Short version :

What is the motivation behind the local condition in the definition of a test category ? Why do we want the slice categories $A/a$ to be weak test ?

Long version :

I start with some background.

In Pursuing stacks, Grothendieck searches for a characterization of modelizers, that is of categories $\mathcal M$ together with a class of arrows $W$ such that the localization $W^{-1}\mathcal M$ is equivalent to the category $\mathsf{Hot}$ of homotopy types. He concentrates on categories $\mathcal M$ where the weak equivalences (the elements of $W$) are arising naturally. Among them, there are the toposes $\mathcal X$, where an arrow $\varphi \colon x \to x'$ is a weak equivalence if and only if the induced geometric morphism $\mathcal X/x \to \mathcal X/x'$ is an equivalence of Artin-Mazur$^{[1]}$. Notably, if $\mathcal A$ is a small category, the weak equivalence of the topos $\widehat{\mathcal A}$ are those maps $F \to F'$ between presheaves such that the induced morphism $$ \widehat{(\mathcal A/F)} \simeq \widehat{\mathcal A}/F \to \widehat{\mathcal A}/F' \simeq \widehat{(\mathcal A/F')} $$ is an equivalence of Artin-Mazur. It can be showed that those are precisely the maps $F \to F'$ such that the induced functor $\mathcal A/F \to \mathcal A/F'$ is in the class $\mathcal W_\infty$ of functors between small categories whose topological realizations of the nerve are weak homotopy equivalences.

For a small category $\mathcal A$, denote $i_{\mathcal A} \colon \widehat{\mathcal A} \to \mathsf{Cat}$ the functor $F \mapsto \mathcal A/F$. The weak equivalence of $\widehat{\mathcal A}$ just described are the elements of $W_{\widehat{\mathcal A}} = i_{\mathcal A}^{-1}(\mathcal W_\infty)$. Moreover $i_{\mathcal A}$ admits a right adjoint $i_{\mathcal A}^\ast$ (namely $\mathcal C \mapsto \hom_{\mathsf{Cat}}(\mathcal A/-,\mathcal C)$). Grothendieck then defines a weak test category as a small category $\mathcal A$ such that the adjunction $i_{\mathcal A} \dashv i_{\mathcal A}^\ast$ respects weak equivalences and induces an equivalence $$ {W_{\widehat{\mathcal A}}}^{-1}\widehat{\mathcal A} \simeq \mathcal {W_\infty}^{-1}\mathsf{Cat}. $$ Notably, $\widehat{\mathcal A}$ together with $W_{\widehat{\mathcal A}}$ is a modelizer. Even better, Grothendieck finds a good charaterization of weak test categories (namely, those are the categories $\mathcal A$ such that for any small category $\mathcal C$ with a final object, the functor $i_{\mathcal A}i_{\mathcal A}^\ast(\mathcal C) \to \mathcal C$ is in $\mathcal W_\infty$).

Now, in every pieces of literature about the subject that I have read so far, the author write something like : "But the notion of weak test category has the inconvenient not to be local". This is the starting point for Grothendieck to refine his notion : a test category is a weak test category $\mathcal A$ such that for every object $a$, the slice category $\mathcal A/a$ is weak test.

But why do we even want the slice categories to be test ? Weak test is already a sufficient condition on $\mathcal A$ for the topos $\widehat{\mathcal A}$ to be a modelizer : why the need to refine the notion ?

[1] I know nothing about cohomology of toposes, but I surely am ready to admit that it is natural to think of a Artin-Mazur equivalence as a weak equivalence.

  • $\begingroup$ It is probably theoretically convenient. For instance, locally cartesian closed categories are nicer than cartesian closed categories, etc. $\endgroup$ – Zhen Lin Sep 7 '14 at 22:02

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