Cocartesian Model Structures in Higher Categories

• Cocartesian model structures are specialized model categories that encode covariant (cocartesian) fibrations, organizing objects over a fixed base with controlled liftings.
• They use precise Quillen-theoretic characterizations with generating cofibrations and trivial cofibrations to ensure compatibility with standard homotopical operations.
• Key applications include the straightening–unstraightening equivalence and Grothendieck constructions in derived algebraic geometry and higher category theory.

A cocartesian model structure is a type of model category structure that encodes the homotopy theory of objects organized above a base category or base object, subject to a cocartesian (covariant) fibration condition. Such structures provide a unified homotopical framework to treat families of (higher) categories, functoriality, and base-change phenomena, arising centrally in Grothendieck's program and in derived algebraic geometry via Lurie's $\infty$-categorical straightening-unstraightening theory. Cocartesian model structures have been constructed on a variety of categories—most notably on slices of marked simplicial sets, marked cubical sets, complex diagrams of model categories, and various enriched and higher-categorical extensions—with precise Quillen-theoretic characterizations ensuring robust compatibility with standard homotopical manipulations. They play a critical role in organizing and comparing homotopy theories varying over families and in presenting higher categorical Grothendieck constructions.

1. Fundamental Definitions and Model-Categorical Frameworks

The cocartesian model structure universally arises in contexts where objects are considered together with a morphism to a fixed base $S$, with structure-preserving morphisms. In the standard setting of marked simplicial sets, the category $\mathrm{sSet}^+_{/(A, E_A)}$ consists of maps $p:(X, E_X)\to (A, E_A)$ where $E_X$ (resp. $E_A$) are sets of marked edges containing all degenerate edges. The cocartesian model structure has:

• Cofibrations: All monomorphisms over $(A, E_A)$.
• Fibrations: Morphisms with the right lifting property against the left anodyne maps (certain marked horn inclusions and related morphisms).
• Fibrant objects: Marked left fibrations, i.e., $p:X\to A$ is an inner fibration and the marked edges in $X$ are precisely the $p$-cocartesian edges over marked edges in $A$.
• Weak equivalences: Morphisms over $(A, E_A)$ that induce isomorphisms in the homotopy category obtained by inverting the left anodyne morphisms or, equivalently, by mapping into fibrant objects (Nguyen, 2019, Nguyen, 2022).

Explicitly, the generating trivial cofibrations include left-anodyne maps, which encode the cocartesian lifting properties central to the model's combinatorial and homotopical behavior (Nguyen, 2019).

Analogously, for marked cubical sets $\mathrm{cSet}^+_{/\overline S}$, the construction parallels the simplicial case: cofibrations are all monomorphisms, and fibrant objects are the cocartesian (covariant) fibrations over $\overline S$ (marked cubical set). The weak equivalences are determined by mapping-space criteria into fibrant objects (Arakawa et al., 16 Nov 2025).

2. Straightening–Unstraightening and Grothendieck Constructions

One of the central applications of cocartesian model structures is to model and present the straightening–unstraightening (Grothendieck construction) equivalence:

• For a base ordinary category $\mathcal{C}$, the straightening functor $\mathrm{St}_\mathcal{C}$ establishes a Quillen equivalence

$$\mathrm{St}_\mathcal{C}: \mathrm{sSet}^+_{/N(\mathcal{C})^\sharp} \longleftrightarrow \mathrm{Fun}(\mathcal{C}, \mathrm{sSet}): \mathrm{Un}_\mathcal{C}$$

under the projective model structure on the functor category, identifying coCartesian fibrations with homotopically coherent diagrams and functors (Nguyen, 2022).

• In $\infty$-categorical settings, applying the homotopy coherent nerve produces equivalences with the $\infty$-categorical Grothendieck construction, with the cocartesian model structure presenting the lax colimit of a diagram in $\mathrm{Cat}_\infty$ (Harpaz et al., 2014).
• For $(\infty,2)$-categories, combinatorial models (via marked-biscaled simplicial sets) yield a straightening–unstraightening equivalence between the $(\infty,2)$-category of local cocartesian fibrations over a scaled simplicial set $S$ and the $(\infty,2)$-category of covariant functors from $S$ to $(\infty,2)$-categories. The construction is combinatorially explicit and fully compatible with oplax functoriality and the oplax Yoneda embedding (Abellán, 2023).

The Grothendieck construction for model categories (Harpaz et al., 2014) establishes an equivalence between proper relative diagrams of model categories and model fibrations, with the integral category $\int_\mathcal{M}\mathcal{F}$ carrying a canonical model structure encoding the cocartesian behavior.

3. Generating Sets, Classes, and Lifting Properties

The combinatorial characterization of cocartesian model structures relies on precisely determined generating cofibrations and trivial cofibrations:

Category	Generating Cofibrations	Generating Trivial Cofibrations (Left Anodyne)
$\mathrm{sSet}^+$	$$(\partial\Delta^n)^\flat\to(\Delta^n)^\flat,\ (\Delta^1)^\flat\to(\Delta^1)^\sharp$$	Horn inclusions $(\Lambda^n_k)^\flat\to(\Delta^n)^\flat, 0<k<n$; $J^\flat\to J^\sharp$; pushout-products
$\mathrm{cSet}^+$	$$\partial\square^n{}^\flat\to\square^n{}^\flat, \{0\}^\sharp\to\square^1{}^\sharp$$	(ML1–ML5): marked inner open box inclusions, equivalence markings, 2-cube scenarios
$(\infty,2)$-MBSC	Boundary inclusions, marking, and scaling maps	MB-anodyne morphisms: inner horn inclusions, 4-simplex relations, marking thin triangles, etc.

