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I say that a category of (say) algebras for a monad[¹] $\text{Alg}(\mathbb T)$ is "uninteresting" if the only model structures on $\text{Alg}(\mathbb T)$ result as transfer of the nine model structures on $\bf Set$.

Clearly $\bf Set$ is uninteresting; also, there are interesting varieties of algebras: Hovey proves that modules over a Frobenius ring carry a nontrivial model structure.

Is there a way to determine what is the difference between these two examples?

Are the categories $\bf Grp, Monoids, Vect, Pos,\dots$ interesting?

It is unbelievable that there is plenty of literature about categories of simplicial objects in $\text{Alg}(\mathbb T)$, and not on $\text{Alg}(\mathbb T)$ itself!

[¹] It is not important that $\text{Alg}(\mathbb T)$ is the category of algebras for a monad; you can rephrase the question in the case of models of a (single-sorted) Lawvere theory, models for a sketch, etc.

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    $\begingroup$ I would expect “uninteresting” varieties to be quite unusual. From the point of view of algebraic theories, any extra non-trivialising axiom that one could impose gives a non-trivial map that one could localise along, giving a model structure that (if I’m not mistaken) won’t be transferred from Set. So any “uninteresting” theory should have a maximality property something along the lines of: every equation either generally holds, or (if generally assumed) trivialises the theory. This seems quite interesting! $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Mar 8, 2017 at 9:51
  • $\begingroup$ AFAICU you are saying something like this: let $Ab \to Grp$ the obvious morphism of theories; then this gives abelianization as the induced $[Ab,{\bf Set}]\leftarrow [Grp, {\bf Set}]$. Now, one can fix a model structure on $\bf Ab$ (they are classified by cotorsion pairs on $\bf Ab$, so I think we can find at least one nontrivial) and define a wk/fib of groups as a wk/fib between the abelianizations. If you choose your cotorsion pair "far enough" from the trivial one this is an interesting model structure. Right? $\endgroup$
    – fosco
    Commented Mar 8, 2017 at 10:21
  • $\begingroup$ Ach, no! The functor above is a left adjoint, not a right. Anyway, you mention a localization; with respect to which map (if abelian groups do not add structure but only property feel free to take semigroups and monoids..), and is it a Bousfield localization, or something else? $\endgroup$
    – fosco
    Commented Mar 8, 2017 at 10:39
  • $\begingroup$ I think all Peter is pointing out is the following. Any category has a model structure where all maps are fibrant and cofibrant, and the weak equivalences are the isomorphisms. As long as the category has cokernel pairs, any reflective subcategory yields a Bousfield localization of this model structure generated by the maps $X \to \tilde X$ along with the maps $\tilde X +_X \tilde X \to \tilde X$ where $\tilde X$ is the reflection of $X$; weak equivalences are maps whose reflection is an isomorphism and fibrant objects are those in the reflective subcategory. $\endgroup$
    – Tim Campion
    Commented Mar 15, 2017 at 2:06
  • $\begingroup$ Of your examples, I think that $\mathsf{Vect}$ is the only one which is "uninteresting"; its model structures can be enumerated by a similar analysis to $\mathsf{Set}$ -- there are fewer of them, I think. By the way, the model structure of Hovey that you mention is not an isolated result -- there's a reasonable amount of literature on "the stable module category." $\endgroup$
    – Tim Campion
    Commented Mar 15, 2017 at 2:11

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