The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of *Higher topos theory*. Can it ever happen that this subcategory is also *coreflective*?



Can hypercomplete objects be coreflective?
Let $\mathcal{X}$ be the $\infty$-category of $1$-excisive functors from (pointed) spaces to (unpointed) spaces: equivalently, the $\infty$-category of pairs $(X, E)$ where $X$ is a space and $E$ is a local system of spectra on $X$. Then $\mathcal{X}$ is an $\infty$-topos (for example, it's a left exact localization of the $\infty$-category of functors from finite pointed spaces to spaces, via the Goodwillie calculus).


The homotopy groups of an object $(X,E) \in \mathcal{X}$ are just the homotopy groups of $X$, and the pair $(X,E)$ is hypercomplete if and only if $E = 0$. The construction $(X,E) \mapsto (X,0)$ is both left and right adjoint to the inclusion of hypercomplete objects into all of $\mathcal{X}$.


