In a [recent paper,](https://arxiv.org/pdf/1605.06794.pdf) 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](https://arxiv.org/pdf/1702.04070.pdf), 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 $\mathcal{E}$).



Inducing a model structure using a cosimplicial object
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](https://ncatlab.org/nlab/show/transferred+model+structure#PathObjects).
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.


