{"id":"c359aaae-c8a0-4a67-99eb-f2091581b463","arxiv_id":"1908.06879","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A general formalism produces directed model structures from a cylinder object, unifying the covariant and contravariant model structures on simplicial sets and the Cartesian and coCartesian model structures on marked simplicial sets.","lead":"This paper constructs model structures whose direction of homotopy is controlled by the choice of a cylinder object, and it shows that several known model structures in higher category theory are special cases. It gives a unified framework for covariant, contravariant, Cartesian, and coCartesian homotopy theories, and it derives notions of final, initial, smooth, and proper maps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Example identification depends on cited Lemma 4.3; if the saturation equality failed, the promised model structures would not be Joyal's.","rationale":"Full-text stress test. Theorems 2.17 and 2.18 are carefully proved: the weak equivalence characterization, the saturation of trivial cofibrations via Proposition 2.32, and the use of Simpson's recognition theorem are all coherent. The slice construction for families is standard and the final/initial map theory follows from the model structures. The examples are where the central claim meets the literature. For simplicial sets the bridge is the cited Lemma 4.3; the paper gives no proof, and the entire identification of Theorem 4.5 with Joyal's structures rests on it. This is not an internal inconsistency, but it is the least locally supported load-bearing step. The marked simplicial set comparison is supported by a lengthy in-paper analysis, so I do not see a comparable internal gap there. The reader's weakest assumption identifies the same step; my check would settle it by an independent derivation of Lemma 4.3. Since the cited result is standard and the abstract theory is sound, the verdict remains ACCEPT.","tokens_in":17762,"tokens_out":43847,"duration_ms":392900,"concrete_test":"Prove Lemma 4.3 from first principles: show that a morphism p:X→A has the right lifting property against every map Δ^1×∂Δ^n ∪ {1}×Δ^n → Δ^1×Δ^n if and only if it has the right lifting property against every horn inclusion Λ^n_k → Δ^n with 0<k≤n. Equivalently, exhibit each horn inclusion as a retract of a pushout of the cylinder maps and each cylinder map as a retract of a pushout of horn inclusions. A counterexample in either direction would break the identification in Theorem 4.5 and invalidate the claimed recovery of Joyal's model structures.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract construction (Theorems 2.17/2.18) is proved in detail and appears internally sound. The recovery of the known examples, however, rests on the identification of the abstract right I-anodyne extensions with the standard right anodyne maps. For simplicial sets this is Lemma 4.3, which asserts that the saturated class generated by Δ^1×∂Δ^n ∪ {1}×Δ^n → Δ^1×Δ^n (n≥0) equals the saturated class generated by the horn inclusions Λ^n_k → Δ^n for 0<k≤n. The lemma is cited from Cisinski and Lurie rather than proved. If this equality failed, the model structures of Theorem 4.5 would exist but their fibrant objects would not be right fibrations, so the claim that they are Joyal's Covariant and Contravariant structures would collapse. The marked simplicial analogue is argued in the paper (Lemmas 4.33–4.34) rather than cited, so the vulnerability is concentrated in the simplicial set example; still, that example is one of the paper's two advertised applications.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a general framework for constructing model structures from a functorial cylinder object equipped with a direction. For a locally presentable category with a cofibrantly generated weak factorization system, the author defines right and left homotopical structures and proves (Theorems 2.17 and 2.18) that each gives rise to a unique model structure, with cofibrations the given left class and weak equivalences detected by bijectivity on I-homotopy classes into the appropriate fibrant objects. The proof uses Simpson's recognition theorem and is carried out in detail. The framework is then applied to simplicial sets and marked simplicial sets: the right (respectively left) anodyne classes recover the standard right (respectively left) anodyne maps, yielding the Covariant and Contravariant model structures on slices of simplicial sets and the Cartesian and coCartesian model structures on slices of marked simplicial sets. The paper also introduces abstract notions of final and initial maps and of smooth and proper maps, and identifies them in the examples.","tokens_in":17971,"tokens_out":55905,"duration_ms":537775,"significance":"If the results stand, this is a useful unifying result: two families of model structures that are normally treated by separate arguments are shown to arise from one elementary homotopical datum. The central theorems are proven in detail, with the proof skeleton via Simpson's recognition theorem being coherent. The paper also gives a new description of marked right