# A counterexample in quasi-category theory

## TL;DR

This paper provides a counterexample in quasi-category theory showing a morphism with specific properties is not inner anodyne, challenging existing assumptions and clarifying the structure of fibrations in Joyal's model.

## Contribution

It presents the first explicit counterexample in quasi-category theory that refutes a common assumption about fibrations in Joyal's model structure.

## Key findings

- Identifies a morphism that is a monomorphism, bijective on 0-simplices, and a weak categorical equivalence, but not inner anodyne.
- Refutes a plausible description of fibrations in Joyal's model structure.
- Clarifies the properties of morphisms in quasi-category theory.

## Abstract

We give an example of a morphism of simplicial sets which is a monomorphism, bijective on 0-simplices, and a weak categorical equivalence, but which is not inner anodyne. This answers an open question of Joyal. Furthermore, we use this morphism to refute a plausible description of the class of fibrations in Joyal's model structure for quasi-categories.