{"id":"b98374c3-92ba-4669-b3df-9756f6956adc","arxiv_id":"2509.02065","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors construct reduced and partial directed flag complexes whose homologies are naturally isomorphic to the singular quiver homologies, enabling efficient persistent homology computation.","lead":"The paper shows that several recently defined homologies of directed quivers can be computed from much smaller flag-type complexes, with natural isomorphisms and persistent homology algorithms. Generalists should care because this makes topological analysis practical for directed networks with multiple edges and loops, common in biology and engineering.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.3/6.4 depend on an arbitrary choice of representative h that is never shown to be a chain map; without face-compatible choices the acyclic carrier argument cannot run.","rationale":"The reader's weakest assumption identifies the same load-bearing concern. The central claims are the isomorphisms in Theorems 5.3 and 6.4, and those proofs require two chain maps, g and h. The map g is defined from the vertex image of a given singular simplex, so its boundary compatibility is checkable. The map h, by contrast, is defined by arbitrary representatives of abstract simplices, and boundary compatibility is not imposed. The acyclic carrier theorem applies only to chain maps, so h must be known to be a chain map before it can be used. The paper's assertion that h is a chain map 'by a similar argument' is not justified: the argument for g relied on g(f) being determined by f, whereas h's value on a face is chosen independently of any parent simplex. The Lemma 4.3/digraph issue is a second manifestation of the same under-specification in the carrier argument. These are proof gaps rather than demonstrated falsehoods; the statements may still be true, and the paper has substantial algorithmic content and a public code repository, so the correct verdict remains conditional rather than reject. I would not change the reader's CONDITIONAL verdict.","tokens_in":46306,"tokens_out":14486,"duration_ms":175930,"concrete_test":"Let G be the quiver on {a,b,c} with edges a→b, b→a, a→c, b→c. Then Fbar(G) contains the 2-simplex {a,b,c} and its three 1-faces. Choose s_{abc}: Δ^2 -> G by f(0)=b, f(1)=a, f(2)=c, so its {a,b}-face is b→a, and choose s_{ab}: Δ^1 -> G to be a→b. Then h({a,b,c}) and h({a,b}) violate ∂h = h∂, showing the h constructed in §5.2 is not automatically a chain map. To settle whether the theorem is nevertheless true, exhaustively check whether some choice of all s_{v0,...,vn} satisfying face compatibility exists for this G; if none exists, Theorem 5.3 requires a new proof, not just a 'similar argument'. Run the analogous check for a two-triangle version and for Ftilde_< in Theorem 6.4.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 5.3, h:C_*(Fbar(G)) -> C^{Δ,m}_*(G) is defined by picking, for every abstract simplex {v0,...,vn} in Fbar(G), an arbitrary singular simplex s_{v0,...,vn}: Δ^n -> G realizing that vertex set. The paper then asserts 'The function h can be shown to be a chain map by a similar argument.' This is exactly where the proof breaks. A chain map requires ∂h({v0,...,vn}) = h∂({v0,...,vn}), i.e. for each face τ_i the chosen s_{v0,...,vn}∘δ_i must coincide, up to the prescribed sign, with the chosen s_{τ_i}. The choices in the definition are independent, so nothing guarantees this. Indeed, different inclusions of Δ^n into the same quiver can assign different edges to the same vertex set, and the chosen representative for a face need not be a face of the chosen representative for the parent simplex. The same gap appears in Theorem 6.4 for h:C_*(Ftilde_<(G)) -> C^{Δ,h}_*(G). Since h is then used with the acyclic carrier theorem to show h∘g is chain homotopic to id, the isomorphisms H^{Δ,m}_* ≅ H_*(Fbar(G)) and H^{Δ,h}_* ≅ H_*(Ftilde_<(G)) are not established as written. A second, related gap is the use of Lemma 4.3 on full subquivers G_{f(0),...,f(n)} that may contain loops and multiple edges, although that lemma is stated for digraphs. The statements may be repairable, but the present proof does not supply the required compatible choices.