Join (category theory)

For other uses, see Join (disambiguation).

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.

Definition

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:

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

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

Right adjoints

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.

Properties

• 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]

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

Literature

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

External links

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

References

1. Joyal 2008, p. 241
2. Kerodon, Corollary 4.3.2.17.
3. Kerdon, Remark 4.3.2.6.
4. Kerodon, Warning 4.3.2.8.
5. Joyal 2008, Corollary 3.3.
6. Kerodon, Example 4.3.3.14.