What sorts of (sets of) equations are "approximately compatible" with the $2$-sphere?
Given a metric space $\mathcal{X}=(X,d)$ and an equational theory (in the sense of universal algebra) $\mathsf{E}$, say that an approximate model of $\mathsf{E}$ on $\mathcal{X}$ is a sequence $(\mathcal{M}_i)_{i\in\mathbb{N}}$ of structures in the signature of $\mathsf{E}$ with domain $X$ such that:
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in each $\mathcal{M}_i$, each function symbol gets interpreted as a continuous function $X^{arity}\rightarrow X$ in the sense of $d$; and
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for each $i\in\mathbb{N}$, each equation $t(x_1,...,x_n)=s(x_1,...,x_n)$ in $\mathsf{E}$, and each $a_1,...,a_n\in X$, we have $d(t(a_1,...,a_n), s(a_1,...,a_n))<2^{-i}.$
Say that $\mathsf{E}$ is approximately compatible with $\mathcal{X}$ iff $\mathsf{E}$ has an approximate model on $\mathcal{X}$. Compare this with the notion of (genuine) compatibility discussed here, where there is no "error" permitted.
In the above-linked paper, a generalization of Adams' theorem that there is no group structure on the $n$-sphere for $n\not\in\{0,1,3,7\}$ is proved: for $n\not\in\{0,1,3,7\}$ there is no "interesting" algebraic structure compatible with the $n$-sphere at all. On the other hand, approximate compatibility seems like an extremely weak notion, and I don't see any nontrivial examples of equational theories which are not approximately compatible with (for example) the $2$-sphere:
Is there an equational theory $\mathsf{E}$ which has a model with more than one element but is not approximately compatible with $S^2$?
Note that for equational theories, "has a model with more than one element" is equivalent to (for example) "has a model of size continuum," so once we rule out the genuinely trivial theories there is no cardinality obstacle.
The linked paper already answers this! Bold emphasis mine:
Operations $\bar F_t$ are said to satisfy an equation $σ ≈ τ$ up to homotopy if, when we substitute $\bar F_t$ for each $F_t,$ the functions associated with $σ$ and $τ$ are homotopic to each other (although not necessarily equal as functions). In a similar way, one speaks of compatibility up to homotopy, and so on. Theorem 1 will be stated and proved for satisfaction up to homotopy.
[...]
Theorem 1. Let A be a path-connected space satisfying one of the following seven hypotheses. If A is compatible with $\Sigma$ up to homotopy, then Σ is undemanding.
- A is homeomorphic to the sphere $S_n$ ($n \neq 1, 3, 7$).
[...]
If functions $f,g:Z\to S^2$ satisfy $d(f(z),g(z))<2$ then they are homotopic, by the homotopy $$h(t,z)=\frac{(1-t)f(z)+t g(z)}{\|(1-t)f(z)+t g(z)\|}.$$ So the answer is: yes, any demanding equational theory with models with more than one element, for example the equational theory of free groups.