The finite stable homotopy type of some topoi (Q580488)

Let \(\epsilon\) be a Grothendieck topos and F a spectrum, in the sense of stable homotopy. The author [Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437-552 (1985; Zbl 0596.14012)] defined the sheaf hypercohomology spectrum of \(\epsilon\), \({\mathbb{H}}(\epsilon;F)\). In this paper he constructs a spectrum, \(h\epsilon\), such that there is a natural homotopy equivalence, \((*)\quad Map(h\epsilon,F)\simeq {\mathbb{H}}(\epsilon;F)\) under suitable cohomological conditions on \(\epsilon\). When (*) holds part of its importance lies in the fact that \(h\epsilon\wedge F\) is a spectrum which represents ``the homology of \(\epsilon\) with coefficients in F''. A well-behaved homology functor on a general Grothendieck topos is unknown.