[Paper Review] Homotopy theory of higher categories

Carlos Simpson|arXiv (Cornell University)|Jan 22, 2010

Homotopy and Cohomology in Algebraic Topology141 references77 citations

TL;DR

This paper develops a homotopy-theoretic framework for higher categories using iterated Segal's method, constructing a tractable, left proper, cartesian model structure on M-precategories for any tractable left proper cartesian model category M. The key contribution is a systematic construction of model categories for (∞,n)-categories through iterative application of this method, providing a foundational tool for higher category theory.

ABSTRACT

This is the first draft of a book about higher categories approached by iterating Segal's method, as in Tamsamani's definition of $n$-nerve and Pelissier's thesis. If $M$ is a tractable left proper cartesian model category, we construct a tractable left proper cartesian model structure on the category of $M$-precategories. The procedure can then be iterated, leading to model categories of $(\\infty, n)$-categories.

Motivation & Objective

• To establish a systematic homotopy-theoretic framework for higher categories using iterative applications of Segal's method.
• To define and study M-precategories in a tractable left proper cartesian model category M.
• To construct a tractable, left proper, cartesian model structure on the category of M-precategories.
• To generalize this construction iteratively to obtain model categories of (∞,n)-categories.
• To provide a coherent, foundational model for higher category theory suitable for advanced homotopical algebra.

Proposed method

• Applying Segal’s method iteratively to define higher categorical structures via simplicial objects.
• Using Tamsamani’s n-nerve and Pelissier’s thesis as foundational references for the iterative construction.
• Defining M-precategories as simplicial objects in M satisfying Segal-type conditions.
• Establishing a model structure on M-precategories that is tractable, left proper, and cartesian.
• Verifying that the model structure is compatible with the iterative process across levels.
• Extending the construction to define model categories of (∞,n)-categories through n-fold iteration.

Experimental results

Research questions

• RQ1 How can Segal’s method be systematically iterated to model higher categories?
• RQ2 What conditions on a model category M ensure the existence of a well-behaved model structure on M-precategories?
• RQ3 Can a tractable, left proper, cartesian model structure be constructed on the category of M-precategories?
• RQ4 How does the iterative application of this construction yield model categories for (∞,n)-categories?
• RQ5 What properties are preserved through the iteration process to ensure coherence in the higher categorical framework?

Key findings

• A tractable, left proper, cartesian model structure is constructed on the category of M-precategories for any tractable left proper cartesian model category M.
• The construction is iterative, allowing the definition of model categories for (∞,n)-categories via n-fold application of the method.
• The framework generalizes Tamsamani’s n-nerve and builds on Pelissier’s thesis to provide a coherent higher categorical model.
• The resulting model categories are suitable for homotopical algebra and provide a foundation for studying (∞,n)-categories.
• The method ensures that key homotopical properties such as left properness and tractability are preserved at each level of iteration.
• The approach provides a systematic and structured pathway to higher category theory using model-theoretic techniques.