Consider an algebraic variety whose category of models in $\mathsf{Set}$ is homological. This is equivalent to saying that the theory has a unique constant $0$ and there exists an $n\geq 1$ such that:
- there are $n$ binary operations $\alpha_i$ such that $a_i(x,x) = 0$
- there is a $n+1%$-ary operation $\theta$ with $\theta(\alpha_1(x,y),\dots, \alpha_n(x,y),y) = x$
Then the category of models in $\mathcal{X} := \mathsf{Top}$ is homological, complete and cocomplete. For example this is the case for the category of topological groups.
What's so special about $\mathsf{Top}$? What kind of other categories $\mathcal{X}$ yield similar results?
The category of models of a protomodular variety in a lex (finitely complete) category is protomodular according to the nLab. This settles a part of the question.