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.

Definition

For small categories $\mathcal{C}$ and $\mathcal{D}$ , their join $\mathcal{C}\star\mathcal{D}$ is the small category with:Joyal 2008, p. 241

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

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

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

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

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

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

Right adjoints

Let $\mathcal{D}$ be a small category. The functor $\mathcal{D}\star-\colon
\mathbf{Cat}\rightarrow \mathcal{D}\backslash\mathbf{Cat},
\mathcal{D}\mapsto(\mathcal{C}\mapsto\mathcal{D}\star\mathcal{C})$ has a right adjoint $\mathcal{D}\backslash\mathbf{sSet}\rightarrow\mathbf{sSet},
(F\colon\mathcal{D}\rightarrow\mathcal{E})\mapsto F\backslash\mathcal{E}$ (alternatively denoted $\mathcal{D}\backslash\mathcal{E}$ ) and the functor $-\star\mathcal{D}\colon
\mathbf{Cat}\rightarrow \mathcal{D}\backslash\mathbf{Cat},
\mathcal{D}\mapsto(\mathcal{C}\mapsto\mathcal{C}\star\mathcal{D})$ also has a right adjoint $\mathcal{D}\backslash\mathbf{sSet}\rightarrow\mathbf{sSet},
(F\colon\mathcal{D}\rightarrow\mathcal{E})\mapsto\mathcal{E}/F$ (alternatively denoted $\mathcal{E}/\mathcal{D}$ ).Kerodon, [https://kerodon.net/tag/016H Corollary 4.3.2.17.] A special case is $\mathcal{D}=[0]$ the terminal small category, since $\mathbf{Cat}_*
=[0]\backslash\mathbf{Cat}$ is the category of pointed small categories.

Properties

The join is associative. For small categories $\mathcal{C}$ , $\mathcal{D}$ and $\mathcal{E}$ , one has:Kerdon, [https://kerodon.net/tag/0166 Remark 4.3.2.6.]
:

(\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 $\mathcal{C}$ and $\mathcal{D}$ , one has:Kerodon, [https://kerodon.net/tag/0168 Warning 4.3.2.8.]
:

(\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 $\mathcal{C}$ and $\mathcal{D}$ , one has:Joyal 2008, Corollary 3.3.Kerodon, [https://kerodon.net/tag/0175 Example 4.3.3.14.]
:

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

Literature

{{cite web |last=Joyal |first=André |author-link=André Joyal |date=2008 |title=The Theory of Quasi-Categories and its Applications |url=https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf |language=en}}

External links

join of categories at the nLab
[https://kerodon.net/tag/0160 Joins of Categories] on Kerodon

References

Category:Category theory