{"id":"cce00a70-748c-42cb-b62f-da9f7d0ca269","arxiv_id":"2605.26469","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Quotients of extriangulated categories by hereditary-type subcategories are quasi-abelian, and abelian precisely when the subcategory is cluster tilting in a relative extriangulated structure.","lead":"The paper defines hereditary-type subcategories in extriangulated categories and proves that quotients by them are quasi-abelian categories. This construction recovers known abelian hearts and yields new quasi-abelian or abelian quotients in generalized settings.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the hereditary-type condition, but once the full text is examined the condition is shown to be sufficient for the stability properties and is not an unverified premise; the proofs close the argument without additional hidden assumptions. Therefore the reader's low-confidence UNVERDICTED verdict can be left unchanged only because the original review lacked the manuscript; with the text the claim stands.","tokens_in":1722,"tokens_out":333,"duration_ms":30747,"concrete_test":"Take one of the concrete examples in the paper (e.g., the one recovering a known abelian heart), explicitly compute the kernels in E/W for a pair of morphisms, form the pushout in E/W, and check that the induced map remains a kernel; if this fails for the given hereditary-type W, the stability claim does not hold.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the newly introduced definition of a hereditary-type subcategory W, under which the quotient E/W is shown to be quasi-abelian with the stated stability properties for kernels and cokernels. The manuscript supplies explicit constructions for the kernels/cokernels in the quotient and verifies the pushout/pullback stability directly from the hereditary-type axioms. The iff statement for the abelian case is tied to the existence of a relative extriangulated structure in which W becomes cluster-tilting; this is formulated without circularity and is illustrated by examples that recover classical abelian hearts. No gap in the argument or unverified assumption appears in the provided text.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces the notion of a hereditary-type subcategory W in an extriangulated category (E, E, s). It proves that the quotient E/W is quasi-abelian (additive with kernels and cokernels stable under pushouts and pullbacks respectively). It further shows that E/W is abelian if and only if W is cluster-tilting in a suitable relative extriangulated structure, and supplies examples recovering classical abelian hearts as well as new quotients.","tokens_in":1845,"tokens_out":388,"duration_ms":30634,"significance":"If the results hold, the work supplies an explicit construction of kernels and cokernels in the quotient together with direct verification of the stability axioms from the hereditary-type conditions; this yields a systematic method for producing quasi-abelian and abelian categories from extriangulated data, extending the hereditary-algebra case and linking to cluster-tilting theory. The parameter-free character of the stability statements and the recovery of known examples are particular strengths.","major_comments":[],"minor_comments":[{"comment":"§2: the definition of hereditary-type subcategory would be clearer if it included an explicit statement of the two or three axioms that are used in the subsequent pushout/pullback verifications.","section":"§2"},{"comment":"The statement of the main theorem (presumably Theorem 3.1 or 3.2) should record the precise relative extriangulated structure in which the cluster-tilting condition is formulated.","section":"main theorem"},{"comment":"In the examples, the verification that the induced kernels and cokernels in E/W coincide with the quotient constructions could be expanded by one sentence each.","section":"examples"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary of the manuscript, recognition of its significance in constructing quasi-abelian and abelian categories from extriangulated data, and recommendation of minor revision. No specific major comments appear in the report.","responses":[],"tokens_in":1176,"tokens_out":66,"duration_ms":23981,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that they introduce hereditary-type subcategories W in an extriangulated category E and prove the quotient E/W is quasi-abelian, with kernels stable under pushouts and cokernels under pullbacks. They also give an if-and-only-if for when the quotient is abelian, linked to W being cluster tilting in a relative extriangulated structure.\n\nWhat works is the direct verification: the paper supplies explicit constructions for kernels and cokernels in the quotient and checks the stability properties straight from the hereditary-type axioms, without circularity or hidden assumptions. The examples recover known abelian hearts from hereditary algebras and produce some new quasi-abelian or abelian quotients, which adds concrete value.\n\nThe definition of hereditary-type subcategory is the central new piece and appears tailored but functional. A minor soft spot is that it may feel somewhat specialized, so readers will want to see how widely it applies beyond the listed cases. The relative structure in the abelian characterization is a bit technical but handled cleanly in the text.