nLab
join of categories

Idea

The join of two categories $C$ and $C\prime$ is obtained from the disjoint union of $C$ with $C\prime$ by throwing in a unique morphism from every object of $C$ to every object of $C\prime$.

Definition

The join of categories $C$ and $C\prime$ is the category with

• objects $\mathrm{Obj}(C\star C\prime ):=\mathrm{Obj}(C)⨿\mathrm{Obj}(C\prime )$;

• morphisms given by

  $${\mathrm{Mor}}_{C\star C\prime }(a,b):=\{\begin{array}{cc}{\mathrm{Mor}}_C(a,b)& \mathrm{if}a,b\in C\\ {\mathrm{Mor}}_{C\prime }(a,b)& \mathrm{if}a,b\in C\prime \\ \varnothing & \mathrm{if}a\in C\prime ,b\in C;\\ \mathrm{pt}& \mathrm{if}a\in C,b\in C\prime ;\end{array}$$Mor_{C \star C'}(a,b) := \left\lbrace \array{ Mor_C(a,b) & if a,b \in C \\ Mor_{C'}(a,b) & if a,b \in C' \\ \emptyset & if a \in C', b \in C; \\ \mathrm{pt} & if a \in C, b \in C'; } \right.

Examples

• The cone below a category $C$ is the join $C\star \mathrm{pt}$. The cone above $C$ is the join $\mathrm{pt}\star C$.

References

See p. 42 of

• J. Lurie, Higher topos theory (arXiv)