# A model structure on semi-simplicial algebraic Kan complexes?

By an algebraic semi-simplicial kan complex I mean a semi-simplicial set (i.e. a presheaf on the category of finite ordered sets and injective order preserving maps), which is a Kan complex (in the sense that any map from a semi-simplicial horn to it can be filled into a cell) and in which a filling has been chosen for each such Horn, and with the morphism between them being the semi-simplicial map which preserves those chosen fillers.

My question is:

Is there a model structure on this category of semi-simplicial algebraic Kan complexes, in which:

• The weak equivalences are the maps that induce bijections on all the $\pi_n$.

• The fibrations are the maps which are "Kan fibrations" as map of semi-simplicial sets.

• the trivial fibrations are the maps which are trivial kan fibration of semi-simplicial sets.

In particular, not all objects would be cofibrant: the cofibrant objects are those which are obtained by freely adding cell gradually.

This would be very similar to the model structure of algebraic kan complex constructed by T.Nikolaus in this paper, with the exception that it can not come from a model structure on semi-simplicial sets.

The reason I'm asking this, is that I have recently observed that there is what Spitzweck called in his thesis a J-semi model structure that fits this description. I.e. I have been able to prove all the axiom of a model structure except that my "trivial cofibrations" (left lifting property with respect to fibrations) are the same as "cofibration and weak equivalences" only if they have a cofibrant domain.

to put it more simply:

The only thing I don't know is whether maps with a non cofibrant domain can be factored as as acyclic cofibration followed by a fibration, or even more simply whether pushout of Horn inclusion are always weak equivalences when the domain is not cofibrant.

Moreover I know that this semi-model structure is Quillen equivalent to the model structure on spaces.

I have obtain this as a corollary of a more general results that I was working on, but my approach does not say anything about non cofibrant objects and so I cannot answer the question.

It would help a lot to know whether this semi-model structure is actually a model structure. So I was hoping that someone with a different point of view might answer this...

• My answer to a previous question mathoverflow.net/questions/133051/… may be relevant. Jul 16 '16 at 1:20
• My view is that a J-semi model structure is probably enough for anything you need. It can even be Quillen equivalent as a semi-model structure to spaces. Some good references are: Fresse's book on operads and modules, Goerss-Hopkins obstruction theory, things Spitzweck has written, and some of my papers. Basically everything I've ever wanted to do in a model category has a version in a semi-model category (often involving cofibrant replacements, but that's usually fine). Why specifically did you want this to have a full model structure? Jul 16 '16 at 4:02
• @DavidWhite: I agree view you, I am perfectly happy with a semi model structures (but thanks for the reference, I didn't know any beside Spitzweck thesis). The short answer (that will fit in a comment) is that I'm annoyed by the fact that I don't know if it is in fact a model structure and I haven't been able to prove it, or if there is actually situation where "Freely adding a cell along a horn" will produce something that is not homotopy equivalent if one start with a non cofibrant object. Jul 16 '16 at 13:54
• And the motivation is that I have obtain very recently a familly of new semi-model structure similar to this (this one being probably the simplest) but there is no example where I can say that if it is or not a model structure. Having at least one example where I know the answer would help me get a more precise picture of what is going on. Jul 16 '16 at 13:59
• I wondered for a long time if there was a semi-model structure that was provably not a model structure. Now I have an example, and it's in my paper with Batanin, put on arxiv in June. Maybe the method we used can help you understand your situation better. Or maybe not, because our situation is chain complexes. Jul 16 '16 at 14:08

I went back to this question a few days ago and found the solution: it is indeed a true model structure.

I have two (related) approaches to this, but anyway the key point is the semi-simplicial approximation theorem:

Theorem: If $A \hookrightarrow B$ is a cofibration (=an inclusion) of semi-simplicial sets, $C$ is a semi-simplicial Kan complex, $f:A \rightarrow C$ is a semi-simplicial map and one has $g: |B| \rightarrow |C|$ a continuous map on the geometric realization that extends $|f|$ then there is a semi-simplicial map $g' : B \rightarrow C$ which extends $f$ and such that $|g'|$ is homotopy equivalent to $g$ relative to $|A|$.

