Stable infinite category counterpart of pathological behaviours around the AB3,AB4 and AB5 axioms of abelian categories

Question

In his 2002 paper "A counterexample to a 1961 theorem in homological algebra" (Invent. Math.) Amnon Neeman exhibited the infamous and scary example of a cocomplete abelian AB4 (colimits are exact) category where there is a string of morphisms

$X_0 \hookrightarrow X_1 \hookrightarrow X_2 \hookrightarrow \ldots \ \operatorname{colim}_{i} X_i$,

where the objects are non-zero, all the arrows are monomorphisms, and yet the colimit is zero. It is an obscenely aggressive reminder that the usual advice given to undergrads to visualize abelian categories as some form of $R$-module category can be misleading, to say the least.

I suppose when considering the (bounded or unbounded, as you like) derived stable $\infty$-category of such a weird abelian category, it should similarly exhibit some highly counter-intuitive behaviour. Or is this in some sense just a weirdness of the heart? After all, the behaviour only feels counter-intuitive because the arrows are monomorphisms and that's not a concept that has a 'meaning' for stable $\infty$-categories.

So, how should one think about this:

• it's a phenomenon specific to the setting of abelian categories...

OR

• there is an $\infty$-categorical sibling of these strange abelian categories....

Do you know of any formulations of similar pathological behaviour on the stable $\infty$-categorical level which do not just merely proceed by returning to an underived viewpoint?

Answer 1

Let $\mathcal{C}$ be an AB4 abelian category. Then the AB5 condition on $\mathcal{C}$ is equivalent to sequential homotopy colimits in $D(\mathcal{C})$ commuting with homology. Here $D(\mathcal{C})$ is the stable $\infty$-category of chain complexes in $\mathcal{C}$. Maybe this is the kind of "stable $\infty$-category counterpart" you asked for.

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Here's a bit more explanation. Suppose you try to calculate the homotopy colimit in $D(\mathcal{C})$ of some sequence $$X_0 \rightarrow X_1 \rightarrow X_2 \rightarrow\dots.$$ A nice general way to approach this is via the Bousfield-Kan spectral sequence $$E^2_{s,t} \cong L_s colim_n(H_t(X_n)) \Rightarrow H_{s+t}(hocolim_n X_n).$$

This BKSS collapses immediately at the $E^2$-page, because there's no room for differentials for silly degree reasons, since the $E^2$-page is concentrated on the $s=0$ and $s=1$-lines; this is a general feature of sequential colimits in AB4 abelian categories. So whether homology commutes with sequential homotopy colimits in $D(\mathcal{C})$, this reduces to the question of whether anything nonzero ever appears above the $s=0$-line of the $E^2$-page of this BKSS.

The AB5 axiom for $\mathcal{C}$ is the exactness of sequential colimits in $\mathcal{C}$. Hence the abelian category $\mathcal{C}$ satisfies AB5 if and only if the Bousfield-Kan SS collapses to the $s=0$-line at the $E^2$-page for every sequence $X_0 \rightarrow X_1 \rightarrow \dots$ in $D(\mathcal{C})$. So the AB5 condition on $\mathcal{C}$ is equivalent to having the isomorphism $$H_*(hocolim_n X_n) \cong colim_n H_*(X_n)$$ for every sequence $X_0 \rightarrow X_1\rightarrow \dots$ in $D(\mathcal{C})$.

You can do the whole argument with the Milnor sequence instead of the BKSS, if you don't like spectral sequences. The spectral sequence argument is nice because it generalizes in all kinds of ways, though.