Join (category theory)

In category theory in mathematics, the join of categories is an operation making the category of small categories into a monoidal category. In particular, it takes two small categories to construct another small category. Under the nerve construction, it corresponds to the join of simplicial sets.

For small categories ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle {\mathcal {D}}}$, their join ${\displaystyle {\mathcal {C}}\star {\mathcal {D}}}$ is the small category with:^[1]

${\displaystyle \operatorname {Ob} ({\mathcal {C}}\star {\mathcal {D}})=\operatorname {Ob} ({\mathcal {C}})\sqcup \operatorname {Ob} ({\mathcal {D}});}$

${\displaystyle \operatorname {Hom} _{{\mathcal {C}}\star {\mathcal {D}}}(X,Y):={\begin{cases}\operatorname {Hom} _{\mathcal {C}}(X,Y);&X,Y\in \operatorname {Ob} ({\mathcal {C}})\\\operatorname {Hom} _{\mathcal {D}}(X,Y);&X,Y\in \operatorname {Ob} ({\mathcal {D}})\\\{*\};&X\in \operatorname {Ob} ({\mathcal {C}}),Y\in \operatorname {Ob} ({\mathcal {D}})\\\emptyset ;&X\in \operatorname {Ob} ({\mathcal {D}}),Y\in \operatorname {Ob} ({\mathcal {C}})\end{cases}}.}$

The join defines a functor ${\displaystyle -\star -\colon \mathbf {Cat} \times \mathbf {Cat} \rightarrow \mathbf {Cat} }$, which together with the empty category as unit element makes the category of small categories ${\displaystyle \mathbf {Cat} }$ into a monoidal category.

For a small category ${\displaystyle {\mathcal {C}}}$, one further defines its left cone and right cone as:

${\displaystyle {\mathcal {C}}^{\triangleleft }:=[0]\star {\mathcal {C}},}$

${\displaystyle {\mathcal {C}}^{\triangleright }:={\mathcal {C}}\star [0].}$

Let ${\displaystyle {\mathcal {D}}}$ be a small category. The functor ${\displaystyle {\mathcal {D}}\star -\colon \mathbf {Cat} \rightarrow {\mathcal {D}}\backslash \mathbf {Cat} ,{\mathcal {D}}\mapsto ({\mathcal {C}}\mapsto {\mathcal {D}}\star {\mathcal {C}})}$ has a right adjoint ${\displaystyle {\mathcal {D}}\backslash \mathbf {sSet} \rightarrow \mathbf {sSet} ,(F\colon {\mathcal {D}}\rightarrow {\mathcal {E}})\mapsto F\backslash {\mathcal {E}}}$ (alternatively denoted ${\displaystyle {\mathcal {D}}\backslash {\mathcal {E}}}$) and the functor ${\displaystyle -\star {\mathcal {D}}\colon \mathbf {Cat} \rightarrow {\mathcal {D}}\backslash \mathbf {Cat} ,{\mathcal {D}}\mapsto ({\mathcal {C}}\mapsto {\mathcal {C}}\star {\mathcal {D}})}$ also has a right adjoint ${\displaystyle {\mathcal {D}}\backslash \mathbf {sSet} \rightarrow \mathbf {sSet} ,(F\colon {\mathcal {D}}\rightarrow {\mathcal {E}})\mapsto {\mathcal {E}}/F}$ (alternatively denoted ${\displaystyle {\mathcal {E}}/{\mathcal {D}}}$).^[2] A special case is ${\displaystyle {\mathcal {D}}=[0]}$ the terminal small category, since ${\displaystyle \mathbf {Cat} _{*}=[0]\backslash \mathbf {Cat} }$ is the category of pointed small categories.

• The join is associative. For small categories ${\displaystyle {\mathcal {C}}}$, ${\displaystyle {\mathcal {D}}}$ and ${\displaystyle {\mathcal {E}}}$, one has:^[3]

  ${\displaystyle ({\mathcal {C}}\star {\mathcal {D}})\star {\mathcal {E}}\cong {\mathcal {C}}\star ({\mathcal {D}}\star {\mathcal {E}}).}$

• The join reverses under the dual category. For small categories ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle {\mathcal {D}}}$, one has:^[1][4]

  ${\displaystyle ({\mathcal {C}}\star {\mathcal {D}})^{\mathrm {op} }\cong {\mathcal {C}}^{\mathrm {op} }\star {\mathcal {D}}^{\mathrm {op} }.}$

• Under the nerve, the join of categories becomes the join of simplicial sets. For small categories ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle {\mathcal {D}}}$, one has:^[5][6]

  ${\displaystyle N({\mathcal {C}}\star {\mathcal {D}})\cong N{\mathcal {C}}*N{\mathcal {D}}.}$

• Joyal, André (2008). "The Theory of Quasi-Categories and its Applications" (PDF).

• join of categories at the nLab
• Joins of Categories on Kerodon