Does the category of semisimplicial Kan complexes form a category of fibrant objects? It is known that the category of Kan complexes form a category of fibrant objects, as remarked here. We obtain obvious notions for fibrations and weak equivalences in the category of semisimplicial sets, being the Kan fibrations and the maps inducing homotopy equivalences. Does the full subcategory of semisimplicial sets, consisting of the semisimplicial Kan complexes, give a category of fibrant objects?
Or, maybe more general, does the forgetful functor from simplicial sets to semisimplicial sets induce a model structure on semisimplicial sets? I've read from various resources that this is not necessarily the case, but to which extend is this structure preserved by the forgetful functor? That is, which of the axioms of a model category are satisfied, and which aren't, by considering this structure inherited from simplicial sets? What is the standard literature on this?
There are some discussions like this, or this, but these do not entirely answer the question with great certainty.
 A: For clarity, let's fix precise definitions.
A Kan fibration of semisimplicial sets is a morphism that has the right lifting property w.r.t. the horn inclusions $\Lambda^n_k \hookrightarrow \Delta^n$, where $\Delta^n$ here means the semisimplicial set represented by $[n]$.
A weak homotopy equivalence of semisimplicial sets is a morphism whose geometric realisation is a homotopy equivalence.
In brief:

*

*There does not exist a model structure on the category of semisimplicial sets where the weak equivalences are the weak homotopy equivalences.

*There does not exist a full subcategory of semisimplicial sets that contains $\Delta^0$ as well as non-discrete semisimplicial sets and admits the structure of a category of fibrant objects where the weak equivalences are the weak homotopy equivalences and pullbacks of fibrations are preserved by the inclusion functor.
(This is not quite an answer to your question since $\Delta^0$ is not a Kan-fibrant semisimplicial set.)


Let $i^*$ be the forgetful functor from simplicial sets to semisimplicial sets.
It has a left adjoint $i_!$ and a right adjoint $i_*$.
It is clear that $i_! (\Lambda^n_k \hookrightarrow \Delta^n)$ is (isomorphic to) the corresponding horn inclusion in simplicial sets, so a morphism $X \to Y$ of simplicial sets is a Kan fibration if and only if $i^* (X \to Y)$ is a Kan fibration of semisimplicial sets.
It is also clear that a morphism $X \to Y$ of semisimplicial sets is a weak homotopy equivlence if and only if $i_! (X \to Y)$ is a weak homotopy equivalence of simplicial sets.
This sounds good, but notice that the functors involved in the two observations are going in opposite directions!
That is basically why we can't use the usual machinery to transfer the model structure from simplicial sets to semisimplicial sets.

In fact, as I claimed above, there is simply no way of constructing a model structure or a category of fibrant objects with the desired properties.
The comments here explain why there is no possible model structure on the category of semisimplicial sets where the weak equivalences are the weak homotopy equivalences defined above.
I reproduce the argument.
Suppose there is such a model structure.
Let $p : X \to Y$ be a trivial fibration in the putative model structure.
As in any model category, the class of trivial fibrations is closed under pullback, so for each $n$-simplex $y : \Delta^n \to Y$ we obtain a trivial fibration $p_y : X_y \to \Delta^n$.
But $\Delta^n$ has no $m$-simplices for $m > n$, so $X_y$ also has no $m$-simplices.
In the case $n = 0$, $p_y : X_y \to \Delta^0$ must therefore be an isomorphism.
By induction on $n$, we deduce that $p : X \to Y$ is an isomorphism on the $n$-skeleton for every $n \ge 0$, hence $p : X \to Y$ is an isomorphism of semisimplicial sets.
Thus every trivial fibration in the putative model structure is an isomorphism.
This implies every morphism is a cofibration, which would mean every pushout is automatically a homotopy pushout – but e.g. the pushout of $\Delta^0 \leftarrow \Delta^0 \amalg \Delta^0 \rightarrow \Delta^0$ is not a homotopy pushout.
Therefore there is no such model structure.
The above argument does not, prima facie, preclude the possibility that there is some full subcategory of semisimplicial sets that admits the structure of a category of fibrant objects containing $\Delta^0$ and where the weak equivalences are the weak homotopy equivalences and the inclusion preserves pullbacks of fibrations.
Since $\Delta^0$ is assumed fibrant, we can apply the earlier argument to deduce that every trivial fibration (of fibrant semisimplicial sets) is an isomorphism of 0-skeleta.
This is already problematic.
Let $X$ be a fibrant semisimplicial set.
It has a path object, i.e. $(P, i, p_0, p_1)$ where $P$ is a fibrant semisimplicial set, $i : X \to P$ is a weak homotopy equivalence, $p_0, p_1 : P \to X$ are morphisms such that $p_0 \circ i = p_1 \circ i = \textrm{id}_X$, and $\langle p_0, p_1 \rangle : P \to X \times X$ is a fibration.
As usual, $p_0, p_1 : P \to X$ are trivial fibrations, so they are isomorphisms of 0-skeleta.
Hence $i : X \to P$ is also an isomorphism of 0-skeleta.
But that means every path in $X$ is constant – in particular, there are no paths between distinct points, nor loops.
So the only objects in our subcategory are discrete sets.

The arguments above don't actually use anything special about simplices.
The key point is that a semisimplicial set has no degeneracies, so there are no semisimplicial maps $X \to Y$ if $X$ has an $n$-simplex and $Y$ does not have any $n$-simplices.
(In particular, $\Delta^0$ is not a terminal object!)
So the same argument applies to e.g. the category of cubical sets without degeneracies, or the category of globular sets without degeneracies, etc.
