nLab
join of quasi-categories

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

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

    C×D p 2 D p 1 C CD \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

    C×D×1 D C×D×0 C×D×Δ 1 C CD \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 CC and DD simplicial sets, the canonical morphism

CDCD C \diamondsuit D \to C \star D

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

Examples

Joins with the point

Let *=Δ[0]* = \Delta[0] be the terminal quasi-category. Then for XX any quasi-category,

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

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

For instance for X=Δ[1]={01}X = \Delta[1] = \{ 0 \to 1 \} be have

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

Related concepts

References

The operation s\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 both constructions is established.

Revised on December 6, 2013 21:11:31 by Tim Porter (2.26.40.111)