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Model Structures on \( \sSet \)

Table of Contents

See Model Categories for a definition of model structures and model categories.

There are two important model structures on the category of simplicial sets: the Kan model structure (or model structure for spaces) and the Joyal model structure (or model structure for \(\infty \)-categories).

1. Model Structure for Spaces

The model structure for spaces (Kan model structure) on \(\sSet \) is given by

  • the cofibrations are exactly the monomorphisms,
  • the weak equivalences are exactly the weak homotopy equivalences,
  • fibrations are exactly the Kan fibrations.

To make it easier to distinguish between different model structures we will write Kan equialence instead of weak homotopy equivalence.

2. Model Structure for \(\infty \)-Categories

The model structure for \(\infty \)-categories (Joyal model structure) on \(\sSet \) is given by

  • the cofibrations are exactly the monomorphisms,
  • the fibrations are exactly the Joyal fibrations (as defined in Land)(??),
  • the weak equivalences are exactly the Joyal equivalences.

To make it easier to distinguish between different model structures we will write Joyal fibration instead of isofibration.

We are not sure that these are the correct fibrations. nLab says that the fibrations (in the model structure for \(\infty \)-categories) between \(\infty \)-categories are exactly the isofibrations. The isofibrations between \(\infty \)-categories are exactly the Joyal fibrations as defined in Land, but Land’s definition of a Joyal fibration makes sense for all simplicial sets. Therefore we suspect that the fibration in the Joyal model structure are exactly the Joyal fibrations. When in doubt we can just define the fibrations as \(\chi_R(\text{trivial Joyal cofibrations}) \). But we are still left with a potentially overloaded notation.

The fibrant objects with respect to the Joyal model structure are precisely the \(\infty \)-categories.

  1. The trivial Joyal fibration are exactly the trivial Kan fibration.
  2. The Joyal fibrations between \(\infty \)-categories are exactly the isofibrations.

Author: Frederik Gebert

Created: 2025-02-10 Mon 21:45