join of quasi-categories
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 and is a quasi-category which looks roughly like the disjoint union of with with one morphisms from every object of to every object of thrown in.
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.
The join of two quasi-categories and is the join of simplicial sets of their underlying simplicial sets.
The alternate join of two quasi-categories should be thought of as something like the pseudopushout
Explicitly (compare mapping cone) it is the ordinary colimit
The join of two simplicial sets that happen to be quasi-categories is itself a quasi-category.
For and simplicial sets, the canonical morphism
is a weak equivalence in the model structure on quasi-categories.
Joins with the point
Let be the terminal quasi-category. Then for any quasi-category,
the join is the quasi-category obtained from by freely adjoining a new initial object;
the join is the quasi-category obtained from by freely adjoining a new terminal object.
For instance for be have
The operation is discussed around proposition 188.8.131.52, p. 43 of
The operation 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 November 20, 2015 12:10:59
by Urs Schreiber