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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?

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    $\begingroup$ I'm confuse by your terminology: the category is left exact reflexive mean that the inclusion functor $i$ from Hypercomplete object to arbitrary object has a left exact left adjoint $H$ and it makes the couple $(H,i)$ into a geometric morphism from the topos of Hypercomplete object to the topos we started from. Now "essential" would mean that $H$ also has a left adjoint, while "coreflective" would mean that $i$ also has a right adjoint. Or am I missing something ? $\endgroup$ – Simon Henry Aug 5 '15 at 8:00
  • $\begingroup$ @SimonHenry You're right, I misused the terminology. Fixed. $\endgroup$ – Mike Shulman Aug 6 '15 at 12:28
  • $\begingroup$ I'm glad to have had this question answered, but the question I meant to ask is mathoverflow.net/questions/213168/…. $\endgroup$ – Mike Shulman Aug 6 '15 at 12:34
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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 $\mathcal{X}$.

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  • $\begingroup$ Thanks! I've heard this fact about excisive functors stated several times before; but what is the best reference for it? Has it appeared in print? $\endgroup$ – Mike Shulman Aug 8 '15 at 3:10
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    $\begingroup$ Tom Goodwillie, "Calculus III: Taylor Series" $\endgroup$ – Jacob Lurie Aug 10 '15 at 6:27
  • $\begingroup$ I meant the fact that it is equivalent to the category of local systems of spectra, not the fact that it is a left exact localization of a functor category. Or is that also in there? $\endgroup$ – Mike Shulman Aug 11 '15 at 4:47
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    $\begingroup$ Not sure if it is explicitly stated there, but it is at least easily derived from what's there. First, reduced 1-excisive functors are spectra (Calculus III). If F is an arbitrary excisive functor then Y=F(*) is a space and $(y \in Y) \mapsto \lambda X. F(X) \times_{F(*)} \{y\}$ is a local system of reduced excisive functors on $Y$. Conversely if $\{ F_y \}_{y \in Y}$ is a local system of reduced excisive functors on a space Y, then $F(X)=\varinjlim_{y \in Y} F_y(X)$ is an excisive functor (not necessarily reduced). These constructions are homotopy inverse to one another. $\endgroup$ – Jacob Lurie Aug 11 '15 at 17:04

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