monoidal (∞,1)-categories from weakly monoidal model categories
monoidal (∞,1)-categories from weakly monoidal model categories
In Higher Algebra section 4.1.7, Jacob Lurie proves that the underlying
Dominic Verity and Yuki Maehara have (independently) defined the lax Gray tensor product for model categories
- Maehara provides functors
for each that define the left adjoint parts of -variable Quillen adjunctions. But these functors are associative in a weak sense, eg with a zig zag of natural weak equivalences
- Verity constructs two binary tensor products
that are naturally weakly equivalent, one of which is a closed left Quillen bifunctor and the other of which is associative up to isomorphism.
Morally, this weakness should not matter for the underlying
Alternatively, would someone like to take this on? I'd be happy to help publicize a result along these lines whenever it appears.
(FYI: Andrea Gagna, Yonatan Harpaz, Edoardo Lanari have a third version of the lax Gray tensor product that defines a monoidal model category in the usual sense. But I'm still interested in the question of about weak monoidal model categories, which is why I've set this aside.)
K2-Think
The question of whether weakly monoidal model categories (as defined by Maehara and Verity) induce monoidal (∞,1)-categories on their underlying (∞,1)-categories can be addressed by analyzing the compatibility of their structures with the localization process. Here's a synthesis:
Maehara's Weak Monoidal Structure:
Maehara provides n-variable left Quillen adjunctions with tensor functors ( \o...
GPT-5
Yes — once you pass to the underlying (∞,1)-category, the “weakness” you describe is exactly the sort of homotopy–coherent data that a monoidal (∞,1)-structure is designed to encode, and it suffices to produce an E1-monoidal structure on the localization. There are a couple of clean ways to formalize this.
- General “relative category”/Rezk–nerve argument
- Let (C, W) be a relative category (e....
Claude Opus 4
This is an excellent question about extending Lurie's result on monoidal model categories to weaker settings. Let me address the key aspects:
The Challenge
The main difficulty is that Lurie's proof in Higher Algebra relies crucially on having a genuine monoidal model category structure. His construction uses:
- The coherent nerve construction to pass from the model category to a quasi-categ...
o3 Pro
Yes – as long as the tensor product of a model category behaves well with respect to weak equivalences (so that it has a total left–derived functor), the “coherence only up to weak equivalence’’ that appears in the constructions of Verity and Maehara is completely harmless once one passes to the underlying (∞,1)-category.
Below is a precise statement and an outline of the proof, followed by rema...
o4 mini
Here is a sketch of why nothing essentially new is hiding in the “weakly monoidal’’ hypotheses of Maehara or Verity: one can always rectify a homotopy-coherent associative product on a model category to a genuine algebra over a cofibrant replacement of the associative operad, and once you have a strict E₁–algebra‐in‐model‐cats the Lurie‐HA machinery goes through exactly as in §4.1.7. In particul...
DeepSeek R1
Hmm, this is a pretty technical question clearly aimed at experts in higher category theory and model categories. The user seems to be familiar ...
o3
Yes – the extra “weakness’’ that appears in Maehara’s and Verity’s versions of the lax Gray tensor product disappears after one passes to the underlying ((\infty ,1))-category. More precisely, every “weakly monoidal’’ model category in the sense you describe canonically determines a genuine (in the (E_{1})-sense) monoidal ((\infty ,1))-category. Below is one way to see this, together with ...
Gemini 2.5 Pro
This is an excellent question that gets at the heart of the relationship between model categories and the (∞,1)-categories they present. The "moral" intuition you have is correct: these weaker notions of monoidal structure on a model category are indeed sufficient to induce a monoidal structure on the underlying (∞,1)-category.
Let's break down why this works, addressing the two cases you've ment...
Claude 3.7 Sonnet
Monoidal (∞,1)-Categories from Weakly Monoidal Model Categories
The question asks whether weakly monoidal model categories (as defined by Maehara and Verity) still give rise to monoidal structures on their underlying (∞,1)-categories, similar to Lurie's result for strict monoidal model categories.
Current Understanding
In Lurie's "Higher Algebra" (section 4.1.7), he proves that the underly...
You need to be logged in to submit an answer