The n-Category Café

I like this characterization of right properness:

Given a model category $M$, every slice category $M/X$ gets a model category structure in a standard way, and to every map $f:X\to Y$ there is a standard Quillen pair $M/X \rightleftarrows M/Y$. In general, if the standard Quillen pair $M/X\rightleftarrows M/Y$ associated to $f$ is a Quillen equivence, then $f$ has to be a weak equivalence.

$M$ is right proper exactly if the converse holds: i.e., if $f$ a weak equivalence implies that $M/X\rightleftarrows M/Y$ is a Quillen equivalence.

If $X$ and $Y$ happen to be fibrant, then the converse statement will hold. It is only for general objects that a problem arises.

In other words, $M$ is right proper iff all slice categories have the “correct” Quillen equivalence type; i.e., if $X\mapsto h(M/X)$ coincides with its derived functor. So I guess you could say an object $X$ is “quasi-fibrant” if $M/X$ has the correct Quillen equivalence type, so that right proper model categories are the ones with all objects quasi-fibrant. I don’t know if this is the sort of thing you are after.

From this point of view, it seems that a nontrivial higher structure probably cannot be presented by any model category in which all objects are both fibrant and cofibrant.

Let’s see. How about the Strøm model structure?

Dear Mike, you are right to think that right proper Cisinski model categories correspond to locally presentable $(\infty,1)$-categories which are locally cartesian closed.

A first result in this direction is the following: a Cisinski model structure $\mathcal{M}$ is right proper if and only if, for any fibration between fibrant objects $p:X\to Y$, the pullback functor $p^*:\mathcal{M}/Y\to\mathcal{M}/X$ is a left Quillen functor (this can be deduced straight away from Theorem 4.8 in my paper THT (for Théories homotopiques dans les topos, J. Pure Appl. Algebra 174 (2002), 43-82; a copy is available on my web page)).

As you noticed yourself, any right proper Cisinski model structure $\mathcal{M}$ defines a locally presentable $(\infty,1)$-category $Ho_\infty(\mathcal{M})$ which is locally cartesian. Let us prove the converse.

Let $\mathcal{C}$ be a locally presentable $(\infty,1)$-category. For any big enough regular cardinal $\kappa$, we have that the category $\mathcal{C}_\kappa$ of $\kappa$-compact objects in $\mathcal{C}$ is closed under finite limits and that $\mathcal{C}=Ind_\kappa(\mathcal{C}_\kappa)$ (i.e. any object of $\mathcal{C}$ is a $\kappa$-filtering colimit of $\kappa$-compact objects). Let $C_\kappa$ be a fibrant simplicial category which is a model for $\mathcal{C}_\kappa$. For a simplicial category $A$, I will write $P(A)$ for the category of simplicial presheaves on $A$ endowed with the injective model structure, and $h:A\to P(A)$ for the Yoneda embedding. For a (cofibrant) $\kappa$-small simplicial category $I$ and a simplicial functor $F:I\to C_\kappa$, there is an object $hocolim F$ in $C_\kappa$ which correspond to the colimit of (the nerve of $F$) in $\mathcal{C}_\kappa$ (which always exists, by virtue of Theorem 5.5.1.1 of Lurie’s book), so that we have a canonical map (at least, homotopy theoretically): $$(\ast)\qquad hocolim h(F)\to h(hocolim F)$$ Let $S$ be the class of maps of shape $(\ast)$. Then the left Bousfield localization $\mathcal{M}$ of $P(C_\kappa)$ by $S$ is a model for $\mathcal{C}$ (whenever $\kappa$ is big enough to imply the identification $\mathcal{C}=Ind_\kappa(\mathcal{C}_\kappa)$). Furthermore, any representable presheaf is $S$-local: indeed, for any object $c$ of $C_\kappa$, and any simplicial functor $F:I\to C_\kappa$ with $I$ $\kappa$-small, the presheaf $Map(-,c)$ sends $hocolim F$ to $holim Map(F,c)$.

Assume from now on that $\mathcal{C}$ is a locally cartesian closed $(\infty,1)$-category. I claim that $\mathcal{M}$ is right proper. Consider a fibration between fibrant objects $p:X\to Y$ of $\mathcal{M}$. Applying Theorem 4.8 in THT in its precise form, it is sufficient to prove that pulling back along $p$ sends any map of the form $(\ast)$ to a weak equivalence.

CLaim 1. For any object $c$ of $C_\kappa$ and any map $s:h(c)\to Y$, the pullback $h(c)\times_Y X$ is an homotopy pullback in $\mathcal{M}$.

To see this, we may factor the map $s$ as a weak equivalence $u:h(c)\to W$ followed by a fibration $q:W\to Y$ in the model category $\mathcal{M}$. The homotopy pullback $h(c)\times^h_Y X$ is then $W\times_Y X$. As the model category $P(C_\kappa)$ is right proper, it is sufficient to prove that $u$ is in fact a weak equivalence of $P(C_\kappa)$ (because any fibration of $\mathcal{M}$ is a fibration of $P(C_\kappa)$): this will imply that the map $h(c)\times_Y X\to W\times_Y X$ is a weak equivalence in $P(C_\kappa)$, whence on $\mathcal{M}$ as well. But $W$ is a fibrant replacement of $h(c)$ in the left Bousfield localization by $S$, and we know that $h(c)$ is $S$-local, from which we deduce that $u$ is a weak equivalence of $P(C_\kappa)$.

Consider now a cofibrant $\kappa$-small simplicial category $I$ and a simplicial functor $F:I\to C_\kappa$ and a map $s:h(hocolim F)\to Y$. For each object $i$ of $I$, we know from Claim 1 that $h(F_i)\times_Y X$ is the homotopy pullback $h(F_i)\times^h_Y X$, which implies that the map $$hocolim_{i\in I} (h(F_i)\times_Y X)\to h(hocolim_{i\in I}F_i)\times_Y X$$ is a weak equivalence: we are expressing here that pulling back along $p$ preserves homotopy colimits, which is true because $\mathcal{C}$ is assumed to be locally cartesian closed as an $(\infty,1)$-category.

Therefore, to finish the proof, it is sufficient to check

CLaim 2. The canonical map $$hocolim_{i\in I} (h(F_i)\times_Y X)\to (hocolim_{i\in I} h(F_i))\times_Y X$$ is a weak equivalence.

As homotopy colimits are preserved by left Bousfield localization, it is sufficient to prove Claim 2 in the context of the minimal Cisinski model structure on the topos $P(C_\kappa)/Y$. But then the class of weak equivalences is closed under finite products (see Corollaire (c) of Proposition 3.12 of THT), which implies that the functor $(-)\times_Y X$ is left Quillen, and thus implies CLaim 2.

Finally, here are some remarks about model categories in which any object is both fibrant and cofibrant. It may happend that a model category has this property and is cofibrantly generated. An easy example is the projective model category for complexes of $k$-modules in the case where $k$ is a field. A variation on the same theme is the stable model category of modules over a Frobenius ring (see Theorem 2.2.12 in Hovey’s book) (these two examples fall into the realm of abelian Frobenius categories). There are higher structures there. I guess that the example of Frobenius rings is a little more convincing, but even in the case of complexes over a field $k$, if $k$ is of positive characteristic, there is a non trivial obstruction to strictify an $E_\infty$-algebra into a genuine commutative monoid in the category of $k$-vector spaces, which is the starting point of all the story of the Steenrod algebra. These examples show that the presence of higher phenomena is not directly related to the problem of the existence of a model in which any object is cofibrant and fibrant.