Very roughly, *universal algebra* is the study of those classes of algebraic structures which can be defined via a set of equations; such a class is called a *variety*. Of course there is far more to the subject than this, but this is the essential starting point; and (in my opinion) the essential first result is the HSP theorem, which characterizes varieties in an often more tractable way:

> **HSP theorem**: $\mathcal{V}$ is a variety iff $\mathcal{V}$ is closed under homomorphic images, substructures, and arbitrary-arity products.

I'm interested in whether there is a similar characterization of *varieties over a space*.

An algebra $A$ with underlying set $X$ is *compatible with* a topology $\tau$ on $X$ if all the operations of $A$ are continuous in the sense of $\tau$. *(We can also look at compatibility* up to homotopy, *but that doesn't obviously make things cleaner here, so I'll use the more restrictive version here, although I'm interested in each.)* For instance, the ring of real numbers $(\mathbb{R}; +, \times, 0, 1)$ is compatible with the usual topology on the reals, and by contrast there is no group compatible with the usual topology on $S^2$. Vastly more generally, [Adams' Hopf invariant one theorem](https://en.wikipedia.org/wiki/H-space) states that there is no unital magma structure compatible with the usual topology on $S^n$ unless $n\in\{0, 1, 3, 7\}$.

[Walter Taylor has studied](http://matwbn.icm.edu.pl/ksiazki/fm/fm164/fm16431.pdf) algebras compatible with given topologies; among other things, he extended Adams' theorem to show that if $n\not\in\{0, 1, 3, 7\}$, then $S^n$ doesn't admit any nontrivial algebraic structure at all, in a precise sense. He also studied the satisfiability problem for algebras over a topological space, from a computability-theoretic perspective.

I'm interested in a different aspect of "topological universal algebra": we can also ask about those classes of algebras with underlying set $X$, compatible with $\tau$, which are defined by equations; and we can define a $(X, \tau)$-variety as such a class of algebras. This is a perfectly reasonable notion; however, each of the operations H, S, P are terribly behaved in this context! My main question here is:

> **Q1.1**. Is there a "nice" alternate characterization of $(X, \tau)$-varieties, for instance in terms of a fixed family of operations which build new $(X, \tau)$-algebras from old ones?

This is hopelessly broad, though, so a reasonable thing to do is look at a more restrictive case:

> **Q1.2**. Is there a "nice" alternate characterization of algebras compatible with the reals (with the usual topology)?

Even this, though, seems intractable to me - there are *individual equations* whose corresponding variety I don't understand at all! Take our language to consist of a single binary function symbol, "$*$". I don't have a good sense of the class of algebras on $\mathbb{R}$ in which $*$ is commutative. However, at least I know a little about the natural topology on this class - it's path-connected, by taking weighted averages of operations: $$a*_{p}b={a*_1b\over p}+{a*_2b\over 1-p}.$$ An equation that by contrast I know nothing about is *associativity*:

> **Q2.1**. What is a good description of the class of continuous associative binary operations on $\mathbb{R}$?

It's easy to check that weighted averages no longer preserve associativity in general. And this brings me to my final, most-concrete question:

> **Q2.2**. Is there any interesting way to combine two continuous associative binary operations on $\mathbb{R}$ and get a third?

Of course "interesting" is a vague term here - I certainly want to rule out trivialities like constant maps and the projection maps, as well as completely ad-hoc constructions. *I could try to make this precise (e.g. asking for continuity with respect to the natural topology on the space of such operations), but rather than do that it feels more natural to leave this subjective.*

**Basically: as natural a notion as it appears to me, I have absolutely no idea what a topological variety is, and would like to.**