nLab join of categories

Idea

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

Definition

The join of categories $C$ and $C'$ is the category with

• objects $Obj(C \star C') := Obj(C) \amalg Obj(C')$;

• morphisms given by

  $$Mor_{C \star C'}(a,b) := \left\lbrace \array{ Mor_C(a,b) & \text{ if }\;\; a,b \in C \\ Mor_{C'}(a,b) & \text{ if }\;\; a,b \in C'\\ \emptyset & \text{ if }\;\; a \in C', b \in C; \\ \mathrm{pt} & \text{ if } \;\; a \in C, b \in C'; } \right.$$

In terms of profunctors

The join of categories $C,C'$ can also be described to be the cograph of the unique profunctor $W\colon C ⇸ \; C'$ sending all objects $(c,c')$ to the terminal set (the definition speaks for itself).

In terms of adjoint functors

Consider the inclusion of the boundary of the standard 1-simplex, $i\colon \{0,1\}\to [1]$ as a functor between the discrete category with two elements and the walking arrow $I=\{0 \leq 1\}$. It induces a functor

$$i^*\colon \Cat / I \to \Cat \times \Cat$$

which admits a right adjoint. This right adjoint is precisely the bifunctor $\star\colon \Cat \times \Cat \to \Cat / I$, once we noticed that the category $C\star C'$ comes naturally equipped with an arrow $C\star C'\to I=1\star 1$ induced by (bi)functoriality of $\star$, starting from the canonical arrows $C\to 1, C'\to 1$ to the terminal category.

Proof

It is quite clear that $i^*$ is defined by sending $C\to I$ to the pair of categories $i^\leftarrow(0)=C_0, C_1=i^\leftarrow(1)$. The bijection

$$\Cat^{\mathbf{2}}\Big(i^*\big( C\to I\big), (A,B)\Big)\cong \Cat / I \Big( C, A\star B \Big)$$

is now rather obvious, since any functor $i^* \Big( \array{ C\\ \downarrow \\ I } \Big) \to (A,B)$ determines a functor $C\to A\star B$ and viceversa.

As a cocomma construction

The join of $A$ and $B$ can be seen as the cocomma? of their product cone:

This makes it easy to understand the join as a kind of directed/lopsided coproduct: indeed, the coproduct $A+B$ would be the colimit of the same diagram but weighted by $[0] \rightarrow \{0,1\} \leftarrow [0]$ rather than $[0] \rightarrow [1] \leftarrow [0]$.

Properties

• If the small categories $C,C'$ are two posets, their join consists of their ordinal sum;
• The monoidal structure induced on $\Cat$ by $\star$ is not symmetric (if it was, then the right cone of $C$ would be equivalent to the left cone, which is blatantly false);
• The monoidal structure induced on $\Cat$ by $\star$ is not closed, since the functor $A\star -$ does not preserve colimits.
• The functor $-\star C'\colon \Cat \to C'\!/\!\Cat\colon A\mapsto (A\to A\star C')$ is a left adjoint, and similarly is the functor $C\star-$; see Joyal, Ch. 3.1.1-2 for a detailed description.

Examples

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

Related concepts

• join of topological spaces

• join of quasi-categories;

• join of simplicial sets.

• ordinal sum

References

See p. 42 of

• Jacob Lurie, Higher topos theory (arXiv)

See also Ch. 3 of

• André Joyal, The theory of quasicategories and its applications lectures at Simplicial Methods in Higher Categories, (pdf)