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In Appendix C of his book in progress Spectral Algebraic Geometry, Lurie defines the unseparated derived category $\check{{\cal D}}({\cal A})$ (see Definition C.5.8.2 loc.cit) associated to a Grothendieck abelian category. The unseparated derived category $\check{{\cal D}}({\cal A})$ is a stable presentable $\infty$-category equipped with a natural t-structure which is compatible with filtered colimits and whose heart is identified with ${\cal A}$. It is related to the usual derived $\infty$-category ${\cal D}({\cal A})$ by a t-exact functor $\check{{\cal D}}({\cal A}) \to {\cal D}({\cal A})$ which exhibits ${\cal D}({\cal A})$ as the "left separation" of $\check{{\cal D}}({\cal A})$. In addition, $\check{{\cal D}}({\cal A})$ itself enjoys the following universal property (see Theorem C.5.8.8 and Corollary C.5.8.9 of loc.cit): if ${\cal C}$ is any other stable presentable $\infty$-category equipped with a $t$-structure which is compatible with filtered colimits, then restriction along the inclusion of the heart $A \to \check{{\cal D}}({\cal A})$ induces an equivalence $$ {\rm LFun^{{\rm t-ex}}}(\check{{\cal D}}({\cal A}),{\cal C}) \stackrel{\simeq}{\to} {\rm LFun^{{\rm ex}}}({\cal A},{\cal C}^{\heartsuit}) $$ where on the left hand side we have colimit preserving t-exact functors $\check{{\cal D}}({\cal A}) \to {\cal C}$ and on the right hand side we have colimit preserving exact functors ${\cal A} \to {\cal C}^{\heartsuit}$. This is indeed a very satisfying universal characterization of $\check{{\cal D}}({\cal A})$ together with its t-structure. However, for various reasons it can be useful to have a universal characterization of $\check{{\cal D}}({\cal A})$ without the t-structure. To see how this might go, note that t-exact colimit preserving functors $\check{{\cal D}}({\cal A}) \to {\cal C}$ send short exact sequences in ${\cal A}$ to cofiber sequences in ${\cal C}$ and filtered colimits in ${\cal A}$ to filtered colimits in ${\cal C}$. One may hence consider the possibility that restriction along ${\cal A} \to \check{{\cal D}}({\cal A})$ induces an equivalences between all colimit preserving functors $\check{{\cal D}}({\cal A}) \to {\cal C}$ on the one hand and and all functors ${\cal A} \to {\cal C}$ which preserve filtered colimits and send short exact sequences in ${\cal A}$ to cofiber sequences in ${\cal C}$ on the other.

Question 1: Is this true?

A positive answer to Question 1 would imply that $\check{{\cal D}}({\cal A})$ admits the following explicit description: we would be able to identify it with the $\infty$-category of presheaves of spectra ${\cal A}^{{\rm op}} \to {\rm Sp}$ which send filtered colimits in ${\cal A}$ to filtered limits and send short exact sequences in ${\cal A}$ to fiber sequences of spectra.

As a variant to Question 1, one may hope that the connective part $\check{{\cal D}}({\cal A})_{\geq 0}$ enjoys the same universal characterization when ${\cal C}$ is now replaced with a Grothendieck prestable $\infty$-category, or maybe even any presentable $\infty$-category. Such a characterization would imply that $\check{{\cal D}}({\cal A})_{\geq 0}$ can be identified with the $\infty$-category of presheaces of spaces ${\cal A}^{op} \to {\cal S}$ which send filtered colimits to cofiltered limits and short exact sequences to fiber sequences of spaces.

Question 2: Is this true?

Remarks: 1) A positive answer to Question 2 would imply a positive answer to Question 1 since $\check{\cal D}({\cal A}) \simeq {\rm Sp}(\check{\cal D}({\cal A})_{\geq 0})$ is the stabilization of $\check{\cal D}({\cal A})_{\geq 0}$.

