Six functor formalism for sheaves with non-presentable coefficients

In this paper we show that the six functor formalism for sheaves on locally compact Hausdorff topological spaces, as developed for example in [KS90], 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 our coefficients to be presentable or restrict to hypercomplete sheaves, our arguments will be not obvious and substantially different from the ones contained in [KS90]. Along the way we also study locally contractible geometric morphisms and prove that, if $f:X\rightarrow Y$ is a continous map which induces a locally contractible geometric morphism, then the exceptional pullback functor $f^!$ satisfies a formula generalizing [KS90, Proposition 3.3.2 (ii)], where it is proven only for topological submersions. At the end of our paper we also show how one can express Atiyah duality by means of the six functor formalism.