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Given a pair of test categories $C_1$ and $C_2$ (in the sense of Grothendieck - weak or strict or otherwise), has anyone defined an interesting notion of morphism between them? Or are ordinary functors enough? Given that, has anyone studied the resulting category? In particular, what effect do properties of the functors have on the resulting presheaf categories, $Set^{C_1^{op}}$ and $Set^{C_2^{op}}$? Is there a characterisation of functor between test categories which induces a Quillen equivalence of presheaf categories, using the model structure as in Cisinski's Asterisque volume?

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I guess that it might be slightly more interesting to look for a notion of morphism of local test categories. A first natural candidate is given by the notion of locally constant functor: a functor $u:A\to B$ is locally constant is it satisfies the assumpions of Quillen's theorem B, namely that, for any map $b\to b'$ in $B$, the induced functor on the comma categories $A/b\to A/b'$ is a weak equivalence. If $A$ and $B$ are local test categories, then a functor $u:A\to B$ is locally constant if and only if the inverse image functor $u^\star : Set^{B^{op}}\to Set^{A^{op}}$ is a left Quillen functor (and such a $u^\star$ is moreover a Quillen equivalence if and only if $u$ is a weak equivalence); see prop 6.4.29 in Astérisque 308.

If we want to look at something which looks like some theory of bimodules, in order to produce a bicategory of local test categories, we might start with spans of shape

$$A\overset{w}{\leftarrow} C \overset{u}{\to} B$$

where $w$ is aspherical (i.e. satisfies the assumptions of Quillen's theorem A), while $u$ is locally constant (with $C$ a local test category as well). From such data, we obtain a left Quillen functor

$$Set^{B^{op}}\overset{u^\star}{\longrightarrow} Set^{C^{op}}$$

as well as a left Quillen equivalence

$$Set^{C^{op}}\overset{w^\star}{\longleftarrow} Set^{A^{op}}.$$

The trouble is that such spans may not compose (because neither aspherical functors nor locally constant functors are stable under pullbacks in general). We may correct this defect by asking furthermore that $w$ is smooth (e.g. that $C$ is fibred over $A$): a reformulation of Quillen's theorem B is that the pullback of a locally constant functor by a smooth map is always homotopy cartesian (see prop 6.3.39 and thm 6.4.15 in Astérisque 308); furthermore, for any smooth functor $p:X\to Y$, if $Y$ is a local test category, so is $X$ (in fact, for such a property, we need only $p$ to be locally aspherical); see thm 7.2.1 in loc. cit. In other words, we obtain a bicategory of local test categories, for which the $1$-cells are the couples $(w,u)$, with $w$ an aspherical smooth functor, while $u$ is locally constant (e.g. any smooth and proper functor is locally constant). This bicategory is in fact another model for homotopy types: if we denote by $S(A,B)$ the category of spans $(w,u)$ from $A$ to $B$ as above, then the classifying space of $S(A,B)$ has the homotopy type of the mapping space from the classifying space of $A$ to the classifying space of $B$ (and any homotopy type is the classifying space of some local test category).