fibrations in Proposition 4.37, which has independent value. A particular strength is that the abstract construction is not circular: the model structures are built from the cylinder and anodyne data, and the known examples are then recovered rather than assumed. The identification with the standard model structures does rely on standard cited results (notably Lemma 4.3), but this is a normal and acceptable use of the literature.","major_comments":[],"minor_comments":[{"comment":"In the first bullet of the definition of left I-anodyne extensions, the notation appears to be incorrect: it reads \"Anr(I) = l(r(Λ))\" but should presumably read \"Anl(I) = l(r(Λ))\".","section":"Definition 2.11"},{"comment":"The citation in the proof is incomplete: \"[Lurie]\" should be a precise reference such as [Lur09] with a location. Since this lemma is the bridge from the abstract construction to the simplicial set examples, a precise citation or a short proof sketch would be helpful.","section":"Lemma 4.3"},{"comment":"The paper asserts that the model structures Cart(A,EA) and coCart(A,EA) are Lurie's Cartesian and coCartesian model structures, but the proof only identifies the cofibrations and the fibrant objects. A brief remark explaining that, since all objects are cofibrant, the weak equivalences are determined by the fibrant objects together with the cylinder homotopy relation would make the identification explicit and complete.","section":"Section 4.2, after Theorem 4.29"},{"comment":"In the 'if' direction, the verification of the right lifting property against the generators (B1) and (B2) is very compressed: the text says that (B1) follows immediately from assumptions (2) and (3) and that (B2) follows from the right cancellation property of Cartesian edges. Please spell out the lifting diagrams or give a precise reference for these two steps.","section":"Proposition 4.37"},{"comment":"The statement of condition (2) uses the same symbol f for the given marked edge in A and for its lift in X. Using a different symbol, such as \\bar f, for the lift would remove ambiguity.","section":"Proposition 4.37, condition (2)"},{"comment":"The conclusion that a map in /Rcal is a weak equivalence uses the implicit observation that a map with a homotopy section induces a bijection on I-homotopy classes into right I-fibrant objects. Adding one sentence making this observation explicit would improve readability.","section":"Section 2.27"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a well-written piece of work, and the central abstract construction appears sound. The main concerns are local: a few citations are imprecise, and the comparison with Lurie's model structures is asserted rather than fully spelled out. These are easy to fix and do not affect the validity of the main theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it is a serious, carefully written piece of formal category theory: the directed variant of Cisinski's axioms, where the right anodyne class only asks for closure under pushout product with the terminal endpoint map, is a genuine novelty, and the author proves the main existence theorems (2.17 and 2.18) in full detail. Second, the advertised payoff—that the abstract machine recovers Joyal's covariant/contravariant and Lurie's Cartesian/coCartesian model structures—is real, but the identification rests on one cited lemma for the simplicial set case, so the paper's value is in the framework rather than in new examples.\n\nWhat is genuinely new: the notion of right/left homotopical structure, the two model structures produced from a single cylinder by selecting a direction, and the abstract theory of final/initial and smooth/proper maps. The author also gives a new proof that coCartesian fibrations are proper (Corollary 4.46), which is a nice bonus. The expositions of Propositions 2.29 and 4.37 are detailed and the proofs look sound. The paper is honest about its debts: the introduction says most of the proofs are due to Cisinski in a less general setting, and the author's contribution is the generalization and the systematic treatment of direction.\n\nThe soft spots are in proportion. Lemma 4.3, which identifies the saturated class generated by the cylinder maps with the right anodyne extensions, is cited from Cisinski and Lurie rather than proved. This is a standard result, so it is not a flaw, but it is load-bearing: if it failed, the model structures from Theorem 4.5 would still exist but would not be Joyal's. The marked simplicial set case is argued internally (Lemmas 4.33–4.34), so the vulnerability is concentrated in the unmarked example. The only other issue is minor: the text has a handful of formatting/rendering artifacts, and the main examples are already in the literature, so the novelty budget is spent on the formalism.\n\nWho should read this: anyone working in Cisinski–Olschok theory or comparing model structures on presheaf categories. It would make a solid reading-group selection for a group comfortable with locally presentable categories and the small object argument. I would send it to a serious referee; it is not a desk reject. Minor revisions, mostly clarifying the reliance on Lemma 4.3 and fixing typos, would be enough.