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops computational reductions for three singular simplicial homology theories of quivers introduced in [33]. For the map-type homology H^{Δ,m}_* it constructs the reduced digraph \\bar R(G) and the reduced directed flag complex \\bar F(G), and claims H^{Δ,m}_*(G) ≅ H_*(\\bar F(G)) with naturality. For the homomorphism-type homology H^{Δ,h}_* it introduces local strong h-homotopies, a partially reduced quiver \\tilde R(G), and a partial directed flag complex \\tilde F_<(G), and claims H^{Δ,h}_*(G) ≅ H_*(\\tilde F_<(G)) with naturality. For inclusion-type homology H^{Δ,i}_* it observes the isomorphism with the directed flag complex F(G). The paper also gives algorithms for computing persistent homology of filtered quivers via these smaller complexes and analyzes their complexity. The main conceptual contribution is that the three singular simplicial homologies can be computed from flag-type complexes that are typically much smaller than the singular chain complexes, with naturality suitable for persistence.","tokens_in":46727,"tokens_out":11043,"duration_ms":127415,"significance":"If the main isomorphisms hold, the paper provides a genuinely useful computational bridge between singular simplicial quiver homologies and ordinary simplicial homology of smaller flag complexes, with persistent homology applications. The naturality claims are central for applying the results to filtered data. The paper also contains explicit algorithms and a demonstration implementation, and the complexity analysis identifies where the proposed reductions improve over direct computation. The novel local strong h-homotopy invariance statement for H^{Δ,h}_* is potentially valuable. However, the proofs of the two principal isomorphisms contain a load-bearing gap: the map h from the smaller complex back to the singular chains is chosen non-constructively and is not proved to be a chain map. Because the acyclic carrier argument requires h to be a chain map, the main theorems are not established as written. The statements appear plausible and likely repairable, but the repair is a substantive technical step rather than a typographical fix.","major_comments":[{"comment":"The chain map h: C_*(\\bar F(G)) → C^{Δ,m}_*(G) is defined by choosing, for each abstract simplex {v_0,...,v_n}, an arbitrary singular simplex s_{v_0,...,v_n} realizing that vertex set. The proof then asserts, without proof, that h is a chain map 'by a similar argument'. A chain map requires ∂h({v_0,...,v_n}) = h∂({v_0,...,v_n}), i.e. the chosen representative of each face must be the corresponding face of the chosen representative of the parent simplex, up to the fixed sign. The arbitrary choices in the definition do not guarantee this. Since h is then used in the acyclic carrier theorem to compare h∘g with id, and since the acyclic carrier theorem only applies to chain maps, the chain homotopy equivalence, and hence the isomorphism H^{Δ,m}_*(G) ≅ H_*(\\bar F(G)), is not proven as written. The argument needs either an explicit face-compatible selection of representatives or a different pr","section":"§5.2, proof of Theorem 5.3"},{"comment":"The same gap occurs for h: C_*(\\tilde F_<(G)) → C^{Δ,h}_*(G). The sentence 'which is a chain map for the same reasons as g in the case of singular simplex inclusions' repeats the problematic step: for each s ∈ \\tilde F_<(G)_n the proof chooses a singular simplex inclusion f_s inducing s, and these choices are independent across simplices. No argument is given that the chosen representative of a face is the face of the chosen representative of the parent. The subsequent acyclic carrier argument for h∘g and id therefore requires an unproved property. This load-bearing issue must be fixed before Theorem 6.4 can be accepted.","section":"§6.2, proof of Theorem 6.4"},{"comment":"The acyclicity of C^{Δ,m}_*(G_{f(0),...,f(n)}) is derived from Lemma 4.3, which is stated only for digraphs. In Theorem 5.3 the subquiver G_{f(0),...,f(n)} is a full subquiver of an arbitrary quiver and may contain loops and multiple edges; it need not be a digraph. Since this acyclicity is the carrier used in the acyclic carrier theorem, the proof needs either a quiver version of Lemma 4.3 or an explicit justification that the digraph statement extends. The argument of Lemma 4.3 looks adaptable to quivers, but the adaptation is not supplied in the manuscript.","section":"§5.2, proof of Theorem 5.3; §4, Lemma 4.3"},{"comment":"The naturality claim for \\bar F uses the rule \\bar F(ϕ)({v_0,...,v_n}) = {ϕ(v_0),...,ϕ(v_n)}. If ϕ collapses two vertices of a simplex, the righthand side is not an n-simplex of the target simplicial complex, so this does not define a chain map C_n(\\bar F(G_1)) → C_n(\\bar F(G_2)) in the usual degree-preserving sense. The proof of the commutative diagram implicitly uses the convention that such a simplex is sent to 0. This convention should be stated explicitly, otherwise the naturality claim is not well-defined as written. The same convention issue appears in the definition of the chain-level partial flag functor in equation (20).","section":"§5.2, equations (12) and the naturality