Here is some explanation of the meaning of this bicategory: this has to do with the description of homotopy types as $\infty$-groupoids. Let us start with the toposic description of $1$-groupoids. There is a $2$-functor from the $2$-category of $1$-groupoids to the $2$-category of (Grothendieck) topoi which associates to a $1$-groupoid $G$ the category of presheaves of sets over $G$. This functor is fully faithful (i.e. induces equivalences of categories at the level of Hom's). Its essential image may be characterized as the $2$-category of topoi which are generated (as cocomplete categories) by their locally constant objects. This embedding of $1$-groupoids into topoi is the starting point of the Grothendieck version of Galois theory (unifying classical Galois theory of fields and topological Galois theory). If we consider homotopy types as $\infty$-groupoids (in the spirit of Toën, Lurie, and others), then one can consider the classifying space of a small category $A$ as the $\infty$-groupoid obtained from $A$ by inverting all the arrows in $A$. The $\infty$-topos associated to this $\infty$-groupoid may be described as the one associated to the model category obtained from the (injective) model category of simplicial presheaves over $A$, by considering the left Bousfield localization by the arrows between representable presheaves. The latter model category is precisely the model structure corresponding to the local test category $A\times \Delta$ (recall that the product of any small category with a (local) test category is a local test category). There is a higher version of higher Galois theory whose first fundamental statement is that the $(\infty,2)$-category of $\infty$-groupoids is embedded fully faithfully in the $(\infty,2)$-category of $\infty$-topoi; see Toën's Vers une interprétation Galoisienne de la théorie de l'homotopie Cahiers de Top. et de Geom. Diff. Cat. 43, No. 4 (2002), 257-312". The bicategory of local test categories as described above is a convenient presentation to see this canonical embedding of homotopy types into $\infty$-topoi explicitely: one associates to a span $(w,u)$ as above the map induced by the left Quillen functors ${w^{\star}}$ and ${u^{\star}}$ (inverting ${w^{\star}}$); see the last paragraph of my paper Locally constant functors, Math. Proc. Camb. Phil. Soc. 147 (2009), 593-614, for a similar description of this embedding. In other words, given a local test category $A$, the corresponding model category on $Set^{A^{op}}$ defines an $\infty$-topos, which is Galois (in the sense that it is generated, as a cocomplete $(\infty,1)$-category, by its locally constant objects), and the corresponding $\infty$-groupoid is simply the one associated to $A$ by inverting all the arrows in $A$. This is how I understand the meaning of these model category structures associated to local test categories.

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Since small categories are models of homotopy types (from Thomason's model structure on Cat), what more does one get by restricting to local test categories? – David Roberts May 22 '10 at 9:02
On a different note, can this bicategory of local test categories be obtained as a bicategorical localisation (a la Pronk, or any other way, for that matter) of any 2-category of test categories? – David Roberts May 22 '10 at 9:06
I added some comments on the meaning of this bicategory of local test categories which (hopefully) answers to your first question. I don't really understand the second question. However, it is clear that the localization à la Pronk of this bicategory of local test categories is biequivalent to the $2$-category whose objects are spaces, and whose categories of morphisms are the groupoids $\pi_1(Map(X,Y))$. – Denis-Charles Cisinski May 22 '10 at 22:06
My second question was more along the lines of: is this bicategory $S$ with spans $(w,u)$ as 1-arrows the localisation of the full sub-2-category $loc.test.Cat \subset Cat$ at the aspherical (resp. aspherical smooth) functors? What if we don't necessarily take the full sub-2-category, say by restricting attention to smooth functors? (for example) – David Roberts May 23 '10 at 12:34

I'm not sure what “interesting” means in this context. It's probably too much to demand that morphisms between $\widehat{C_1} = \mathbf{Set}^{{C_1}^{\mathrm{Op}}}$ and $\widehat{C_2} = \mathbf{Set}^{{C_2}^{\mathrm{Op}}}$ arise only from functors $C_2 \to C_1$. This would eliminate functors such as the simplicial realization of a cubical set. Better, we should take the morphisms among left adjoints $\widehat{C_1} \to \widehat{C_2}$, i.e., diagrams $C_1 \times {C_2}^\mathrm{Op} \to \mathbf{Set}$. Not all of these give Quillen adjunctions, but those corresponding to suitably cofibrant resolutions of the terminal object in ${\widehat{C_2}}^{C_1}$ probably do.

In the special case that a left adjoint $\widehat{C_1}\to\widehat{C_2}$ is given by restriction $f^\ast$ along an aspherical functor $f:C_2 \to C_1$, the corresponding functor $C_2 \downarrow f^\ast X \to C_1 \downarrow X$ is a weak equivalence for all $X\in\widehat{C_1}$. This is one implication of Proposition 1.2.9 in Maltsiniotis' Asterisque volume. Now, if $f^\ast$ is to be a left Quillen equivalence, you need it to preserve all weak equivalences (since everything in $\widehat{C_1}$ is cofibrant). This forces your hand: since the representables in $\widehat{C_1}$ are weakly contractible you need $f^\ast C_1({-},x)$ to be weakly contractible, i.e., the nerve of $f\downarrow x$ should be weakly equivalent to a point.

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