REVIEW 6 minor 10 references
Covariant & Contravariant Homotopy Theories
T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read 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…
desk verdict A careful and valuable formalization of directed model structures, but the promised identification with the standard examples depends on a cited lemma. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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.
minor comments (6)
- [Definition 2.11] 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))".
- [Lemma 4.3] 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 4.2, after Theorem 4.29] 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.
- [Proposition 4.37] 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.
- [Proposition 4.37, condition (2)] 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 2.27] 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.
Circularity Check
No significant circularity: the abstract model-structure construction is self-contained, and the identification with Joyal's and Lurie's examples rests on independent external results, not on self-citation or fitted inputs.
full rationale
The paper's central construction is not circular. An elementary homotopical datum (a functorial cylinder plus a set S) generates right and left I-anodyne extensions by an explicit inductive saturation in Construction 1, and Theorems 2.17 and 2.18 prove, via Simpson's recognition theorem and a sequence of lemmas, that any such structure yields a model structure with cofibrations precisely the given class L and weak equivalences characterized by bijectivity of f^*: [B,W]_I -> [A,W]_I for all right (or left) I-fibrant objects W. The weak equivalences are not defined as 'whatever makes the construction work'; they are defined through the I-homotopy category and then shown to match Simpson's hypotheses. No parameter is fitted to a subset of data, and no output is a renamed input. The recovery of Joyal's Covariant/Contravariant model structures and Lurie's (co)Cartesian model structures depends on external comparisons, most notably Lemma 4.3, which identifies the saturated class generated by the cylinder maps with the standard right anodyne horn inclusions and is cited to Cisinski and Lurie. This is an ordinary reliance on established external mathematics, not a self-citation or an ansatz smuggled in by the present author; it is also the kind of independent support that hard rule 4 explicitly treats as real evidence. The marked simplicial set comparison is argued in the paper itself (Lemmas 4.33-4.34 and Proposition 4.37). Thus there is no circular step, no load-bearing self-citation chain, and no definitional equivalence between a theorem and its input.
Assumptions & free parameters
assumptions (5)
- domain assumption C is a locally presentable category with a cofibrantly generated weak factorization system (L,R), and for every object X the canonical map from the initial object to X lies in L.
- domain assumption The cylinder I is exact with respect to (L,R): it commutes with small colimits, and pushout products of maps in L with the cylinder remain in L.
- domain assumption A right or left homotopical structure includes a class of anodyne extensions generated by a small set and closed under the directed pushout product rules.
- standard math Simpson's recognition theorem for combinatorial model categories is applicable.
- standard math Known results about simplicial sets and marked simplicial sets, including Lemma 4.3 on right anodyne generators and Propositions 4.14 through 4.17 from Lurie, are assumed.
Cite this review
Pith. "Pith review of Covariant & Contravariant Homotopy Theories." pith.science (2026). https://pith.science/paper/4CXAQO62
@misc{pith2026190806879,
author = {Pith},
title = {Pith review of: Covariant & Contravariant Homotopy Theories},
year = {2026},
howpublished = {\url{https://pith.science/paper/4CXAQO62}},
note = {Machine review of arXiv:1908.06879}
}
read the original abstract
Given a locally presentable category together with a suitable functorial cylinder object, we construct model structures which are sensitive to the `direction' of the cylinder. We show that the Covariant and Contravariant model structures on simplicial sets as well as the coCartesian and Cartesian model structures on marked simplicial sets are examples of our formalism. In this setting, notions of final and initial maps and smooth and proper maps arise very naturally and we will identify these maps in the examples.
Reference graph
Works this paper leans on
-
[1]
Th \'e ories homotopiques dans les topos
Denis-Charles Cisinski. Th \'e ories homotopiques dans les topos. J. Pure Appl. Algebra , 174(1):43--82, 2002
work page 2002
-
[2]
Les pr \'e faisceaux comme mod \`e les des types d'homotopie
Denis-Charles Cisinski. Les pr \'e faisceaux comme mod \`e les des types d'homotopie. Ast \'e risque , (308):xxiv+390, 2006
work page 2006
-
[3]
Higher categories and homotopical algebra , volume 180 of Cambridge studies in advanced mathematics
Denis-Charles Cisinski. Higher categories and homotopical algebra , volume 180 of Cambridge studies in advanced mathematics . Cambridge University Press, Cambridge, 2019
work page 2019
- [4]
-
[5]
The theory of quasi-categories and its applications
Andr\' e Joyal. The theory of quasi-categories and its applications. preprint , 2008
work page 2008
-
[6]
Higher topos theory , volume 170 of Annals of Mathematics Studies
Jacob Lurie. Higher topos theory , volume 170 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 2009
2009
-
[7]
Jacob Lurie. Higher algebra. http://www.math.harvard.edu/ lurie/papers/HA.pdf , 2017
work page 2017
-
[8]
Left determined model structures for locally presentable categories
Marc Olschok. Left determined model structures for locally presentable categories. Appl. Categ. Structures , 19(6):901--938, 2011
work page 2011
Show all 10 references
-
[9]
Homotopy theory of higher categories , volume 19 of New Mathematical Monographs
Carlos Simpson. Homotopy theory of higher categories , volume 19 of New Mathematical Monographs . Cambridge University Press, Cambridge, 2012
2012
-
[10]
D. R. B. Verity. Weak complicial sets. I . B asic homotopy theory. Adv. Math. , 219(4):1081--1149, 2008
2008
Reviewed August 14, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.