Inducing a model structure using a cosimplicial object

Question

In a recent paper, Hiroshi Kihara induced a model structure on the category of diffeological spaces. He generates the classes of fibrations, cofibrations, and weak equivalences by constructing a functor $d:\Delta \to \mathcal{D}$ (where $\mathcal{D}$ is the category of diffeological spaces, and $\Delta$ is the simplex category), horns $\Lambda^p_k$ are constructed as smooth deformation retracts of the cells $\Delta^p$. Weak equivalences are those maps $f:X \to Y$ so that the $d$-nerve $N_d(f) = \mathcal{D}(d-,f)$ is a weak equivalence of simplicial sets, and using the set of horn inclusions $\Lambda^p_k \to \Delta^p$ as the generating cofibrations.

Based on the results claimed in this arxiv preprint, it seems like he is using the nerve of the functor $d: \Delta \to \mathcal{D}$ to transfer the model structure from simplicial sets to differential objects. This makes me think that there is some notion of a "good enough" cosimplicial object that induces a model structure on a category (somewhat similar to how tangent category can be encoded as a nice functor from $\mathsf{Weil}^{op}$ into a cartesian closed category $\mathcal{E}$).

Answer 1

A somewhat general statement along these lines would be as follows. Suppose $\mathcal{D}$ is a cartesian closed locally presentable category and $d : \Delta \to \mathcal{D}$ is a cosimplicial object with associated geometric realization adjunction $|{\cdot}| : \mathrm{sSet} \to \mathcal{D}$, satisfying the following properties.

1. The maps $|\Lambda^n_k| \to |\Delta^n|$ have retracts for each $n \ge 1$ and $0 \le n \le k$.

   Then $\mathrm{Sing} X$ is a fibrant simplicial set for any $X \in \mathcal{D}$, so every object will be fibrant in the transferred model structure.

2. The functor $|{\cdot}|$ preserves finite products.

   Then $X^{|\Delta^1|}$ is a path object for $X = X^{|\Delta^0|}$, for any $X \in \mathcal{D}$.

This is enough to ensure that the transferred model structure exists, as described on the nLab. In fact, this is essentially how Quillen originally constructed the classical model structure on Top--except that the category of all topological spaces does not quite satisfy most of these conditions; but it comes close enough for similar arguments to go through.

Of course condition 1 (and to a lesser extent, condition 2) is far from formal, and the kinds of model structures produced in this way are rather special. As a trivial example, the Yoneda embedding $\Delta \to \mathrm{sSet}$ doesn't satisfy condition 1 (otherwise it would be automatic!) but obviously the model category structure on $\mathrm{sSet}$ can still be transferred across the identity functor.

We can be somewhat more precise about which model categories can be constructed by a variation of the above argument. Condition 1 is equivalent to every object of the transferred model structure being fibrant, because if $|\Lambda^n_k|$ is fibrant then the acyclic cofibration $|\Lambda^n_k| \to |\Delta^n|$ must have a retraction. Condition 2 is basically meant to ask for a simplicial structure on $\mathcal{D}$ to use for building path objects. One way to get this, not necessarily the only one, is for $\mathcal{D}$ to be cartesian closed and set $K \otimes X = |K| \times X$, and ask for $|{\cdot}|$ to preserve finite products. So, the model categories that can be built in this way are the simplicial ones transferred along a simplicial adjunction $\mathrm{sSet} \rightleftarrows \mathcal{D}$ with every object fibrant.

Having looked at the referenced paper more closely, it seems that the geometric realization functor considered there does not preserve finite products. Instead the acyclicity condition is checked in another way using deformation retracts, analogous to Lemma 2.4.8 in Mark Hovey's book. I'm not aware of a general framework for this kind of argument but it does seem relatively formal as well.