Complicial simple-minded collections

Abstract: We consider the problem of characterizing derived endomorphism algebras of simple objects in length categories up to quasi-isomorphism. We give such a characterization for module categories, abelian categories, exact categories, as well as, for certain differential graded analogues of them. It turns out that the property of being $d$-complicial ($d\geq 1$), in the sense of Lurie, of the involved simple-minded collections plays a central role. We also explain how this characterization can be interpreted as a coherent generation property for any minimal $A_{\infty}$-model of the derived endomorphism algebra. Along the way, we propose a notion of length exact differential graded categories and explain how they relate to length abelian $d$-truncated differential graded categories, generalizing results of Enomoto.

• The paper characterizes derived endomorphism dg algebras of simples as Koszul duals whose extended hearts are 1-generated, equivalently whose minimal A∞ models are coherently generated.
• It proves that a realization functor reconstructs a pretriangulated dg category from its d-extended heart exactly when the associated t-structure is (d−1)-complicial, linking silting and simple-minded collections.
• It establishes correspondences among length exact d-categories, abelian d-truncated dg categories, proper connective dg algebras, and smooth coconnective dg algebras, while identifying perfect-field and admissible-Schur limitations.

Motivation and main problem

The paper studies the following reconstruction question: given a length category $\mathcal{A}$ (a module category, or more generally an abelian or exact length category), which dg algebras arise, up to quasi-isomorphism, as the derived endomorphism algebra $\mathrm{RHom}_{\mathcal{A}}(S)$ of the direct sum $S$ of the simple objects? The classical Koszul duality picture identifies $D^b(\mathcal{A})$ with $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$, so a characterization of such endomorphism algebras amounts to an intrinsic characterization of one side of a Koszul duality correspondence. The author works over a field $k$, considers proper connective dg algebras $A$ (finite-dimensional total cohomology concentrated in non-positive degrees), and asks when $H^*(A)$ is concentrated in degrees $(-d,0]$ for some $d \geq 1$.

The central technical notion is that of a $\mathrm{RHom}_{\mathcal{A}}(S)$0-extended heart. For a pretriangulated dg category $\mathrm{RHom}_{\mathcal{A}}(S)$1 with a $\mathrm{RHom}_{\mathcal{A}}(S)$2-structure, this is

$\mathrm{RHom}_{\mathcal{A}}(S)$3

an extension-closed subcategory and hence an exact dg category in the sense of Chen. The key observation is that $\mathrm{RHom}_{\mathcal{A}}(S)$4 is always 1-generated inside $\mathrm{RHom}_{\mathcal{A}}(S)$5: every morphism $\mathrm{RHom}_{\mathcal{A}}(S)$6 between objects of the extended heart factors as a composition of degree-one morphisms between objects of the heart. This generation property turns out to be exactly what characterizes the Koszul duals of connective dg algebras with cohomology bounded by $\mathrm{RHom}_{\mathcal{A}}(S)$7.

Dg realization functors

The main tool is a generalization to exact dg categories of the classical criterion (due to Psaroudakis–Vitoria) for realization functors to be fully faithful. Given a pretriangulated dg category $\mathrm{RHom}_{\mathcal{A}}(S)$8 and a full extension-closed dg subcategory $\mathrm{RHom}_{\mathcal{A}}(S)$9, setting $S$0 yields a realization functor $S$1. The paper proves that $S$2 is quasi fully-faithful if and only if any of several equivalent conditions holds: it induces isomorphisms (or merely epimorphisms) on Hom-spaces of degree $S$3; the functors $S$4 are weakly effaceable for $S$5 (or dually); or $S$6 is 1-generated in $S$7. The proof uses Chen's identification of higher extension groups $S$8 with morphisms in $S$9 via $D^b(\mathcal{A})$0-functors, together with weak effaceability arguments in the spirit of Grothendieck–Neeman–Popescu. When these conditions hold, the essential image of $D^b(\mathcal{A})$1 is precisely $D^b(\mathcal{A})$2.

Combining this with a density argument adapted from Chen–Han–Zhang–Zhou, the paper proves that for a pretriangulated dg category with a bounded $D^b(\mathcal{A})$3-structure, the realization functor $D^b(\mathcal{A})$4 is a quasi-equivalence if and only if the $D^b(\mathcal{A})$5-structure is $D^b(\mathcal{A})$6-complicial in Lurie's sense: every object of $D^b(\mathcal{A})$7 admits an epimorphic cover from an object of $D^b(\mathcal{A})$8. Strict compliciality corresponds to the realization functor being a quasi-equivalence at level $D^b(\mathcal{A})$9 but not at level $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$0. Consequently, a $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$1-complicial $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$2-structure allows one to reconstruct the ambient pretriangulated dg category up to quasi-equivalence from its $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$3-extended heart alone. For silting $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$4-structures with length hearts, this specializes to the clean statement that the silting collection lies in degrees $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$5 if and only if the $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$6-structure is $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$7-complicial if and only if the extended heart is 1-generated.