\n\nMy verdict: accept.","headline":"A careful and valuable formalization of directed model structures, but the promised identification with the standard examples depends on a cited lemma.","tokens_in":18461,"tokens_out":2529,"would_cite":true,"duration_ms":25762,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18N40"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a functorial cylinder object with a chosen direction yields two model structures, one covariant and one contravariant, and that these recover the standard directed homotopy theories of simplicial and marked…","keywords":["model categories","locally presentable categories","functorial cylinder","anodyne extensions","covariant model structure","contravariant model structure","simplicial sets","marked simplicial sets"],"falsifier":"Compute the weak saturation of the set $\\{\\Delta^1\\times\\partial\\Delta^n\\cup\\{1\\}\\times\\Delta^n\\to\\Delta^1\\times\\Delta^n:n\\ge0\\}$ in simplicial sets and compare it with the saturation of $\\{\\Lambda^n_k\\to\\Delta^n:0<k\\le n\\}$: equality is exactly Lemma 4.3, so finding a horn inclusion with $k=0$ or $k=n$ in the first class, or a cylinder map outside the second, would disconnect the paper's model structures from the standard Covariant and Contravariant ones.","tokens_in":17580,"feed_emoji":"🧭","tokens_out":17208,"duration_ms":148913,"temperature":0.7,"pith_summary":"This paper claims that the direction of a cylinder object can be made into a structural feature of homotopy theory. Starting from a locally presentable category with an exact functorial cylinder $I$, it constructs two model structures: a Contravariant one built from the second endpoint $\\partial_1$ of the cylinder, and a Covariant one built from the first endpoint $\\partial_0$. The main theorem states that such a structure is unique—cofibrations are the given left class, weak equivalences are exactly the maps inducing bijections $f^*:[B,W]_I\\to[A,W]_I$ on $I$-homotopy classes into every right (or left) $I$-fibrant object, and fibrant objects are the right (or left) $I$-fibrant objects. Applied to the cylinder $\\Delta^1\\times(-)$ on simplicial sets, the two constructions become the Contravariant and Covariant model structures on slices $sSet/A$, with fibrant objects the right and left fibrations; applied to the marked cylinder $(\\Delta^1)^\\sharp\\times(-)$ on marked simplicial sets, they become the Cartesian and coCartesian model structures. The same formalism defines final and initial maps and smooth and proper maps, and identifies them in the examples.","feed_headline":"Two model structures from one directed cylinder","feed_subtitle":"One directed cylinder yields covariant and contravariant model structures for simplicial and marked simplicial sets.","key_machinery":"The load-bearing mechanism is the exact functorial cylinder object $I$ on a locally presentable category, together with a chosen class of right (or left) $I$-anodyne extensions. A functorial cylinder is an endofunctor $I$ with two natural sections $\\partial_0,\\partial_1:\\mathrm{id}\\Rightarrow I$ and a projection $\\sigma:I\\Rightarrow \\mathrm{id}$; exactness means $I$ preserves colimits and carries the left class of the weak factorization system into itself under the pushout-product operations $\\partial_I\\boxtimes(-)$ and $\\partial_i\\boxtimes(-)$. The direction enters by requiring only one of the two endpoint inclusions to generate anodynes, making the $I$-homotopy category $[X,W]_I$ sensitive to the cylinder's orientation. The proof then uses a recognition theorem for combinatorial model categories: the weak equivalences defined by bijectivity of $f^*$ are shown to fit the hypotheses, and the small-object argument supplies the factorizations. In the examples, the cylinder-generated anodyne class is identified with the classical right anodyne horn inclusions and with the marked analogue, which is what turns the abstract model structures into the known ones.","core_discovery":"The central discovery is Theorem 2.17 and its dual: a right homotopical structure—an exact functorial cylinder together with a chosen class of right $I$-anodyne extensions inside the left class $\\mathcal L$—determines a unique model structure on $\\mathcal C$. A morphism $f:A\\to B$ is a weak equivalence precisely when $f^*:[B,W]_I\\to[A,W]_I$ is bijective for every right $I$-fibrant object $W$; the cofibrations are exactly $\\mathcal L$, the fibrant objects are exactly the right $I$-fibrant objects, and fibrations between fibrant objects are exactly right $I$-fibrations. The asymmetry is the point: for right anodynes only $\\partial_1\\boxtimes i$ is required to be anodyne for every cofibration $i$, and for left anodynes only $\\partial_0\\boxtimes i$, so the homotopy theory remembers which end of the interval is used. In the examples this reproduces the standard Covariant and Contravariant model structures on $sSet/A$ and the Cartesian and coCartesian model structures on $sSet^+/(A,E_A)$, with right fibrations and marked right fibrations as the fibrant objects.","pith_inferences":["Because the construction