proof"}],"minor_comments":[{"comment":"Typo: 'principle underlying structure' should be 'principal underlying structure'; likewise 'demision zero' in the introduction should be 'dimension zero'.","section":"Abstract"},{"comment":"Typo: 'see Prostitution 3.1' should be 'see Proposition 3.1'.","section":"§2.5"},{"comment":"The text says 'from Remark 18'; this should be 'Remark 6.1'.","section":"§6.2, paragraph before Definition 15"},{"comment":"The notation for the homomorphism-type homology is inconsistent: H^{Δ,h}_* is standard, but Theorem 4.1 and Theorem 6.5 write H^{h,Δ}_* or H^{h,Δ}. Please unify.","section":"Throughout"},{"comment":"Typo: 'tipples' should be 'triples'.","section":"Example 5.2"},{"comment":"Minor wording issues: 'fuctoriality' should be 'functoriality'; 'known complexity' should be 'known complexity'; 'strongly contactable' and 'strongly contractable' are used inconsistently.","section":"§6.2 and Appendix A.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript depends heavily on [33], which is listed as 'to appear' and shares authors with the present paper. The main proofs also have overlap with work in [41] and [50], though these are cited. I do not see these as circularity problems, but the referee should perhaps confirm the status of [33]. The central defect is the h chain-map issue; it is localized but load-bearing, so I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague — here's my take on arXiv:2509.02065. The headline: this is a substantial, well-motivated extension of singular simplicial quiver homologies, and the new constructions (reduced and partial directed flag complexes, local strong h-homotopy) are worth taking seriously. But the two central isomorphism theorems, 5.3 and 6.4, have a genuine proof gap, and I would not cite the main results as proven in this version.\n\nWhat is actually new: for quivers, the isomorphisms between H^{Δ,m}, H^{Δ,h}, H^{Δ,i} and the homologies of \\bar F, \\tilde F_<, and F are new, and the local strong h-homotopy invariance of H^{Δ,h} is a genuinely new tool. The paper is honest about overlap for digraphs (Turner, Milićević–Scoville) and about the fact that \\bar F is not a faithful or full functor. The persistent homology algorithms and complexity analysis in the appendix are concrete and appear usable; code is provided. For digraphs, H^{Δ,m} has known models, so the quiver extension is the real contribution.\n\nThe soft spot is exactly the one flagged: in the proof of 5.3, h: C_*(\\bar F(G)) → C^{Δ,m}_*(G) is defined by choosing, for each vertex set, an arbitrary singular simplex realizing it. A chain map requires the chosen simplex for a face to be the corresponding face of the chosen simplex for the parent. That is not ensured, and \"can be shown by a similar argument\" doesn't work, because g's argument used the specific simplex f and its faces. The same gap appears for h in 6.4. This is load-bearing: without h being a chain map, the acyclic carrier argument can't go through. I also agree with the second, lesser issue: Lemma 4.3 is stated and proved for digraphs but applied to full subquivers of a quiver. That is likely fixable by first passing to \\bar R(G) or \\tilde R(G), and it is separate from the h issue, but it is still a gap.\n\nAre these fatal? I don't think so. The theorems are plausible; the h map will probably be repaired with a standard acyclic-model or compatible-choice argument, possibly with a more careful definition. There is enough substance here, and the algorithms are concrete enough, that this deserves a serious referee. As written, I would label it conditional: the main computational claims for persistent H^{Δ,m} and H^{Δ,h} should not be taken off the shelf until the proof is fixed.\n\nIf I were editing, I would send it to review and ask the authors to repair the two theorems before publication. I would bring it to a reading group after that fix, not before; and I wouldn't cite the theorems in their current form.","headline":"Plausible and useful quiver homology paper, but the two main isomorphism theorems (5.3 and 6.4) are not proven as written because the chosen chain map h is not shown to be a chain map and Lemma 4.3 is applied beyond its stated domain.","tokens_in":47188,"tokens_out":3170,"would_cite":false,"duration_ms":35009,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55U10","05C20","55N31"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that three singular simplicial homologies of quivers—directed graphs with loops and multiple edges—are isomorphic to the homologies of smaller flag complexes, making their persistent homology computationally tractable.","keywords":["singular simplicial homology","quivers","digraphs","directed