2) If ${\cal A} = {\rm Ind}(A_0)$ with $A_0$ an abelian category with enough projectives in which every object has finite projective dimension then $\check{\cal D}({\cal A}) \simeq {\cal D}({\cal A})$ (Proposition C.5.8.12 in loc.cit) and can also be described using complexes of projective objects. In this case $\check{\cal D}({\cal A})_{\geq 0} \simeq {\cal P}_{\Sigma}((A_0)_{{\rm proj}})$ is the $\infty$-category obtained from $(A_0)_{{\rm proj}}$ by freely adding sifted colimits. In particular, $\check{\cal D}({\cal A})_{\geq 0}$ admits a universal characterization as a presentable $\infty$-category and $\check{\cal D}({\cal A}) \simeq {\rm Sp}(\check{\cal D}({\cal A})_{\geq 0})$ admits a universal characterizations as a stable presentable $\infty$-category (without the t-structure). However, even in this case, I don't know how to deduce from this particular universal characterization the other universal characterization I'm interested in (the missing part is to show that a coproduct preserving functors $(A_0)_{{\rm proj}} \to {\cal C}$ extends in an essentially unique way to a functor ${\cal A}_0 \to {\cal C}$ which sends short exact sequence to cofiber sequences).

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Yes, both of these statements are true (I thought they were in the book, but I can't seem to find them now).

Here is a proof sketch. Let's start with the case described in 2). Let $\mathcal{C}$ be any presentable $\infty$-category. As noted in the question, what you need to identify are functors $F: \mathcal{A}_0 \rightarrow \mathcal{C}$ that preserve finite coproducts and carry short exact sequences to cofiber sequences to functors $F_0: \mathcal{A}^{proj}_0 \rightarrow \mathcal{C}$ that preserve finite coproducts. There's an obvious restriction functor in one direction, and there's a functor of left Kan extension in the other (the universal property of $\mathcal{D}( \mathcal{A} )_{\geq 0}$ guarantees that this left Kan extension yields a functor that behaves well on exact sequences). The thing you need to check is that any $F: \mathcal{A}_0 \rightarrow \mathcal{C}$ as above agrees with the left Kan extension of its restriction to $\mathcal{A}_0^{proj}$; let's denote that functor by $F'$, so there's a natural transformation $F' \rightarrow F$ which is an equivalence on projective objects. Since both $F$ and $F'$ carry short exact sequences to cofiber sequences, the collection of objects $X$ for which $F'(X) \rightarrow F(X)$ is an equivalence is closed under taking cokernels of injective maps, and therefore (by induction) contains all objects of finite projective dimension, which is all objects of $\mathcal{A}_0$ by assumption.

To handle the general case, we can use the fact that every Grothendieck abelian category can be obtained as a quotient $\mathcal{A} / \mathcal{B}$, where $\mathcal{A}$ is as above and $\mathcal{B}$ is some localizing subcategory. Let $\mathcal{D}_0$ be the smallest localizing subcategory of $\check{\mathcal{D}}( \mathcal{A})_{\geq 0}$ which contains $\mathcal{B}$, so that we can identify $\check{\mathcal{D}}( \mathcal{A} / \mathcal{B} )_{\geq 0}$ with the quotient $\check{\mathcal{D}}( \mathcal{A} )_{\geq 0} / \mathcal{D}_0$. The first part of the proof shows that the following data are equivalent:

(1) Functors $F: \mathcal{A} \rightarrow \mathcal{C}$ which preserve finite coproducts, carry short exact sequences to cofiber sequences, and preserve filtered colimits.

(2) Functors $F^{+}: \check{\mathcal{D}}(\mathcal{A})_{\geq 0} \rightarrow \mathcal{C}$ which preserve small colimits.

We wish to show that if $F$ and $F^+$ correspond under this equivalence, then the following conditions are equivalent:

(a) For every map $u: X \rightarrow Y$ in the abelian category $\mathcal{A}$, if the kernel and cokernel of $u$ belong to $\mathcal{B}$, then $F(u)$ is an equivalence (so that $F$ factors through $\mathcal{A} / \mathcal{B}$).