The weak equivalences are detected via mapping into (marked) cocartesian fibrations, and fibrations are those with the right lifting property relative to the trivial cofibrations. Properness (left and right) of the cocartesian model structure is established for these contexts (Nguyen, 2022, Nguyen, 2019, Arakawa et al., 16 Nov 2025).

4. Integral Model Structures and Model Fibrations

The "integral" cocartesian model structure (following Harpaz–Prasma (Harpaz et al., 2014)) is constructed on the Grothendieck construction of a proper relative diagram

$\mathcal{F}: \mathcal{M} \to \mathrm{ModCat}$

with objects pairs $(A, X)$ for $A\in \mathcal{M}$, $X \in \mathcal{F}(A)$ and morphisms $(f, \varphi): (A, X) \to (B, Y)$ as pairs of morphisms in base and fiber with compatibility. The key distinguished classes are:

• Weak equivalences: $f$ is a weak equivalence in $\mathcal{M}$, and the induced $$\varphi\circ f_!(q_X): f_!(X^{\mathrm{cof}})\rightarrow Y$$ is a weak equivalence in $\mathcal{F}(B)$.
• Cofibrations: $f$ is a cofibration in $\mathcal{M}$, and $\varphi: f_! X \rightarrow Y$ is a cofibration in $\mathcal{F}(B)$.
• Fibrations: $f$ is a fibration in $\mathcal{M}$, and the adjoint $\varphi^\sharp: X \to f^* Y$ is a fibration in $\mathcal{F}(A)$.

This structure organizes variable model categories and forms a Quillen presentation of Lurie's homotopy-coherent Grothendieck construction (Harpaz et al., 2014).

5. Main Examples and Applications

Several critical classes of examples admit cocartesian model structures:

• Simplicial and Cubical Models: Both marked simplicial sets and marked cubical sets slices admit cocartesian model structures. There exist Quillen equivalences connecting the cubical and simplicial theories, including through geometric realization and via monoidal left Quillen functors satisfying certain unit axioms (Arakawa et al., 16 Nov 2025).
• Group Actions: The category of simplicial sets with a group action, via diagrams over simplicial groups, carries a global cocartesian model structure encoding both strict and weak actions, with push-forward and pull-back functors being Quillen adjoints. The integral model structure lifts the classical action theory to the homotopical setting (Harpaz et al., 2014).
• Modules over Algebras: Categories of modules over associative or commutative algebras in a symmetric monoidal model category, under Schwede–Shipley and related axioms, have natural cocartesian model structures on slices indexed by the category of algebras. The fibers carry transferred model structures, and the push-pull mechanisms along algebra maps are Quillen (Harpaz et al., 2014).
• Higher-Categorical Presentations: The theory extends to marked-biscaled simplicial sets for modeling local cocartesian fibrations over scaled simplicial sets, giving a combinatorial presentation for Grothendieck-type constructions and key results in $(\infty,2)$-category theory including oplax straightening (Abellán, 2023).

6. Compatibility, Properness, and Product Structures

Cocartesian model structures are left and right proper in all major settings: pushouts along cofibrations and pullbacks along fibrations preserve weak equivalences, an essential feature for stability and base change. The model structures are generally combinatorial, simplicial, and left proper, supporting small-object arguments and further Quillen-theoretic developments (Nguyen, 2022, Nguyen, 2019, Arakawa et al., 16 Nov 2025).

Importantly, straightening–unstraightening commutes with cartesian products; for nerves of categories, cocartesian model structures on products of base categories correspond to products of straightened diagrams: $$\mathrm{St}_{\mathcal{C} \times \mathcal{D}}(F) \cong \mathrm{St}_{\mathcal{C}}(F|_{\mathcal{C}}) \times \mathrm{St}_{\mathcal{D}}(F|_{\mathcal{D}})$$ and similarly for cubical and higher-categorical analogs (Nguyen, 2022).

7. Advanced Directions and Homotopical Consequences

The interaction of cocartesian model structures with monoidal model categories allows for robust applications, such as the cubical Bousfield–Kan formula for homotopy colimits in terms of geometric products of cubes and left Quillen actions, contingent on Muro’s unit axiom (Arakawa et al., 16 Nov 2025). In higher dimensional category theory, the precise combinatorics of (local) cocartesian fibrations drive the development of oplax Yoneda embeddings and present explicit structures required for enriched Yoneda lemmas in the $(\infty,2)$-setting (Abellán, 2023).

A plausible implication is that the abundance of Quillen equivalent presentations (simplicial, cubical, model-categorical, $(\infty,2)$-bicategorical) underscores the robust invariance properties of cocartesian model structures under categorical equivalence of the base and of the fibers, making them an essential invariant in modern homotopy theory and higher category theory.

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References:

• (Harpaz et al., 2014) The Grothendieck construction for model categories
• (Nguyen, 2022) A note on coCartesian fibrations
• (Nguyen, 2019) Covariant & Contravariant Homotopy Theories
• (Arakawa et al., 16 Nov 2025) Cubical models of $\infty$-presheaves and the Bousfield-Kan formula
• (Abellán, 2023) On local fibrations of $(\infty,2)$-categories