Koszul duality for proper connective dg algebras

Building on Fushimi's Koszul duality between locally finite connective dg algebras and coconnective locally finite dg algebras with semisimple degree-zero cohomology, the paper establishes two refinements. First, over any field, properness of $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$8 corresponds to the Hfd-closed condition on the Koszul dual $\mathrm{per}(\mathrm{RHom}_{\mathcal{A}}(S))$9 (i.e., $k$0). Second, over a perfect field, properness corresponds to homological smoothness of $k$1. These results complete a diagram relating smoothness, Hfd-closedness, and regularity; notably, the paper exhibits a regular coconnective dg algebra (the Koszul dual of a graded Kronecker quiver with arrows in all negative degrees) that is neither smooth nor Hfd-closed, showing regularity does not imply either finiteness condition.

The main theorem then states: over a perfect field, Koszul duality gives a bijection between quasi-isomorphism classes of connective proper dg algebras with $k$2 concentrated in degrees $k$3, and coconnective locally finite smooth dg algebras $k$4 with $k$5 semisimple such that $k$6 is 1-generated. Moreover, the standard $k$7-structure on $k$8 is $k$9-complicial.

Coherently generated $A$0-algebras

The 1-generation condition on $A$1 is translated into a condition on the minimal $A$2-model of $A$3: the algebra must be coherently generated in degrees $A$4, meaning that closed degree-$A$5 matrices between twisted complexes built from shifted simples factor through degree-one morphisms up to $A$6-boundaries, with the twisted differentials constrained to lie in degrees $A$7. This refines Keller's theorem that the Yoneda algebra $A$8 of a finite-dimensional algebra is generated in degrees 0 and 1 as an $A$9-algebra. The paper emphasizes that naive degree-wise generation is insufficient: it constructs a graded quiver algebra $H^*(A)$0 of type $H^*(A)$1 ($H^*(A)$2) that is generated in degrees 0 and 1 and homologically smooth, yet cannot be the Koszul dual of any finite-dimensional algebra, because $H^*(A)$3 forces non-trivial higher cohomology. This example concretely illustrates why coherence — not mere generation — is the correct invariant, and the author notes candidly that no simpler, more explicit reformulation of coherent generation is currently available.

Abelian $H^*(A)$4-truncated dg categories and length exact $H^*(A)$5-categories

The paper establishes that abelian $H^*(A)$6-truncated dg categories in the sense of Mocanu are precisely the $H^*(A)$7-extended hearts of $H^*(A)$8-structures on pretriangulated dg categories. The converse direction relies on Stefanich's $H^*(A)$9-categorical result that the functor assigning to a prestable category its truncation below $(-d,0]$0 has a fully faithful left adjoint whose essential image consists of the $(-d,0]$1-complicial objects; the dg-level proof proceeds via the dg nerve and Kleiner's stable hull, showing that the canonical comparison $(-d,0]$2 is an equivalence.

The paper then introduces length exact $(-d,0]$3-categories: $(-d,0]$4-filtered exact $(-d,0]$5-truncated dg categories generated under extensions by shifts of simples of the discrete subcategory $(-d,0]$6. For these, it proves a generalization of Ringel's theorem and Enomoto's theorem: there are mutually inverse bijections between $(-d,0]$7-semibricks and length $(-d,0]$8-wide subcategories, given by $(-d,0]$9 and $d \geq 1$0. As a corollary, a length exact $d \geq 1$1-category is abelian $d \geq 1$2-truncated if and only if its simples form a $d \geq 1$3-semibrick (equivalently, a semibrick in the discrete part). The paper also notes that the notion of "simple" used here deliberately differs from existing definitions of simple objects in extriangulated categories, since shifts of simples must be allowed as building blocks.

Enough projectives and the four-fold correspondence

For length abelian $d \geq 1$4-truncated dg categories with finitely many simples, the paper proves a dg analogue of the classical Gabriel/Årdal–Buan result: having enough projectives, having finite height (a Loewy-length-type invariant defined via filtrations by shifted simples), and being quasi-equivalent to $d \geq 1$5 for a proper connective dg algebra $d \geq 1$6 are equivalent conditions. Two notions of Loewy length are introduced and shown to be comparable within a factor of $d \geq 1$7, with equality holding exactly for objects concentrated in a single cohomological degree.