flag complex","persistent homology","acyclic carrier theorem","quiver homotopy","reduced directed flag complex"],"falsifier":"Take the quiver on two vertices with a double edge (and, for the h-case, a loop at one vertex), compute $H^{\\Delta,m}_*(G)$ and $H^{\\Delta,h}_*(G)$ directly from the singular definitions, and compare with $H_*(\\bar F(G))$ and $H_*(\\tilde F_<(G))$. One mismatch would disprove Theorems 5.3 and 6.4 as stated; alternatively, exhibit a choice of the maps $s_{v0,...,vn}$ for which $h$ fails to commute with $\\partial$, showing the published proof is incomplete.","tokens_in":46200,"feed_emoji":"🕸","tokens_out":10383,"duration_ms":109235,"temperature":0.7,"texified_at":"2026-08-05T20:21:45.190409+00:00","pith_summary":"The paper aims to make three singular simplicial homology theories for quivers practical to compute. It proves that each of the three homologies $H^{\\Delta,m}$, $H^{\\Delta,h}$, and $H^{\\Delta,i}$ is isomorphic to the homology of a smaller, flag-type complex: the reduced directed flag complex, the partial directed flag complex, and the generalized directed flag complex, respectively. Because the isomorphisms are natural, persistent homology of a filtered quiver can be read off from a filtration of the smaller complex. These homologies have theoretical advantages over the widely used directed flag complex—strong homotopy invariance and, in the m-case, stability—but previously had no efficient computational route. The paper also supplies algorithms and complexity bounds showing the smaller complexes are at least as convenient as the directed flag complex to build.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":4949,"prompt_tokens":838,"completion_tokens":4111,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":838,"completion_tokens_details":{"reasoning_tokens":3319}},"feed_headline":"Quiver homologies become computable via smaller complexes","feed_subtitle":"Persistent features of weighted directed networks are computed from reduced flag complexes instead of huge singular chains.","key_machinery":"The mechanism is a pair of simplification functors plus a classical chain-homotopy criterion. The reduced digraph $\\bar R(G)$ collapses all loops and multiple edges to a single edge, and its reduced directed flag complex $\\bar F(G)$ carries $H^{\\Delta,m}$. The partially reduced quiver $\\tilde R(G)$ collapses only multiple edges incident to loop-bearing vertices, and with a loop-maximal total order $<$ it produces the partial directed flag complex $\\tilde F_<(G)$, a $\\Delta$-set carrying $H^{\\Delta,h}$. The acyclic carrier theorem—a criterion that makes two chain maps homotopic when each simplex is carried into an acyclic subcomplex—is the tool that converts the evident algebraic comparison maps $g$ and $h$ into chain homotopy","core_discovery":"The central claim: for every quiver $G$, $H^{\\Delta,m}_*(G) \\cong H_*(\\bar F(G))$, $H^{\\Delta,h}_*(G) \\cong H_*(\\tilde F_<(G))$, and $H^{\\Delta,i}_*(G) \\cong H_*(F(G))$, each isomorphism natural in the relevant morphisms. Here $\\bar F$ is the reduced directed flag complex after collapsing loops and multiple edges; $\\tilde F_<$ is the partial directed flag complex that keeps loop information; $F$ is the directed flag complex generalized to quivers. The proofs use acyclic carriers to make the singular chain complexes chain-homotopy-equivalent to the smaller simplicial chains. Fewer generators and naturality make persistent $H^{\\Delta,m}$ and $H^{\\Delta,h}$ computable by filtering the smaller complexes, and $H^{\\Delta,h}$ is shown invariant under weak loc","pith_inferences":["Editorial: the collapse results imply H^{Δ,m} is blind to whether interactions are multiple or looped; for applications where multiplicity encodes real information, persistent H^{Δ,h} may be the better choice, since its partial complex keeps loop data.","If the missing verification that h is a chain map cannot be supplied, the main isomorphisms might still be true for a different choice of chain map or a slightly different carrier; checking this directly is a concrete first step.","A similar acyclic-carrier reduction may work for cubical singular homologies of quivers, provided the face-compatibility issue is solved at the level of choices.","The partial directed flag complex depends on a loop-maximal total order; practical implementations must handle the sign and order bookkeeping in the chain functor automatically, or users may see different barcode conventions for the same quiver."],"forward_implications":["H^{Δ,m} of any quiver is computable from the reduced digraph: loops and multiple edges do not affect it, so persistent H^{Δ,m} can be computed