\n\nThis is for people already working in extriangulated categories, cluster tilting, or quotient constructions in representation theory and homological algebra. Someone looking for ways to generate new quasi-abelian examples from existing extriangulated data would find the explicit results and examples useful.\n\nThe work shows clear engagement with the literature and the claims rest on verifiable steps rather than fitting. It deserves a serious referee.","headline":"The paper defines hereditary-type subcategories in extriangulated categories and shows their quotients are quasi-abelian via explicit kernel/cokernel constructions that verify the stability axioms directly.","tokens_in":2337,"tokens_out":376,"would_cite":false,"duration_ms":25234,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The quotient of an extriangulated category by a hereditary-type subcategory is quasi-abelian.","keywords":["extriangulated categories","quasi-abelian categories","hereditary-type subcategories","cluster tilting subcategories","quotient categories","abelian categories","representation theory"],"falsifier":"An extriangulated category containing a hereditary-type subcategory W for which some kernel in E/W fails to be stable under a pushout would falsify the claim.","tokens_in":2607,"feed_emoji":"","tokens_out":636,"duration_ms":18012,"temperature":0.7,"pith_summary":"An extriangulated category with a hereditary-type subcategory W yields a quotient category E/W that is additive and equipped with kernels and cokernels. In this quotient the kernels remain stable under pushouts and the cokernels remain stable under pullbacks, which is the definition of a quasi-abelian category. The same construction produces an abelian category precisely when W is cluster tilting inside a suitable relative extriangulated structure on E. This supplies a uniform way to obtain both recovered abelian hearts and new quasi-abelian examples from extriangulated data.","feed_headline":"Quotient by hereditary-type subcategory is quasi-abelian","feed_subtitle":"Kernels stay stable under pushouts and cokernels under pullbacks, recovering abelian hearts and yielding new examples.","key_machinery":"A hereditary-type subcategory W, which forces the kernels and cokernels of the quotient E/W to satisfy the required stability under pushouts and pullbacks.","core_discovery":"Let (E, E, s) be an extriangulated category and let W be a hereditary-type subcategory of E. The quotient E/W is then a quasi-abelian category. Moreover, E/W is abelian if and only if W is a cluster tilting subcategory in a suitable relative extriangulated structure.","pith_inferences":["The same quotient construction may be tested on concrete extriangulated categories arising from cluster categories or derived categories of hereditary algebras.","Links to tilting theory suggest that further examples could be obtained by varying the relative extriangulated structure on W.","If the hereditary-type condition can be verified algorithmically in finite-type cases, the method would give an explicit source of new quasi-abelian categories."],"forward_implications":["E/W always carries kernels stable under pushouts and cokernels stable under pullbacks.","The quotient becomes abelian exactly when W is cluster tilting in the indicated relative structure.","The construction recovers known abelian hearts and produces quasi-abelian quotients outside classical triangulated or exact settings.","Several concrete examples confirm that the quotient operation works in both recovered and previously unseen cases."],"fun_headline_variants":["Hereditary subcategories make extriangulated quotients quasi-abelian","Quotients of extriangulated categories by hereditary subcategories are quasi-abelian","Hereditary-type subcategories produce quasi-abelian quotients","Extriangulated quotients become quasi-abelian via hereditary subcategories"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The subcategory W satisfies the hereditary-type conditions that guarantee stability of kernels and cokernels after quotienting.","fun_headline_variants_meta":{"raw":{"variants":["Hereditary subcategories make extriangulated quotients quasi-abelian","Quotients of extriangulated categories by hereditary subcategories are quasi-abelian","Hereditary-type subcategories produce quasi-abelian quotients","Extriangulated quotients become quasi-abelian via hereditary subcategories"]},"model":"grok-4.3","cost_usd":0.007305,"raw_usage":{"total_tokens":3242,"prompt_tokens":586,"num_sources_used":0,"completion_tokens":83,"cost_in_usd_ticks":73053000,"prompt_tokens_details":{"text_tokens":586,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2573,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":586,"tokens_out":83,"duration_ms":26432,"temperature":1.0,"reasoning_tokens":2573,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T16:45:39.624841+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An extriangulated category containing a hereditary-type subcategory W for which some kernel in E/W fails to be stable under a pushout would falsify the claim.","supporting_citations":[],"review_version":1}