This can be found in the first paper of Rourke and Sanderson on semi-simplicial sets as theorem 5.3

From this one can deduce a rather direct proof that the trivial cofibrations of the model structure mentioned in the question are weak equivalences:

Lemma: A map of algebraic semi-simplicial Kan complexes is an equivalence if and only if it is an equivalence on the geometric realization of the underlying semi-simplicial sets.

Indeed, because of the semi-simplicial approximation mentioned above, the $\pi_n$ of a Kan complex and of its semi-simplicial geometric realization are the same.

Now if $A$ is a semi-simplicial algebraic Kan complex, then gluing a horn inclusion to $A$ is done as follows: first, one glues the horn inclusion on the underlying semi-simplicial set and then one iteratively glues all the horn inclusions that do not already have a chosen filling (and one never needs to collapse anything, since the gluing of horn inclusions are monomorphism). In particular, the final Kan complex $A'$ is obtained from $A$ only by gluing horn inclusions and hence $A \rightarrow A'$ is clearly a weak equivalence on the semi-simplicial geometric realization, which proves the claim.

But one can also see the whole model structure mentioned in the question in a nicer way:

Theorem: There is a 'right' semi-model structure on semi-simplicial sets, whose fibrations and trivial fibrations are the Kan fibrations and Kan trivial fibrations and weak equivalences are the maps that induce weak equivalences on the geometric realizations (cofibrations are the monomorphisms).

I am not sure of the utilization of the word "right" here, what I mean is the dual of the notion mentioned in the question, i.e. all trivial cofibrations are weak equivalences, but only the trivial fibrations of fibrant codomain are weak equivalences.

From this theorem, one concludes by observing that Nikolaus' construction of model categories of algebraically fibrant objects also works for a semi-model category and produces a semi-model category. But if one applies it to a right semi-model category one gets a right semi-model category in which every object is fibrant and this is an ordinary model category.

Proof: One has two weak factorization systems by the small object argument, and weak equivalences obviously satisfies $2$-out-of-$3$ and even $2$-out-of-$6$. Trivial cofibrations are obviously weak equivalences. Using the approximation theorem above, weak equivalences between fibrant objects can be characterized as bijections on the semi-simplicial $\pi_n$, from which one easily deduces that the trivial fibrations between fibrant objects are weak equivalences. We conclude by proving that fibrations between fibrant objects which are also weak equivalences are trivial fibrations:

Let $f :X \rightarrow Y$ be a fibration and a weak equivalence, let $a : \partial \Delta_n \rightarrow X$ be a boundary of $X$ and $b : \Delta_n \rightarrow Y$ be a filling of $f(a)$ in $Y$. As $f$ is an equivalence on the geometric realizations, $a$ has a filling on the geometric realization. Morevoer, as $Y$ is fibrant, there is a map $a' : \Delta_n \rightarrow X$ such that $f(a')$ is homotopy equivalent to $b$ relative to the boundary.

The point is then that with a bit of work on simplicial combinatorics one can construct a semi-simplicial set $I$ such that: $I$ contains $\Delta_n \coprod_{\partial \Delta_n} \Delta_n$ as a sub-complex, the map $\Delta_n \hookrightarrow I$ is a trivial cofibration and the geometric realization of $I$ is (at least up to homotopy) $\Delta_n \coprod_{\partial \Delta_n} \Delta_n$ with a filled interior.

At this point, one has a map from $I$ to $Y$ at the level of geometric realizations that extends our already existing map from $\Delta_n \coprod_{\partial \Delta_n} \Delta_n$. Hence by the approximation theorem one has such a map as a semi-simplicial map, the inclusion of $\Delta_n$ into $I$ is a trivial cofibration and therefore $I$ can be lifted to $X$ and the other inclusion of $\Delta_n$ into $I$ provides our lifting. This concludes the proof of the theorem.

• I think you've got the term "right semi-model category" correct - these are used heavily in Barwick's paper "On left and right model categories and left and right localizations." May 4 '18 at 23:31