# Cellular model structures on continuous functors

The category of enriched functors from finite based CW complexes to based topological spaces has a projective model structure. The fibrations are the objectwise Serre fibrations and the weak equivalences are the objectwise weak homotopy equivalences. The cofibrations are defined by the left lifting property. In particular every cofibration is an objectwise Hurewicz cofibration of based spaces, but probably not an objectwise Serre cofibration.

Is this model structure cellular in the sense of Definition 12.1.1 of Hirschhorn's Model Categories and Their Localisations? It is known to be cofibrantly generated.

• Hi. Welcome to MathOverflow. I posted an answer, but realized it was saying something stupid and now have pulled it back. I will repost if I can sort things out in my head. For now, I recommend checking out this question mathoverflow.net/questions/76160/… and the answers there, which delve into Hirschhorn's Theorem 11.6.1 in detail, and might be useful in transferring cellularity in the way you need. Do you have a good sense of the generating (trivial) cofibrations in your case? Jun 20, 2013 at 17:23
• Is it true that if we take all functors rather than just enriched functors that $Fun(CW_\ast,Top_\ast)$ is cellular, e.g. by Proposition 12.1.5? So the problem I guess is that you're looking at a subcategory (now the enriched functors) and a priori there might be enriched functors which are cofibrant here but are not cofibrant in $Fun$ when you forget the enrichment. Still, because this is a transferred model structure you should have (1) and (2) of Hirschhorn for free, leaving us only to wonder if objectwise Hurewicz cofibrations are effective monomorphisms. Am I missing something? Jun 20, 2013 at 18:17
• Thanks for the welcome! I agree that the category of all functors should be cellular. But when working in the enriched setting, I can't seem to show the compactness conditions. The problem is that evaluating a cell complex (in the category of enriched functors) doesn't seem to give a cell complex of spaces. Instead it is just a sequence of $h$-cofibrations. Jun 21, 2013 at 8:12
• In the usual situation of diagram categories the projective generating (triv) cofibrations are just products of the old ones. Is that the case here? If so, shouldn't the domains just be products of spheres and hence compact as topological spaces? Then you can apply 2.4.2 in Hovey's book to say they're compact relative to closed $T_1$ inclusions. Does that help at all? I feel like a sequence of $h$-cofibrations should be a closed $T_1$ inclusion. If this doesn't help, maybe you could get away with just working in Jeff Smith's $\Delta$-generated spaces, which is a combinatorial model of spaces. Jun 21, 2013 at 15:57
• The domains of the generating trivial cofibrations have form $CW_*(X,-) \wedge S^n_+$ where $CW_*(X,-)$ is an enriched functor from $CW_*$ to based spaces whose value at $Y$ is the based space of based maps from $X$ to $Y$ (with the compact-open topology). This morning I found Proposition A.8. of Hovey's "Spectra and symmetric spectra in general model categories". This should be applicable to general topological diagram categories, but only when the projective cofibrations are objectwise Serre cofibrations of spaces. Which is not true in my case. Jun 21, 2013 at 16:35

I am glad that by putting our heads together in Munster this week we finally have a proof and can answer this question (a full 2 years after it was asked!). The answer is yes, and I want to sketch here the email I sent you in case it might be of use to others who are trying to prove things are cellular. Let Top mean compactly generated spaces. Let I (and J) denote the generating (trivial) cofibrations of Top$_\ast$. Let W(I) (and W(J)) denote the generating (trivial) cofibrations of your category of functors, so these sets are generated by functors of the form CW$_\ast(X,-) \wedge i^n_+$ where $i^n_+$ runs through the maps in I (resp. J). We have three conditions to check.

(1) We must show the domains of maps in W(J) are small (for some $\kappa$) relative to cofibrations in Fun(CW$_\ast$,Top$_\ast$). This is easy, because Top is cellular, so domains of maps in J are small relative to cofibrations and the representable functors CW$_\ast(X,-)$ preserve filtered colimits (since finite CW complexes are compact, i.e. small relative to $\kappa = \aleph_0$). So smallness of domains of J relative to I-cell implies smallness of domains of W(J) relative to WI-cell. An easier way to see this is to note that maps in W(J) are objectwise inclusions, colimits are computed objectwise in Fun(CW$_\ast$, Top$_\ast$), and all spaces are small relative to inclusions.

(2) We must show cofibrations are effective monomorphisms. We know the generators are objectwise closed inclusions, i.e. for all $Y$, $CW_\ast(X,Y)\wedge i_\ast^n$ is a closed inclusion, since it's a compact space smashed with a closed inclusion. But in Top, closed inclusions are effective monomorphisms (indeed, these two classes coincide!) so we're done.