Assembling these pieces, the paper obtains a four-fold bijection over a perfect field between:

Class	Description
(1)	Smooth locally finite pretriangulated dg categories with strictly $d \geq 1$8-complicial bounded $d \geq 1$9-structures whose hearts have enough projectives
(2)	Locally finite, Ext-finite length abelian $\mathrm{RHom}_{\mathcal{A}}(S)$00-truncated dg categories with finitely many simples and finite height
(3)	Proper basic dg algebras with cohomology strictly concentrated in degrees $\mathrm{RHom}_{\mathcal{A}}(S)$01
(4)	Smooth locally finite coconnective basic dg algebras with semisimple $\mathrm{RHom}_{\mathcal{A}}(S)$02 and 1-generated $\mathrm{RHom}_{\mathcal{A}}(S)$03

The bijections are explicit: passage to extended hearts, derived endomorphism algebras of simples, minimal right approximations of simples by projectives, and Koszul duals. Specializing to $\mathrm{RHom}_{\mathcal{A}}(S)$04 recovers the equivalence between $\mathrm{RHom}_{\mathcal{A}}(S)$05-term silting collections, $\mathrm{RHom}_{\mathcal{A}}(S)$06-compliciality of the associated $\mathrm{RHom}_{\mathcal{A}}(S)$07-structure, and $\mathrm{RHom}_{\mathcal{A}}(S)$08-term simple-minded collections, along with the corollary that $\mathrm{RHom}_{\mathcal{A}}(S)$09 is 1-generated for every $\mathrm{RHom}_{\mathcal{A}}(S)$10-term simple-minded collection $\mathrm{RHom}_{\mathcal{A}}(S)$11.

Koszul duality for length exact $\mathrm{RHom}_{\mathcal{A}}(S)$12-categories

Finally, the paper axiomatizes the dg endomorphism categories of simples directly. It defines, for a coconnective dg category $\mathrm{RHom}_{\mathcal{A}}(S)$13, the subcategory $\mathrm{RHom}_{\mathcal{A}}(S)$14 of twisted complexes filtered by shifts of objects of $\mathrm{RHom}_{\mathcal{A}}(S)$15, together with an "admissible Schur" condition on admissible morphisms. Under the assumptions that $\mathrm{RHom}_{\mathcal{A}}(S)$16 is 1-generated, $\mathrm{RHom}_{\mathcal{A}}(S)$17, and the Schur condition holds, the resulting category is a length exact $\mathrm{RHom}_{\mathcal{A}}(S)$18-category. The main theorem gives mutually inverse bijections between such dg categories and length exact $\mathrm{RHom}_{\mathcal{A}}(S)$19-categories, restricting successively to length abelian $\mathrm{RHom}_{\mathcal{A}}(S)$20-truncated dg categories and, over perfect fields, to the locally finite, Ext-finite, enough-projectives setting. Whenever $\mathrm{RHom}_{\mathcal{A}}(S)$21 and $\mathrm{RHom}_{\mathcal{A}}(S)$22 correspond, there is a quasi-equivalence $\mathrm{RHom}_{\mathcal{A}}(S)$23.

One limitation is acknowledged here: the "admissible Schur" condition is extrinsic, phrased via conflations in the twisted complex category rather than intrinsic to $\mathrm{RHom}_{\mathcal{A}}(S)$24 itself. The author explains why any purely intrinsic replacement is obstructed — for $\mathrm{RHom}_{\mathcal{A}}(S)$25 there are no restrictions on $\mathrm{RHom}_{\mathcal{A}}(S)$26 at all, since no nonzero morphism is admissible — and leaves the search for an intrinsic formulation open.

Limitations and open questions

Several hypotheses carry real weight throughout. The perfect-field assumption is essential for identifying properness with smoothness on the Koszul dual side, so the smooth variant of the main correspondence is unavailable over arbitrary fields. The equivalence between coherent generation and concentration of cohomology depends on the devissage arguments available for proper algebras, and the paper provides no simplified combinatorial description of coherent generation beyond its definition via 1-generation of twisted complexes. The relationship between the two Loewy-length notions is only controlled up to a factor of $\mathrm{RHom}_{\mathcal{A}}(S)$27, with sharpness examples showing the bounds cannot be improved uniformly. Finally, whether the admissible Schur condition can be made intrinsic to the dg category remains unresolved.

Conclusion

The paper gives a systematic answer to the question of which dg algebras arise as derived endomorphism algebras of simples in length categories: they are exactly the Koszul duals of coconnective dg algebras whose perfect derived categories have 1-generated extended hearts, equivalently whose minimal $\mathrm{RHom}_{\mathcal{A}}(S)$28-models are coherently generated. Along the way it develops a flexible theory of dg realization functors, connects compliciality of $\mathrm{RHom}_{\mathcal{A}}(S)$29-structures to reconstructibility from extended hearts, extends Enomoto's semibrick–wide correspondence to the $\mathrm{RHom}_{\mathcal{A}}(S)$30-truncated setting, and completes the foundational framework relating length exact $\mathrm{RHom}_{\mathcal{A}}(S)$31-categories, abelian $\mathrm{RHom}_{\mathcal{A}}(S)$32-truncated dg categories, and extended hearts of proper connective dg algebras.