# nLab join of quasi-categories

Contents

### Context

#### $(\infty,1)$-Category theory

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# Contents

## Idea

The join of two quasi-categories is the generalization of the join of categories from ordinary categories to quasi-categories.

The join of quasi-categories $C$ and $C'$ is a quasi-category $C \star C'$ which looks roughly like the disjoint union of $C$ with $C'$ with one morphisms from every object of $C$ to every object of $C'$ thrown in.

## Definition

Two different definitions are used in the literature, which are not isomorphic, but are weakly equivalent with respect to the model structure on quasi-categories.

1. The join $C \star C'$ of two quasi-categories $C$ and $C'$ is the join of simplicial sets of their underlying simplicial sets.

2. The alternate join $C \diamondsuit D$ of two quasi-categories should be thought of as something like the pseudopushout

$\array{ C \times D &\stackrel{p_2}{\to}& D \\ {}^{\mathllap{p_1}} &\swArrow& \downarrow \\ C &\to& C \diamondsuit D }$

Explicitly (compare mapping cone) it is the ordinary colimit

$\array{ && C \times D \times {1} &\to& D \\ &&\downarrow && \downarrow \\ C \times D \times {0} &\to& C \times D \times \Delta^1 \\ \downarrow &&&& \downarrow \\ C &\to& &\to& C \diamondsuit D }$

in sSet.

The join of two simplicial sets that happen to be quasi-categories is itself a quasi-category.

## Properties

For $C$ and $D$ simplicial sets, the canonical morphism

$C \diamondsuit D \to C \star D$

is a weak equivalence in the model structure on quasi-categories.

## Examples

### Joins with the point

Let $* = \Delta$ be the terminal quasi-category. Then for $X$ any quasi-category,

• the join $X^{\triangleleft} := (*)\star X$ is the quasi-category obtained from $X$ by freely adjoining a new initial object;

• the join $X^{\triangleright} := X \star (*)$ is the quasi-category obtained from $X$ by freely adjoining a new terminal object.

For instance for $X = \Delta = \{ 0 \to 1 \}$ be have

$X^{\triangleright} = \left\{ \array{ 0 &&\to&& 1 \\ & \searrow &\swArrow& \swarrow \\ && \bottom } \right\} \,.$

The operation $\star_s$ is discussed around proposition 1.2.8.3, p. 43 of

The operation $\diamondsuit$ is discussed in chapter 3 of

• André Joyal, The theory of quasicategories and its applications lectures for the advanced course, at the conference on Simplicial Methods in Higher Categories, (pdf)

and in section 4.2.1 of

where also the relation between the two constructions is established.