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 $𝒞$ and $𝒟$, their join $𝒞\star 𝒟$ is the small category with:

$\mathrm{Ob}(𝒞\star 𝒟)=\mathrm{Ob}(𝒞)\bigsqcup \mathrm{Ob}(𝒟);$

${\mathrm{Hom}}_{𝒞\star 𝒟}(X,Y):=\{\begin{array}{cc}{\mathrm{Hom}}_{𝒞}(X,Y);& X,Y\in \mathrm{Ob}(𝒞)\\ {\mathrm{Hom}}_{𝒟}(X,Y);& X,Y\in \mathrm{Ob}(𝒟)\\ \{\ast \};& X\in \mathrm{Ob}(𝒞),Y\in \mathrm{Ob}(𝒟)\\ \mathrm{\varnothing };& X\in \mathrm{Ob}(𝒟),Y\in \mathrm{Ob}(𝒞)\end{array}.$

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

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

${𝒞}^{◃}:=[0]\star 𝒞,$

${𝒞}^{▹}:=𝒞\star [0].$

Right adjoints

Let $𝒟$ be a small category. The functor $𝒟\star -:𝐂𝐚𝐭\to 𝒟\mathrm{\setminus }𝐂𝐚𝐭,𝒟\mapsto (𝒞\mapsto 𝒟\star 𝒞)$ has a right adjoint $𝒟\mathrm{\setminus }𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭,(F:𝒟\to ℰ)\mapsto F\mathrm{\setminus }ℰ$ (alternatively denoted $𝒟\mathrm{\setminus }ℰ$) and the functor $-\star 𝒟:𝐂𝐚𝐭\to 𝒟\mathrm{\setminus }𝐂𝐚𝐭,𝒟\mapsto (𝒞\mapsto 𝒞\star 𝒟)$ also has a right adjoint $𝒟\mathrm{\setminus }𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭,(F:𝒟\to ℰ)\mapsto ℰ/F$ (alternatively denoted $ℰ/𝒟$). A special case is $𝒟=[0]$ the terminal small category, since ${𝐂𝐚𝐭}_{\ast }=[0]\mathrm{\setminus }𝐂𝐚𝐭$ is the category of pointed small categories.

Properties

• The join is associative. For small categories $𝒞$, $𝒟$ and $ℰ$, one has:

  $(𝒞\star 𝒟)\star ℰ\cong 𝒞\star (𝒟\star ℰ).$

• The join reverses under the dual category. For small categories $𝒞$ and $𝒟$, one has:

  $(𝒞\star 𝒟{)}^{\mathrm{o}\mathrm{p}}\cong {𝒞}^{\mathrm{o}\mathrm{p}}\star {𝒟}^{\mathrm{o}\mathrm{p}}.$

• Under the nerve, the join of categories becomes the join of simplicial sets. For small categories $𝒞$ and $𝒟$, one has:

  $N(𝒞\star 𝒟)\cong N𝒞\ast N𝒟.$

Literature

• Template:Cite web

External links

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