Hypercovers and simplicial presheaves.

*(English)*Zbl 1045.55007Motivated by examples arising in algebraic \(K\)-theory, many authors have studied the homotopy theory of simplicial presheaves on a Groethendieck site \({\mathcal C}\). The basic paper remains [J. F. Jardine, J. Pure Appl. Algebra 47, No. 1, 35–87 (1987; Zbl 0624.18007)]. In that work, Jardine imposed a model category structure on simplicial presheaves where a morphism \(X \to Y\) is a cofibration if it induces a cofibration of simplicial sets \(X(U) \to Y(U)\) for all objects \(U\) of \({\mathcal C}\) and a weak equivalence if the map of associated homotopy sheaves \(\pi_\ast X \to \pi_\ast Y\) is an isomorphism. It is in the latter condition that the topology of the site appears; for example, a presheaf is always weakly equivalent to its associated sheaf and, if the site has enough points, then \(X \to Y\) is a weak equivalence if and only if it induces a weak equivalence of stalks \(X_a \to Y_a\) for all points \(a\) of \({\mathcal C}\).

It is more difficult to describe what it means to be fibrant in this model category, although the situation is not dire. It is fairly straightforward to describe what it would mean for an Eilenberg-MacLane object to be fibrant, using the usual notions of injective sheaves. From this one can come to terms with a general simplicial presheaf using a Postnikov tower. This is discussed in Jardine’s original paper.

In the paper under review, the authors offer another characterization of what it means to be fibrant. There are two requirements. First, an object \(X\) must be fibrant in the model category structure obtained by imposing the discrete topology on \({\mathcal C}\). This is a standard model category structure on diagrams. Second, and more interesting, the presheaf \(X\) must satisfy a “homotopy Čech descent” condition for all hypercovers in \({\mathcal C}\). Specifically, \(A\) is an object of \({\mathcal C}\) and \({\mathcal U} = \{U_i\}\) is a hypercover, with Čech nerve \(\text{Č} {\mathcal U}\), then natural map \[ X(A) \to \text{holim} X(\text{Č} {\mathcal U}) \] must be a weak equivalence. The authors then go on to discuss Quillen equivalent model categories adapted to exploring this condition.

It is more difficult to describe what it means to be fibrant in this model category, although the situation is not dire. It is fairly straightforward to describe what it would mean for an Eilenberg-MacLane object to be fibrant, using the usual notions of injective sheaves. From this one can come to terms with a general simplicial presheaf using a Postnikov tower. This is discussed in Jardine’s original paper.

In the paper under review, the authors offer another characterization of what it means to be fibrant. There are two requirements. First, an object \(X\) must be fibrant in the model category structure obtained by imposing the discrete topology on \({\mathcal C}\). This is a standard model category structure on diagrams. Second, and more interesting, the presheaf \(X\) must satisfy a “homotopy Čech descent” condition for all hypercovers in \({\mathcal C}\). Specifically, \(A\) is an object of \({\mathcal C}\) and \({\mathcal U} = \{U_i\}\) is a hypercover, with Čech nerve \(\text{Č} {\mathcal U}\), then natural map \[ X(A) \to \text{holim} X(\text{Č} {\mathcal U}) \] must be a weak equivalence. The authors then go on to discuss Quillen equivalent model categories adapted to exploring this condition.

Reviewer: Paul Goerss (Evanston)

##### MSC:

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |

18G40 | Spectral sequences, hypercohomology |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |