$n$-truncation of a Simplicial Model Category

Question

I'm working in the category of rational $CDGAs$ and trying to find a reference/construction of a natural $2$-categorical structure via truncation of the mapping spaces.

In my head, the key point is that $CDGA$s over $\mathbb{Q}$ form a simplicial model category, where the $n$-simplices are given by $$Map(A,B)_n = Hom_{CDGA}(A, B \otimes \mathcal{A}_{PL}(\Delta^n)).$$ My guess is that there should be an $n$-truncation, where we truncate each of the mapping spaces, that turns $CDGA$ into an $(n-1)$-category.

The kind of truncation I'm imagining is that from Lurie's Higher Topos Theory Proposition 2.3.4.12 where $n$-truncation roughly corresponds to replacing $Map(A,B)$ with $h_n Map(A,B)$, defined by:

• the $m$-simplices are $h_n Map(A,B)(m) = [\Delta^m, Map(A,B)]_n / \sim$ where
• $[\Delta^m, Map(A,B)]_n$ is the image of the restriction map $$i^* : Fun(Sk^{n+1} \Delta^m, Map(A,B)) \to Fun(Sk^{n} \Delta^m, Map(A,B))$$ induced by the inclusion of skeleta, and
• for maps $f,g \in [\Delta^m, Map(A,B)]_n$, we say $f \sim g$ if $f$ and $g$ are homotopic relative to $sk^{n-1} \Delta^m$.

with the proposition stating that $h_n(Map(A,B))$ is indeed an $n$-category.

The points I'm confused about are:

1. I'm fairly sure this only works for Kan complexes - whereas not all mapping spaces in a simplicial model category are Kan complexes. My feeling is that this is resolved somehow using the extra condition on simplicial model categories about the pullback powering being a Kan Fibration (Definition 2.1 - Condition 3)
2. How do we move from truncating individual mapping spaces $Map(A,B)$, to a $2$-categorical structure on $CDGA$? I would expect that applying $h_n(Map(A,B))$ on all mapping spaces can be done directly on $CDGA$ by applying some other $\tilde{h}_{n+1}$ on CDGA directly.

I've been able to find a lot of references about this kind of truncation for quasi-categories, but nothing about simplicial model categories specifically.

In the end, what I would hope for is that doing this kind of operation at level $1$ in the mapping space would result in $2$-category (say something called $\tilde{h}_2(CDGA)$), where the objects and morphisms are the same and the $2$-cells are homotopy classes of homotopies. I tried proving this directly, but had a lot of trouble working out how the vertical and horizontal composition can be interchanged appropriately.

I was hoping to find a reference that deals with truncation in simplicial model categories. In general, I am quite confused about the passage from simplicially enriched things to quasi-categories, especially when we have simplicial model categories - which are simplicially enriched with a number of extra conditions.

Answer 1

The OP wrote "I was hoping to find a reference that deals with truncation in simplicial model categories." In 2022, Michael Batanin and I published a paper, Homotopy theory of algebras of substitudes and their localisation that included this concept as Definition 9.4.1. A model category is $n$-truncated if for any objects $X,Y$, the derived mapping space $Map(X,Y)$ has homotopy groups vanishing above $n$ (as pointed out in the comments, your issue (1) can be ignored; these objects are automatically (co)fibrantly replaced when we compute $Map(X,Y)$). In Section 14, we construct such model structures as left Bousfield localizations of other combinatorial, simplicial models.

You could do the same for the category CDGA over $\mathbb{Q}$, because I proved the right-induced model structure on CDGA is combinatorial and left proper in Theorems 3.2 and 4.17 of my paper Model Structures on Commutative Monoids in General Model Categories (see also Example 5.1, for the context of CDGAs). Hence, you can apply truncation to the category CDGA, as you suggested to do in your (2).