles both these $\mathrm{Top}$-like categories and $\mathrm{Loc}$. These two conditions are strong and having both of them would be nice, but there seems to be a degree of incompatibility between them. Specifically, as mentioned in AHS Example 23.6, if a topological category $(A,G)$ is algebraic (or just essentially algebraic), then $U$ is an equivalence of categories. Since I'm asking about the dual, this might not be an immediate problem, but the property of being a topological category is self-dual in the sense that $(A,G)$ is topological (over $X$) if and only if $(A^{\mathrm{op}},G^{\mathrm{op}})$ is topological (over $X^{\mathrm{op}}$). That said, $\mathrm{Top}^{\mathrm{op}}$ is a quasi-variety, which makes it surprisingly close to being algebraic over $\mathrm{Set}$. Note also that $\mathrm{Set}^{\mathrm{op}}$ is actually monadic over $\mathrm{Set}$, since it is equivalent to the category of completely distributive complete Boolean algebras.

All of the concrete categories in my questions are over $\mathrm{Set}$ (although a particularly interesting example where this is not the case would also be welcome).

Question 1. Is there a topological category $(A,G)$ such that $G$ is not an equivalence of categories and $A^{\mathrm{op}}$ is monadic? Algebraic? Essentially algebraic?

Given that topological categories don't need to resemble $\mathrm{Top}$ particularly, it's also natural to ask the following more specific question.

Question 2. Is there a topological category $(A,G)$ with $A^{\mathrm{op}}$ monadic or algebraic such that any of the common categories of spaces (e.g., compact Hausdorff spaces, compactly generated weak Hausdorff spaces, topological spaces, locales, etc.) is a full subcategory of $A$? In particular, are either of $\mathrm{Conv}^{\mathrm{op}}$ or $\mathrm{PsTop}^{\mathrm{op}}$ monadic or algebraic (where $\mathrm{Conv}$ is the category of convergence spaces and $\mathrm{PsTop}$ is the category of pseudotopological spaces)?