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I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been great, so I'm hoping the (higher) category theory people here can help me out!

Categories Of Functors: Let $\cal{C}$ be a (closed) model category and let $\mathcal{S}$ be a finite poset.

The category of functors $[\mathcal{S},\mathcal{C}]$ can be imbued with the Reedy model structure (see [1]). A functor of posets $f:\mathcal{S} \to \mathcal{T}$ induces a pullback functor $$f^*:[\mathcal{T},\mathcal{C}] \to [\mathcal{S},\mathcal{C}]$$ This functor fits into a Quillen adjunction with the left Kan extension $f_!$ (see Barwick [2]). $$ f_!:[\mathcal{S},\mathcal{C}] \leftrightarrow [\mathcal{T},\mathcal{C}]:f^* $$ Furthermore, if $\iota:\mathcal{S} \to \mathcal{T}$ is the inclusion of a sub-poset that is downward closed (i.e. $s \in \mathcal{S}$ and $s' \prec s$ implies $s' \in \mathcal{S}$) then

$$ \iota^*:[\mathcal{T},\mathcal{C}] \to [\mathcal{S},\mathcal{C}] $$ preserves cofibrations.

Maybe I should mention for clarity that we are viewing $\mathcal{S}$ as a directed category, i.e. a Reedy category where $\mathcal{S}_+ = \mathcal{S}$ and $\mathcal{S}_-$ is the trivial sub-category with every object.

Let me fix some notation for a special sub-category of the functor category $[\mathcal{S},\mathcal{C}]$. I haven't totally settled on the definition yet, the conditions are mostly coming from the situation I'm in.

Definition 1: The category $\text{Ch}[\mathcal{S},\mathcal{C}]$ of $\mathcal{S}$-chains in $\mathcal{C}$ is the full sub-category of $[\mathcal{S},\mathcal{C}]$ consisting of functors $x:\mathcal{S} \to \mathcal{C}$ such that

  • (a) $x$ is a cofibrant diagram with respect to the Reedy model structure.
  • (b) $x_S \to x_T$ is a quasi-isomorphism for each $S \to T$ in $\mathcal{S}$.

I may want to enhance the assumptions above as so, if it helps prove the result that I'm looking for (Proposition 1 below).

  • (a') $x$ is cofibrant and fibrant with respect to the Reedy model structure.
  • (b') $x_S \to x_T$ is a trivial cofibration for each $S \to T$ in $\mathcal{S}$.

I would also be happy to use the hypothesis that $x$ is cofibrant in the projective model structure, which is just object-wise cofibrance.

Categories Of Strata: One nice class of posets arises from simplicial complexes.

Definition 2: Let $X$ be a simplicial complex. The category of strata $X\mathcal{S}$ is the poset whose objects are simplices $S$ in $X$ and where there is a morphism $S \to T$ if $S$ is contained in $T$.

Clearly the map $X \to X\mathcal{S}$ is functorial. Any map of simplicial complexes $f:X \to Y$ induces a map of posets $f:X\mathcal{S} \to X\mathcal{T}$.

Main Question: The result that I've been trying to prove is the following horn filling property.

Proposition 1 (?): Let $\iota:\Lambda^{n,k} \to \Delta^n$ denote the standard inclusion of the horn $\Lambda^{n,k}$ into the $n$-simplex $\Delta^n$, and let $\iota:\Lambda^{n,k}\mathcal{S} \to \Delta^n\mathcal{S}$ also denote the induced functor on strata categories. Then the corresponding pullback functor

$$\iota^*:\text{Ch}[\Delta^n\mathcal{S},\mathcal{C}] \to \text{Ch}[\Lambda^{n,k}\mathcal{S},\mathcal{C}]$$ admits a section, i.e. a functor $\sigma:\text{Ch}[\Lambda^{n,k}\mathcal{S},\mathcal{C}] \to \text{Ch}[\Delta^n\mathcal{S},\mathcal{C}]$ with $\iota \circ \sigma = \text{Id}$.

My main question is the following.

Question: Is Proposition 1 true? What about if I implement some of the possible modifications to Definition 1 suggested above?

Ideas For Proof: Here's a sketch of the proof that I had in mind.

You can extend a functor $x \in \text{Ch}[\Lambda^{n,k}\mathcal{S},\mathcal{C}]$ to a functor that $\bar{x} \in [\Delta^n\mathcal{S},\mathcal{C}]$ by filling in the two faces $T_0,T_1$ of $\Delta^n$ that are missing in $\Lambda^{n,k}$ with the colimit $\text{colim}(x)$ and the inclusions $S \to T_i$ with the colimit maps $x_S \to x_{T_i}$. A map $x \to y$ in $\text{Ch}[\Lambda^{n,k}\mathcal{S},\mathcal{C}]$ induces a map $\bar{x} \to \bar{y}$ in an obvious way, and this defines a functor $\sigma$ as in the Proposition. We need to show that $\bar{x}$ has properties (a) and (b) from Definition 1 .

To show Property (b) from Definition 1, we note that since the nerve of $\Lambda^{n,k}\mathcal{S}$ is the barycentric sub-division of $\Lambda^{n,k}$ (thus contractible) and the maps $x_S \to x_T$ are quasi-isomorphisms, the map $x_S \to \text{colim}(x)$ is a quasi-isomorphism.

Property (a) is the issue: The colimit $\text{colim}(x)$ is cofibrant, because the the colimit is the left Kan extension of the pullback by $\Lambda^{n,k}\mathcal{S} \to *$ and this is a left Quillen adjoint. However, there seems to be no reason for the extended diagram to be cofibrant. You could try cofibrant replacement, but this will ruin the property that $\iota^*\bar{x} = x$.

I'm not sure if (a') and/or (b') help at all, and switching to the projective model structure (so, just assuming that $x$ is object-wise cofibrant) seems to ruin the property that the colimit will be cofibrant, which is bad. Anyway, this is where I'm stuck.

One last remark is, if I just use the (pointwise) left Kan extension $\iota_!$ then the quasi-isomorphism property (b) of Definition 1 isn't satisfied in general, as far as I can tell.

Thanks: For reading the long question, and for any help or advice you might have!

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I think the following works, if I’ve followed your terminology conventions correctly: Define each $T_i$ (first the missing codim-1 face, then the main simplex itself) as the fibrant replacement of the colimit of sub-faces you tried, so that the map $\varinjlim (S_x) \to T_i$ is an acyclic cofibration? Being acyclic means that your property (b), homotopy-constancy, is inherited from the colimit; being a cofibration ensures that (a), Reedy cofibrancy, is preserved.

This construction works more generally for any inclusion of a downset in a finite poset (and more generally still for discrete fibrations of direct categories). I’m pretty sure it’s in the literature, though I’m not sure where off the top of my head — perhaps try looking in Karol Szumiło’s thesis.

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  • $\begingroup$ Awesome, that's very simple. I agree that this totally seems to work. Thanks a lot! $\endgroup$ – Julian Chaidez Aug 10 at 21:51
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    $\begingroup$ Also you're right, something like this is in Szumilo's thesis. Section 1.3 of arxiv.org/abs/1411.0303 for anyone who's interested. $\endgroup$ – Julian Chaidez Aug 10 at 21:53

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