needs only an exact cylinder and a chosen class of anodyne maps, the same two-model-structure recipe should work in any locally presentable category with a directed interval, not just presheaf categories; checking a new example means verifying exactness and the saturation identity for the anodyne class.","The examples hinge on Lemma 4.3, which identifies the cylinder-generated class with classical right anodyne maps; replacing $\\Delta^1$ with a different directed interval would define new Covariant and Contravariant homotopy theories, and their fibrant objects could be compared with known classes—the paper mentions such a variant only in passing.","The theorem's description of fibrations between fibrant objects is a general payoff of the abstract approach: in any new instance, fibrations are known as soon as the right $I$-fibrations are understood, which is typically the hardest data of a model structure."],"forward_implications":["Any locally presentable category with an exact cylinder and a set of generating maps carries two model structures, Contravariant and Covariant, with cofibrations the given left class; the cylinder alone determines the weak equivalences.","On $sSet/A$, the two model structures have precisely the right fibrations and left fibrations over $A$ as fibrant objects, recovering the standard Covariant and Contravariant homotopy theories of quasi-categories.","On marked simplicial sets, the formalism produces the Cartesian and coCartesian model structures; a map is a marked right fibration exactly when its underlying map is an inner fibration, it lifts marked edges, and its marked edges are precisely the Cartesian edges over marked edges, so $X\\to A$ is a Cartesian fibration iff $X^\\natural\\to A^\\sharp$ is a marked right fibration.","A map is final exactly when it factors as a right anodyne extension followed by a trivial fibration, and initial dually; consequently left fibrations are proper, right fibrations are smooth, and coCartesian fibrations are proper while Cartesian fibrations are smooth with respect to the simplicial datum."],"supporting_citations":[{"why":"Provides the topos-level cylinder/anodyne machine whose axioms are reoriented so that the cylinder direction matters.","marker":"[Cis02]"},{"why":"Establishes the saturated-class theory of I-anodyne extensions that the paper adapts to right and left classes.","marker":"[Cis06]"},{"why":"Supplies the proof framework for Section 2 and the lemma identifying the cylinder-generated class with right anodyne maps.","marker":"[Cis19]"},{"why":"Introduces the Covariant and Contravariant model structures on slices of simplicial sets that Theorem 4.5 recovers.","marker":"[Joy08b]"},{"why":"Defines Cartesian and coCartesian fibrations and the corresponding model structures on marked simplicial sets, recovered here.","marker":"[Lur09]"},{"why":"Extends the topos-level cylinder/anodyne construction to locally presentable categories, providing the ambient setting.","marker":"[Ols11]"},{"why":"Contains the recognition theorem for combinatorial model categories used as the existence criterion.","marker":"[Sim12]"},{"why":"Supplies exactness and colimit-universality facts for marked simplicial sets used in the cylinder argument.","marker":"[Ver08]"}],"fun_headline_variants":["Directed cylinder yields dual model structures","One cylinder, two homotopy theories","Covariant and contravariant from a single cylinder","Right and left anodynes shape model structures"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the saturated class generated by the cylinder maps $\\Delta^1\\times\\partial\\Delta^n\\cup\\{1\\}\\times\\Delta^n\\to\\Delta^1\\times\\Delta^n$ is exactly the classical class of right anodyne horn inclusions (and that the corresponding equality holds on the marked side); if that combinatorial equality failed, the abstract construction would still produce model structures, but they would not be the standard Covariant, Contravariant, Cartesian, and coCartesian ones.","fun_headline_variants_meta":{"raw":{"variants":["Directed cylinder yields dual model structures","One cylinder, two homotopy theories","Covariant and contravariant from a single cylinder","Right and left anodynes shape model structures"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000209,"raw_usage":{"total_tokens":1375,"prompt_tokens":882,"completion_tokens":493,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":436}},"tokens_in":498,"tokens_out":493,"duration_ms":4701,"temperature":1.0,"reasoning_tokens":436,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:32:52.389157+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the weak saturation of the set $\\{\\Delta^1\\times\\partial\\Delta^n\\cup\\{1\\}\\times\\Delta^n\\to\\Delta^1\\times\\Delta^n:n\\ge0\\}$ in simplicial sets and compare it with the saturation of $\\{\\Lambda^n_k\\to\\Delta^n:0<k\\le n\\}$: equality is exactly Lemma 4.3, so finding a horn inclusion with $k=0$ or $k=n$ in the first class, or a cylinder map outside the second, would disconnect the paper's model structures from the standard Covariant and Contravariant ones.","supporting_citations":[],"review_version":1}