The six operations in topology

Abstract

In this paper we show that the six functor formalism for sheaves on locally compact Hausdorff topological spaces, as developed for example in Kashiwara and Schapira's book Sheaves on Manifolds, can be extended to sheaves with values in any closed symmetric monoidal $\infty$-category which is stable and bicomplete. Notice that, since we do not assume that our coefficients are presentable or restrict to hypercomplete sheaves, our arguments are not obvious and are substantially different from the ones explained by Kashiwara and Schapira. Along the way we also study locally contractible geometric morphisms and prove that, if $f:X\rightarrow Y$ is a continuous map which induces a locally contractible geometric morphism, then the exceptional pullback functor $f^!$ preserves colimits and can be related to the pullback $f^*$. At the end of our paper we also show how one can express Atiyah duality by means of the six functor formalism.