TITLE: Application of Lawvere Theories
QUESTION [3 upvotes]: This is supposed to be more of a reference request than an actual question, but anyway...
A Lawvere theory is a category $\mathcal{T}$ with finite products, st. every object is a power of some distinguished object $X$. The category of models is the full subcategory $[\![\mathcal{T},\mathsf{Set}]\!]_\times$ of product preserving functors. Most algebraic categories (like groups, modules, rings etc.) are in fact realized as a category of models of a Lawvere theory. So far so good.
When reading about Lawvere theories it is frequently mentioned that they are special in the sense that they allow us to do algebra in any category $\mathcal{C}$ with finite products, giving a common framework for say topological groups, Lie groups and group schemes etc. Yet I am unable to find any resources on such applications of Lawvere theories beyond the fact that the concept of a $\mathcal{C}$-valued Lawvere theory applies.
I would expect that questions like „under which conditions on $\mathcal{C}$ do colimits exists in the category of models in $\mathcal{C}$? How do we compute them?“ or „how do abelian groups in a good category $\mathcal{C}$ compare to ordinary abelian groups? Under which conditions do we have a calculus of short exact sequences? What diagram lemmas hold true in $\mathcal{C}$?“ are already studied in this context, but I could not find a reference.
PS: I guess that by Kelly‘s transfinite constructions internal algebraic constructions on locally presentable categories are monadic, as we can write down a presentation of the monad. Hence in particular the categories of models are monadic and we know how to compute colimits. Yet my question arose from trying to prove things about topological groups and compact Hausdorff groups and both $\mathsf{Top}$ and $\mathsf{CHaus}$ are far from being locally presentable.

REPLY [2 votes]: The best result I'm aware of is the following:

let $T$ be an algebraic theory such that the category of $\sf Set$-models is protomodular; then the category of models of $T$ in any finitely complete category $\cal E$ is protomodular.

If you call "pointed protomodular" an algebraic theory such that its category of $\sf Set$-models is pointed and protomodular, then the category of topological $T$-algebras is homological, complete and cocomplete. It is, however, almost never a Barr exact category.
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