# nLab join of quasi-categories

### Context

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

(∞,1)-category theory

# 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 $C\prime$ is a quasi-category $C\star C\prime$ which looks roughly like the disjoint union of $C$ with $C\prime$ with one morphisms from every object of $C$ to every object of $C\prime$ 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 $C\star C\prime$ of two quasi-categories $C$ and $C\prime$ is the join of simplicial sets of their underlying simplicial sets.

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

$\begin{array}{ccc}C×D& \stackrel{{p}_{2}}{\to }& D\\ {}^{{p}_{1}}& ⇙& ↓\\ C& \to & C♢D\end{array}$\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

$\begin{array}{ccccc}& & C×D×1& \to & D\\ & & ↓& & ↓\\ C×D×0& \to & C×D×{\Delta }^{1}\\ ↓& & & & ↓\\ C& \to & & \to & C♢D\end{array}$\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

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

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

## Examples

### Joins with the point

Let $*=\Delta \left[0\right]$ be the terminal quasi-category. Then for $X$ any quasi-category,

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

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

For instance for $X=\Delta \left[1\right]=\left\{0\to 1\right\}$ be have

${X}^{▹}=\left\{\begin{array}{ccccc}0& & \to & & 1\\ & ↘& ⇙& ↙\\ & & \perp \end{array}\right\}\phantom{\rule{thinmathspace}{0ex}}.$X^{\triangleright} = \left\{ \array{ 0 &&\to&& 1 \\ & \searrow &\swArrow& \swarrow \\ && \bottom } \right\} \,.

## References

The operation ${\star }_{s}$ is discussed around proposition 1.2.8.3, p. 43 of

The operation $♢$ is discussed in chapter 3 of

and in section 4.2.1 of

where also the relation between both constructions is established.

Revised on November 4, 2010 08:31:58 by Urs Schreiber (87.212.203.135)