How to construct a Segal category from a quasicategory?

Question

The following are both models for $(\infty,1)$-categories:

• quasicategories;
• Segal categories.

Given the above, I was wondering how to set up an equivalence (isomorphism?) between quasicategories and Segal categories? At least, given a quasicategory, what is its corresponding Segal category?

Would anyone know of a source/reference that discusses this? Any help would be much appreciated.

Answer 1

The canonical reference is the paper

Quasi-categories vs Segal spaces

by André Joyal and Myles Tierney.

It works in the setting of Segal spaces, but there are standard tools to move between Segal spaces and Segal categories. For example, see the book

The Homotopy Theory of (∞,1)-Categories

by Julia E. Bergner.

Answer 2

The most natural way of constructing Segal categories from quasicategories is to go via (complete) Segal spaces.

Let $I^n$ be the (nerve of) the contractible groupoid $\{ 0 \leftrightarrows 1 \cdots \leftrightarrows n \}$. Given a simplicial set $X$ we may define a bisimplicial set $t^! (X)$ by the following formula: $$t^! (X)_{m, n} = \textrm{Hom} (\Delta^m \times I^n, X)$$

Joyal and Tierney [2007, Quasi-categories vs Segal spaces] proved the following:

Theorem. The functor $t^! : \textbf{sSet} \to \textbf{ssSet}$ is a right Quillen equivalence from the model structure for quasicategories to the model structure for complete Segal spaces. In particular, if $X$ is a quasicategory, then $t^! (X)$ is a complete Segal space.

To extract a Segal category from a Segal space is easy. Given a bisimplicial set $Y$, let $R (Y)$ be the bisimplicial subset of $Y$ where $y \in Y_{m, n}$ is in $R (Y)_{m, n}$ iff every iterated face operator $Y_{m, n} \to Y_{0, n}$ sends $y$ into the image of the iterated degeneracy operator $Y_{0, 0} \to Y_{0, n}$. This is the largest bisimplicial subset of $Y$ such that the simplicial set $R (Y)_0$ is discrete, and $Y \mapsto R (Y)$ is easily seen to be a functor.

Recall that a Segal precategory is a bisimplicial set $Z$ such that the simplicial set $Z_0$ is discrete. Let $\textbf{PC}$ be the full subcategory of Segal precategories. Bergner [2006, Three models for the homotopy theory of homotopy theories] proved the following:

Theorem. The functor $R : \textbf{ssSet} \to \textbf{PC}$ is a right Quillen equivalence from the model structure for complete Segal spaces to the model structure for Segal categories. In particular, if $Y$ is a complete Segal space, then $R (Y)$ is a Segal category.

Putting the two theorems together, we obtain a right Quillen equivalence $R t^! : \textbf{sSet} \to \textbf{PC}$ from the model structure for quasicategories to the model structure for Segal categories.