(I'm going to try to use definitions from [*Abstract and Concrete Categories: The Joy of Cats* by Adámek, Herrlich, and Strecker](http://katmat.math.uni-bremen.de/acc/), 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](http://katmat.math.uni-bremen.de/acc/acc.pdf#section.0.21), [nLab](https://ncatlab.org/nlab/show/topological+concrete+category)) 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](http://katmat.math.uni-bremen.de/acc/acc.pdf#section.0.23), [nLab](https://ncatlab.org/nlab/show/algebraic+category)), 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.

-------

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](http://katmat.math.uni-bremen.de/acc/acc.pdf#section.0.23), 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](https://eudml.org/doc/91559), 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)?