nLab
join of quasi-categories

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

   $$\begin{array}{ccc}C\times D& \stackrel{p_2}{\to }& D\\ {}^{p_1}& ⇙& \downarrow \\ C& \to & C♢D\end{array}$$\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

   $$\begin{array}{ccccc}& & C\times D\times 1& \to & D\\ & & \downarrow & & \downarrow \\ C\times D\times 0& \to & C\times D\times {\Delta }^1\\ \downarrow & & & & \downarrow \\ C& \to & & \to & C♢D\end{array}$$\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 $*=\Delta [0]$ be the terminal quasi-category. Then for $X$ any quasi-category,

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

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

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

$$X^{▹}=\left\{\begin{array}{ccccc}0& & \to & & 1\\ & \searrow & ⇙& \swarrow \\ & & \perp \end{array}\right\}.$$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

• Jacob Lurie, Higher Topos Theory .

The operation $♢$ is discussed in chapter 3 of

• André Joyal, The theory of quasicategories and its applications lectures at Simplicial Methods in Higher Categories, (pdf)

and in section 4.2.1 of

• Jacob Lurie, Higher Topos Theory ,

where also the relation between both constructions is established.