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 $C'$ 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 $* = \Delta[0]$ 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 $X = \Delta[1] = \{ 0 \to 1 \}$ be have

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

Related concepts

• join of topological spaces

• join of categories;

• join of simplicial sets.

References

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

• Jacob Lurie, Higher Topos Theory .

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

• Jacob Lurie, Higher Topos Theory ,

where also the relation between the two constructions is established.