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I have two reference questions

  1. What should be required of a category with finite products so that a (multi-sorted, finitary) Lawvere theory induces a monadic adjunction on it? This should be perfectly standard (I think I saw it a month ago), but now I can't find it in books, nlab or MO.

  2. What should be required from a model category so that the category of algebras of a (multi-sorted, finitary) Lawvere theory in it has the right-transferred model structure?

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    $\begingroup$ For your first question, see this discussion by Todd Trimble. $\endgroup$
    – varkor
    Commented Feb 6, 2023 at 18:26
  • $\begingroup$ Thank you very much, this is very helpful! Then "bicomplete cartesian-closed category" is my answer to the first question (because it is natural sufficient condition for being a monoidally cocomplete category as defined by Todd Trimble). $\endgroup$
    – Arshak Aivazian
    Commented Feb 6, 2023 at 18:43
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    $\begingroup$ Although the Cartesian-closure condition is somewhat restrictive, since excludes Abelian categories. Does the category of chain complexes in a good abelian category satisfy the condition under discussion? $\endgroup$
    – Arshak Aivazian
    Commented Feb 6, 2023 at 19:44
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    $\begingroup$ The obstruction to monadicity is, almost always in practice, the existence of the left adjoint. Aside from that it is not hard to ensure that the conditions of the strict monadicity theorem are satisfied: so, indeed, the category of algebras of a single sorted Lawvere theory in abelian category will be monadic over the abelian category iff the left adjoint exists. The issue with multi sorted Lawvere theories is that there is no canonical candidate for a forgetful functor, so your question does not quite make sense in that context. $\endgroup$
    – Zhen Lin
    Commented Feb 6, 2023 at 22:59
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    $\begingroup$ @ZhenLin Of course, I mean the adjunction between $T[M]$ and $M^S$, where $T$ is an algebraic theory with a set of sorts $S$. The forgetting functor returns a tuple of all carriers. I really hoped that adjunction and monadicity were the same here, thanks, glad to hear that! $\endgroup$
    – Arshak Aivazian
    Commented Feb 7, 2023 at 18:57

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The first question is already answered in the comments, so let me focus on the second question. Note first that every (colored) Lawvere theory is in particular a (colored) PROP. This is easy to see from the definitions of a Lawvere theory and of a PROP, and is also proven as Proposition 4.2 in the paper Lawvere Categories as Composed PROPs by Bonchi et al. The idea is that if you're given a Lawvere theory then you map to a PROP where the required strict symmetric monoidal product of the PROP is the given cartesian product in the Lawvere theory. I previously wrote about this here.

So, if $M$ is a model category where every PROP yields a right-transferred model structure on its category of algebras, then the same is true for Lawvere theories. And, a criterion to guarantee this is given in the paper On homotopy invariance for algebras over colored PROPs by Mark Johnson and Donald Yau, in Theorem 1.4. The idea is to use the path object argument in the category of $P$-algebras, so a symmetric monoidal fibrant replacement functor is required, and a cocommutative coalgebra interval can be used to produce functorial path objects for fibrant $P$-algebras (though, this can also be done in other ways). Examples of $M$ satisfying the requirements include simplicial sets, chain complexes over a field of characteristic zero, and simplicial modules over a commutative ring (page 2 of that paper). Donald Yau and I also investigated functorial path object data in our paper Right Bousfield Localization and Operadic Algebras, which includes many more examples like equivariant spaces and spectra, the stable module category, and the category of small categories.

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