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 $\mathrm{Obj}\left(C\star C\prime \right):=\mathrm{Obj}\left(C\right)⨿\mathrm{Obj}\left(C\prime \right)$;

• morphisms given by

${\mathrm{Mor}}_{C\star C\prime }\left(a,b\right):=\left\{\begin{array}{cc}{\mathrm{Mor}}_{C}\left(a,b\right)& \mathrm{if}a,b\in C\\ {\mathrm{Mor}}_{C\prime }\left(a,b\right)& \mathrm{if}a,b\in C\prime \\ \varnothing & \mathrm{if}a\in C\prime ,b\in C;\\ \mathrm{pt}& \mathrm{if}a\in C,b\in C\prime ;\end{array}$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)
Revised on November 25, 2009 20:10:36 by Toby Bartels (173.60.119.197)