Classically, the (non-local-coefficients) homology Whitehead theorem says that if $X \xrightarrow f Y$ is a map of simple spaces, and if the induced map $H_\ast(X;\mathbb Z) \to H_\ast(Y;\mathbb Z)$ is an isomorphism, then $f$ is a weak homotopy equivalence.

Conceptually for me, the essence of this theorem is that we have a *stable* invariant ($H_\ast(-;\mathbb Z)$), and we identify a (reasonably large) class of spaces (the simple spaces) such that our invariant detects equivalences when restricted to this class. I'm wondering how generally a statement of this form holds in a fairly general $\infty$-category $\mathcal C$ in place of $Spaces$.

For my purposes, I'm not particularly concerned with *which* stable invariant we use, so we might as well restrict attention to the universal case. Moreover, there are two general forms of "stabilization" which come to mind -- the category of spectrum objects $Sp(\mathcal C) = \varprojlim (\cdots \xrightarrow \Omega \mathcal C_\ast \xrightarrow \Omega \mathcal C_\ast)$, and the Spanier-Whitehead category $SW(\mathcal C) = \varinjlim (\mathcal C_\ast \xrightarrow \Sigma \mathcal C_\ast \xrightarrow \Sigma \cdots)$ (I use $\mathcal C_\ast$ to denote the $\infty$-category of pointed objects in $\mathcal C$). But we're eventually passing to some subcategory anyway, so we can reduce the $SW$ notion to the $Sp$ notion if we start out by replacing $\mathcal C$ with $Ind(\mathcal C)$ via the equation $Ind(SW(\mathcal C)) = Sp(Ind(\mathcal C))$.

Thus we are led to the following formulation:

Question:Let $\mathcal C$ be a presentable $\infty$-category. Can we identify a (reasonably large) full subcategory $\mathcal D \subseteq \mathcal C$ such that the composite functor $\mathcal D \to \mathcal C \xrightarrow {\Sigma^\infty_+} Sp(\mathcal C)$ is conservative? In particular, is this the case for $\mathcal D$ being one of the following?

The 1-fold suspension objects?

The 1-fold loop objects?

The 1-connected objects?

Here, a *1-fold suspension object* is simply an object of the form $X = \Sigma Y$ for some $Y \in \mathcal C$; a *1-fold loop object* is an object of the form $X = \Omega Y$ where $Y \in \mathcal C_\ast$ is a pointed object of $\mathcal C$. A *1-truncated* morphism $W \to Z$ is a morphism such that for every $C \in \mathcal C$, the map $\mathcal C(C,W) \to \mathcal C(C,Z)$ has 1-truncated fibers, a morphism is *1-connected* if it is left orthgonal to the 1-truncated morphisms, and an object $X$ is *1-connected* if the map $X \to 1$ is 1-connected, where $1$ is the terminal object.

As a sanity check, I think each of my candidates for $\mathcal D$ are trivial when $\mathcal C$ has discrete hom-spaces, which is a good thing because in this case $Sp(\mathcal C)$ is also trivial.

hypercomplete1-connected objects. But I don't know how to do better right now. $\endgroup$ – Mike Shulman Feb 27 at 0:49