on a much smaller flag complex.","H^{Δ,h} is invariant under weak local strong h-homotopy, so degenerate loops and certain multiple-edge structures can be removed before computation without changing the output.","For loopless quivers H^{Δ,h} equals H^{Δ,i}, and for digraphs without double edges all three homologies coincide; the exceptional behavior comes precisely from loops, multiple edges, and double edges.","The algorithms in the appendix compute filtered versions of F, \\bar F, and \\tilde F_<, so persistent homology packages can be applied directly to weighted directed networks.","The computational reductions are natural, so persistence barcodes for H^{Δ,m} and H^{Δ,h} respect filtered quiver maps, not just individual quivers."],"supporting_citations":[{"why":"Defines the three singular simplicial homologies and their strong homotopy invariance, which the reductions rely on.","marker":"[33]"},{"why":"Supplies the acyclic carrier theorem used to prove the chain homotopy equivalences in Theorems 5.3 and 6.4.","marker":"[42]"},{"why":"Introduced the directed clique/flag complex that the paper generalizes to quivers and benchmarks against.","marker":"[39]"},{"why":"Provides the memory-efficient parallel algorithm for directed flag complexes that the new algorithms extend.","marker":"[35]"},{"why":"Gives the ordered tuple complex and stability results for the m-case that motivate computing H^{Δ,m} persistently.","marker":"[50]"},{"why":"Established homotopy theory for digraphs, the source of the strong homotopy notions extended here.","marker":"[20]"}],"fun_headline_variants":["Quiver homologies shrink to smaller chain complexes","Persistent quiver homology from reduced flag complexes","Efficient quiver homology: natural isomorphisms at work","Quiver homology made practical with compact simplicial chains","New functors cut quiver homology computation cost"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The load-bearing premise is that the arbitrarily chosen singular simplex for each vertex set in the proofs of Theorems 5.3 and 6.4 can be chosen compatibly with face maps, so that the map $h$ is actually a chain map; the paper asserts this by a 'similar argument' without proving it.","fun_headline_variants_meta":{"raw":{"variants":["Quiver homologies shrink to smaller chain complexes","Persistent quiver homology from reduced flag complexes","Efficient quiver homology: natural isomorphisms at work","Quiver homology made practical with compact simplicial chains","New functors cut quiver homology computation cost"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00033,"raw_usage":{"total_tokens":1707,"prompt_tokens":807,"completion_tokens":900,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":824}},"tokens_in":551,"tokens_out":900,"duration_ms":9108,"temperature":1.0,"reasoning_tokens":824,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T11:55:09.650795+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the quiver on two vertices with a double edge (and, for the h-case, a loop at one vertex), compute $H^{\\Delta,m}_*(G)$ and $H^{\\Delta,h}_*(G)$ directly from the singular definitions, and compare with $H_*(\\bar F(G))$ and $H_*(\\tilde F_<(G))$. One mismatch would disprove Theorems 5.3 and 6.4 as stated; alternatively, exhibit a choice of the maps $s_{v0,...,vn}$ for which $h$ fails to commute with $\\partial$, showing the published proof is incomplete.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the three singular simplicial homologies and their strong homotopy invariance, which the reductions rely on."},{"cited_title":"Munkres,Elements of Algebraic Topology, Addison Wesley Publishing Company, 1984","cited_arxiv_id":null,"evidence_quote":"Supplies the acyclic carrier theorem used to prove the chain homotopy equivalences in Theorems 5.3 and 6.4."},{"cited_title":"Masulli and P","cited_arxiv_id":null,"evidence_quote":"Introduced the directed clique/flag complex that the paper generalizes to quivers and benchmarks against."},{"cited_title":"L¨ utgehetmann, D","cited_arxiv_id":null,"evidence_quote":"Provides the memory-efficient parallel algorithm for directed flag complexes that the new algorithms extend."},{"cited_title":"Turner,Rips filtrations for quasimetric spaces and asymmetric functions with stability results, Algebraic and Geometric Topology19(3)(2019), 1135–1170","cited_arxiv_id":null,"evidence_quote":"Gives the ordered tuple complex and stability results for the m-case that motivate computing H^{Δ,m} persistently."},{"cited_title":"4, 919–674","cited_arxiv_id":null,"evidence_quote":"Established homotopy theory for digraphs, the source of the strong homotopy notions extended here."}],"review_version":1}