(b) For every map $u: X \rightarrow Y$ in $\check{\mathcal{D}}(\mathcal{A})_{\geq 0}$, if the cofiber belongs to $\mathcal{D}_0$, then $F^{+}(u)$ is an equivalence in $\mathcal{C}$ (so that $F^{+}$ factors through $\check{\mathcal{D}}( \mathcal{A} )_{\geq 0} / \mathcal{D}_0$).

The implication $(b) \Rightarrow (a)$ is immediate. To proceed in the reverse direction, let $u: X \rightarrow Y$ be as in $(b)$ and choose an object $Y'$ in $\mathcal{A}$ and a map $Y' \rightarrow Y$ which is surjective on $\pi_0$, and set $X' = Y' \times_{Y} X$. Then $u$ is a pushout of the projection map $u': X' \rightarrow Y'$, so it suffices to show that $F^{+}(u')$ is an equivalence. The map $u'$ factors as a composition $X' \xrightarrow{w} Y'_0 \xrightarrow{v} Y'$ where $v$ is a monomorphism in $\mathcal{A}$ and $w$ is surjective on $\pi_0$. Using the assumption that the cofiber of $u$ belongs to $\mathcal{D}_0$, it is not hard to check that $Y' / Y'_0 \in \mathcal{B}$ and that the fiber of $w$ belongs to $\mathcal{D_0}$. Then $F^{+}(v)$ is an equivalence by virtue of $(a)$, and $w$ is a pushout of the map $fib(w) \rightarrow 0$. We are therefore reduced to proving that the functor $F^{+}$ annihilates every object of $\mathcal{D}_0$, which is not hard to check (using the fact that it annihilates every object of $\mathcal{B}$).

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  • $\begingroup$ This is very cool. Can we derive a universal property for $\mathcal{D}(\mathcal{A})$ out of this? Said differently, can we describe which functors $\mathcal{A}\to \mathcal{C}$ correspond to functors from $\check{\mathcal{D}}(\mathcal{A})$ sending acyclic complexes to 0? $\endgroup$ – Denis Nardin Oct 8 '17 at 8:45
  • $\begingroup$ @Denis: I don't see how say anything about that (except in the case where $\mathcal{C}$ is stable with a nice t-structure, and the functors are required to be right t-exact). $\endgroup$ – Jacob Lurie Oct 8 '17 at 14:55
  • $\begingroup$ Thanks! This is indeed cool. I suppose the same thing happens with $\infty$-topoi, no? i.e., if ${\cal X}$ is a topos, or a 1-topos, then I imagine that the $\infty$-topos ${\cal X}_{\infty}$ "generated" from ${\cal X}$ has a universal property both as an $\infty$-topos and as a presentable $\infty$-category: colimit preserving functors from ${\cal X}_{\infty}$ to a given presentable $\infty$-category ${\cal C}$ are the same (?) as functors ${\cal X} \to {\cal C}$ which preserve filtered colimits and send hypercoverings to geometric realizations. $\endgroup$ – Yonatan Harpaz Oct 8 '17 at 18:40
  • $\begingroup$ @Yonatan I would say that the $\infty$-topos analogue of the unseparated derived category is the construction which carries a $1$-topos $\mathcal{X}$ to the $\infty$-category of space-valued sheaves on $\mathcal{X}$, where you equip $\mathcal{X}$ with the canonical Grothendieck topology. So the universal property is: coproducts go to coproducts, and Cech nerves of effective epimorphisms go to colimit diagrams. (If you ask this for general hypercoverings, you get the $\infty$-topos of hypercomplete sheaves. This is more like the analogue of the usual derived category.) $\endgroup$ – Jacob Lurie Oct 9 '17 at 0:30
  • $\begingroup$ Right, I see what you mean about hypercoverings. But what about filtered colimits? Shouldn't we take only sheaves which send filtered colimits to limits (or is this built into the definition of the canonical Grothendieck topology on ${\cal X}$)? $\endgroup$ – Yonatan Harpaz Oct 9 '17 at 11:15

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