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starting a stub here, for an entry which we should have had long ago, to complete the pattern of entries listed under “Related entries”.
What I was really after is seeing if the construction in section 2.4 of
is genuinely new, or a special case of an existing theorem. The category $Rep(\mathcal{A})$ considered by these authors should equivalently be that of modules over a monoid in the monoidal category of copresheaves with values in a fixed monoidal model category. Phrased this way, the model structure on this category which these authors present might be expected to be a special case of the general construction of Schewede & Shipley. I haven’t checked yet.
Let me turn this question around:
For $\mathbf{M}$ a monoidal model category and $\mathbf{C}$ any small category, $Func(\mathbf{C}, \mathbf{M})$ should be a monoidal category under tensor product in $\mathbf{M}$ pointwise over $\mathbf{C}$, and for $\mathcal{A} \,\in\, Mon\big(Func(\mathbf{C}, \mathbf{M})\big)$ it should follow that $\mathcal{A} Mod\big(Func(\mathbf{C}, \mathbf{M}) \big)$ is a model category with objectwise weak equivalences and fibrations.
Is this special case of Theorem 4.1 in Schwede & Shipley 2000 explicitly citable from anywhere?
Because this is – it seems to me – what Sec. 2 in arXiv:2201.06464 re-proves from scratch.
Re #2: What is the model structure on Func(C,M)?
For the projective model structure, C should have finite products (at least up to retracts). Otherwise, the monoidal product of two cofibrant objects y(c_1)⊗m_1 and y(c_2)⊗m_2 is y(c_1)⨯y(c_2)⊗(m_1⊗m_2), which need not be cofibrant if y(c_1)⨯y(c_2) is not representable (or a retract of a representable).
If C does have finite products, then Func(C,M) is indeed a monoidal model category once we equip it with the projective model structure. This is trivial to check using generating (acyclic) cofibrations.
For A-modules to admit a model structure, you need an additional condition, e.g., the monoid axiom.
Thanks. True, these authors allow any small $\mathcal{C}$ and the intended example (Ex. 2.2) does not have products.
So maybe these mathematical field theorists filled an actual gap in the model category literature then! :-)
What is the exact theorem in Section 2 that is being discussed? I see them repeating a lot of well-known facts with proofs, e.g., the construction of the projective model structure.
Cor. 2.13 claims the model structure on $\mathcal{A}Mod \big( Func(\mathbf{C}, \mathbf{M}) \big)$ and Prop. 2.16 the corresponding base change Quillen adjunction.
Re #6: The model structure of Corollary 2.13 is indeed a special case of Schwede–Shipley’s Theorem 4.1, part (1). I really don’t understand why they rederived it from scratch, and in much more restrictive setting: they require all objects to be fibrant, which is completely unnecessary.
The model structures are clearly the same, but does Schwede&Shipley’s proof apply? I thought you had a point in #3 that checking the assumptions for S&S’s theorem may be nontrivial.
If one wants only a model structure on modules, as in Corollary 2.13, then the monoid axiom alone is sufficient, there is no need for the model structure to be monoidal.
To construct a model structure on monoids, or a monoidal model structure on modules over a commutative monoid, one does need the model structure to be monoidal.
Oh, I see. Thanks for pointing this out!
I have added (here) a sentence to the entry to clarify this.
Regarding my original question: I take it that the example of the model structure on modules in a functor category you have not seen discussed elsewhere? It does seem like a fundamental class of examples that deserve to be highlighted.
I take it that the example of the model structure on modules in a functor category you have not seen discussed elsewhere? It does seem like a fundamental class of examples that deserve to be highlighted.
The monoidality of functor category has definitely been discussed before, for example, see the second paragraph of the proof of Proposition 7.9 in my paper https://arxiv.org/abs/1410.5675.
I am not sure whether I have seen the particular statement about modules in a functor category before, though.
Thanks! Have recorded that here
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