[Paper Review] Fibrations in $\infty$-category theory

Clark Barwick, Jay Shah|arXiv (Cornell University)|Jul 14, 2016

Homotopy and Cohomology in Algebraic Topology4 references3 citations

TL;DR

This paper provides a comprehensive, expository overview of eight types of fibrations in quasicategory theory—left, right, Kan, inner, isofibration, cocartesian, cartesian, and flat fibrations—demonstrating how they enable explicit, computationally tractable constructions in $∞$-category theory. The key contribution is the systematic use of these fibrations to realize powerful, concrete constructions that complement universal characterizations, particularly through the use of categorical patterns and homotopy-invariant functors.

ABSTRACT

In this short expository note, we discuss, with plenty of examples, the bestiary of fibrations in quasicategory theory. We underscore the simplicity and clarity of the constructions these fibrations make available to end-users of higher category theory.

Motivation & Objective

• To clarify and systematize the diverse types of fibrations used in quasicategory theory, which are essential for explicit constructions in $∞$-category theory.
• To highlight the underappreciated power of fibrations in enabling concrete, computable constructions that avoid complex homotopy coherence issues.
• To demonstrate how fibrations—especially cocartesian and cartesian fibrations—facilitate the construction of functors and adjunctions in $∞$-categories.
• To establish a bridge between abstract universal properties (e.g., free generation of $∞$-categories) and explicit, functorial constructions via fibrations.
• To provide a practical guide for researchers by illustrating how fibrations simplify and clarify complex constructions in higher category theory.

Proposed method

• The paper classifies and compares eight types of fibrations in quasicategories, including left, right, Kan, inner, isofibration, cocartesian, cartesian, and flat fibrations, with detailed examples.
• It employs categorical patterns $π^*\rho_*$ and $(ρ')_*(π')^*$ to model functors between $∞$-categories and analyze their behavior under homotopy equivalences.
• The authors use left Quillen functors and homotopy equivalences between simplicial sets to show that pullbacks and pushforwards preserve weak equivalences under suitable conditions.
• They apply results from Lurie’s work [7] and [6, 1.4.4(b)] to prove that induced natural transformations between functors are weak equivalences when the underlying map is a homotopy equivalence.
• The construction of fibrations is grounded in the theory of quasicategories, using inner fibrations and the lifting properties of cocartesian and cartesian morphisms.
• The paper leverages the fact that homotopy equivalences in $s\textbf{Set}^+_{/\mathfrak{P}_{C'}}$ induce weak equivalences on associated functors, ensuring stability of constructions under homotopy.

Experimental results

Research questions

• RQ1 How do different types of fibrations in quasicategories facilitate explicit constructions in $∞$-category theory?
• RQ2 What conditions ensure that a natural transformation induced by a homotopy equivalence between simplicial sets is a weak equivalence on the level of functors?
• RQ3 How do categorical patterns $π^*\rho_*$ and $(ρ')_*(π')^*$ model functors between $∞$-categories and preserve weak equivalences?
• RQ4 In what way do left and right Quillen functors interact with fibrations to ensure homotopy-invariant behavior?
• RQ5 How do cocartesian and cartesian fibrations support the construction of adjoint functors in the $∞$-categorical setting?

Key findings

• The natural transformation $\rho_!\pi^* \to (\rho')_!(\pi')^*$ induced by a homotopy equivalence $f$ is a weak equivalence on all objects, ensuring homotopy-invariance of the construction.
• The adjoint natural transformation $(\pi')_*\rho'^* \to \pi_*\rho^*$ is a weak equivalence on all fibrant objects, confirming stability under duality.
• Homotopy equivalences $f$ induce homotopy equivalences $\text{id}_X \times_C f$ on fiber products, preserving the structure of the fibration over $C$.
• The functors $\pi^*\rho_*$ and $(\rho')_*(\pi')^*$ are left Quillen with respect to the model structures defined by categorical patterns $\mathfrak{P}_C$ and $\mathfrak{P}_{C'}$, ensuring compatibility with homotopy theory.
• The pullback of a homotopy equivalence along any map $X \to C$ results in a homotopy equivalence $X \times_C K \to X \times_C L$, which is crucial for preserving weak equivalences in fibered constructions.
• The use of fibrations allows for explicit, computable constructions in $∞$-category theory that avoid the need for intricate homotopy coherence machinery.