(3) We must show the domains and codomains of $W(I)$ are compact in the sense of Hirschhorn relative to $W(I)$. This is somewhat painful. It means that there is some regular cardinal $\kappa$ such that for every relative $W(I)$-cell complex $\eta:F\to T$ (a map in our category, hence a natural transformation) and for every presentation of $\eta$ by a chosen collection of cells then every natural transformation $CW(X,-)\wedge K \to T$, where $K$ is a sphere or disk, factors through a subcomplex of size at most $\kappa$. A presentation of $\eta$ is a realization of $\eta$ as the colimit of a $\lambda$-sequence of maps which are pushouts of coproducts of cells (i.e. maps in $W(I)$). A subcomplex of the given presentation of $\eta$ is a $\lambda$-sequence formed by pushouts of coproducts of a subset of cells. The size is the cardinality of the set of cells. For us $\kappa$ will be taken to be larger than $\gamma_+$ where $\gamma$ is the cardinal making Top cellular. We'll follow the roadmap given by Hovey's proof of Proposition A.8 in his paper on Spectra and Symmetric Spectra in General Model Categories. In particular, we'll use his reduction that it is sufficient to consider cells which are transfinite compositions of pushouts of maps in $W(I)$ rather than coproducts of maps in $W(I)$.

The idea is to reduce to considering maps in Top, use the cellularity of Top to identify which cells we need in our sub-presentation of $T$, then prove that $\eta$ actually factors through that sub-presentation. First, evaluate $\eta$ on a given $Y$ and get $\eta_Y:CW(X,Y)\wedge K \to T(Y)$. Be careful here: it is NOT true that CW(X,Y) is cofibrant (see e.g. this MO question). However, with the compact open topology these CW(X,Y), are compact (this follows from Ascoli's theorem, for example), hence so are the spaces $CW(X,Y)\wedge K$, since K is also compact.

Write $T(Y)$ as the filtered colimit of the cells that make up $T$, evaluated at $Y$, e.g. as a transfinite composition of pushouts of maps $C_{\beta}(Y)\to D_{\beta}(Y)$. Note that Hovey demonstrates it's enough to consider such pushouts rather than pushouts of coproducts of maps in $W(I)$. The filtered colimit building $T(Y)$ consists of closed inclusions because cofibrations in Fun(CW$_\ast$,Top$_\ast$) are objectwise closed inclusions. Since we're working in compactly generated spaces, closed inclusions are automatically closed $T_1$-inclusions, so Hovey's 2.4.2 tells us that maps into such a colimit $T(Y)$ from a compact space factors through a finite subcomplex. This implies only finitely many cells have been used.

Use Hovey's method of writing the $\lambda$-sequence as a retract of another $\lambda$-sequence with the same cells (technically, a set of cells in bijection with your cells), but now formed out of maps in $I$ rather than maps in $W(I)$ evaluated at $Y$. This relies on Hirschhorn 10.5.25 (beware: Hovey was referencing a different draft of Hirschhorn's book than the published one, so his numbering does not match Hirschhorn's), but at the end allows you to apply cellularity in Top to find a subcomplex with lesser cardinality through which $\eta_Y$ factors, though verifying this carefully requires a transfinite induction following the same steps as in Proposition A.8. Let $S_Y$ be the set of $W(I)$ cells which appear (evaluated at $Y$) in the subcomplex for $\eta_Y$.

Next, form a subcomplex $R$ of $T$ which contains all the cells $S_Y$. We know this is a subcomplex, because for each $Y$ there were fewer than $\kappa$ many cells and only had the cardinality of $CW$ many Y’s, which is much less than $\kappa$. Next, use that $R(Y)$ contains all the cells in $S_Y$ so contains the subcomplex through which $CW(X,Y)\wedge K \to T(Y)$ factored through. This provides a map of functors $CW(X,-) \wedge K \to R$, though we don't automatically know it's a natural transformation. We can make it natural by "going further along if necessary", which I now make rigorous. Let $f:Y \to Z$ in CW, i.e. $f$ is a cellular map. We have to check that the naturality square commutes in Top (i.e. that $\eta_Y \circ CW(X,f) \wedge K = R(f) \circ \eta_X$). Because we started with a natural transformation, we know that the requisite naturality diagram commutes eventually in the colimit defining $T$ (i.e. replacing $R$ by $T$ above). Because f has a bounded number of cells (finite even), we can make this occur in $R$ by adding in finitely many more cells to $R$ from our presentation of $T$ (using here that we know the square will commute eventually). Even if you have to add these finitely many more cells for each map f, you still won’t reach the regular cardinal $\kappa$, so $R$ will still be a subcomplex and by construction will admit a natural transformation from $CW(X,-)\wedge K$ factoring $\eta$. This proves the domains are compact as required.

Incidentally, I think this proof method can be used to prove something more general, namely a statement of the form: "if $F:\mathcal{M} \leftrightarrows \mathcal{N}:G$ is an adjunction through which the model structure on $\mathcal{M}$ is transferred to $\mathcal{N}$, and if the model structure on $\mathcal{M}$ is cellular, and if $F$ and $G$ behave sufficiently well with respect to filtered colimits, then $\mathcal{N}$ is cellular.'' I'll try to write this up formally and fold it into a future paper.