In addition to the space \({\mathbb{R}}\) we consider the reals augmented with a coarser topology.

We can pushforward cosheaves on \({\mathbb{R}}\) to cosheaves on \(\bar{{\mathbb{R}}}\) via \({\operatorname{id}}\).

Similar to defining the pullback for sheaves, we take two steps to define the pullback of a cosheaf via \({\operatorname{id}}\).

With this definition \({\operatorname{id}}^+ F\) is merely a precosheaf for all we know. Yet we have the following.

By the previous lemma \({\operatorname{id}}^{-1} F\) as defined above is a cosheaf.

  1. Proposition. \({\operatorname{id}}_*\) and \({\operatorname{id}}^{-1}\) form a pair of adjoint functors \({\operatorname{id}}_* \dashv {\operatorname{id}}^{-1}\) with unit \(\eta'\) and whose counit \(\varepsilon'\) is an isomorphism.