Exact homotopy squares and derivators (Q2910255)

A square of small categories is a diagramNEWLINENEWLINENEWLINE\[NEWLINE \begin{pmatrix} A' \rightarrow [d]_{u'} \rightarrow [r]^{v} & A \rightarrow [d]^u \\ B' \rightarrow [r]_w & B \end{pmatrix} NEWLINE\]NEWLINE which commutes up to a natural transformation \(uv \Rightarrow w u'\). Such a square is called exact in the sense of Guitart if the corresponding ``base change'' morphisms are isomorphisms. It turns out that Guitart exact squares can be seen as exact squares associated to a fundamental localizer. For example, any model category \(\mathcal C\) determines a fundamental localizer \(\mathcal W_{\mathcal C}\). It consists in all functors \(A \rightarrow B\) inducing an isomorphism \(\text{hocolim}_A X \rightarrow \text{hocolim}_B X\) in the homotopy category for all constant diagrams~\(X\). Next one associates to each fundamental localizer \(\mathcal W\) a class of \(\mathcal W\)-exact squares. Using the same notation as above one requires \(v\) to induce a \(\mathcal W\)-equivalence, i.e., a functor in \(\mathcal W\), on comma categories \(A' / b' \rightarrow A / w(b')\). When \(\mathcal W\) is the fundamental localizer of functors \(u\) inducing an isomorphism \(\pi_0 u\) on connected components, one gets back the Guitart exact squares.NEWLINENEWLINEThe first and second sections of this clear and carefully written article deals with these constructions. The author shows in particular that comma squares and Beck-Chevalley squares are \(\mathcal W\)-exact for any fundamental localizer \(\mathcal W\). Nice closure and initialization properties are also shown to hold. This is the subject of the third section and the last one is devoted to derivators.NEWLINENEWLINEJust like a model category determines a fundamental localizer, so does any derivator \(\mathbb D\). Let us denote by \(p: A \rightarrow \ast\) and \(q: B \rightarrow \ast\) the functors to the trivial category (with one object). A functor \(u: A \rightarrow B\) is a \(\mathcal W_{\mathbb D}\)-equivalence if \(q_\ast q^\ast \rightarrow p_\ast p^\ast\) is an isomorphism in \(\mathbb D(\ast)\); this is the translation of the model theoretical analogue for derivators. The aim of the article is to deduce most basic properties of derivators from formal properties of exact squares. For example, \({\mathbb D}\)-exact squares satisfy cohomological base change, in other words \(\mathbb D\) sends any \(\mathbb D\)-exact square to a Beck-Chevalley square. This happens in fact if and only if the square is \textit{weakly} \(\mathbb D\)-exact, a slightly weaker notion than \(\mathbb D\)-exactness which appears naturally in the context of derivators.