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 $\mathcal$ and $\mathcal$, their ''join'' $\mathcal\star\mathcal$ is the small category with:Joyal 2008, p. 241

: $\operatorname(\mathcal\star\mathcal) =\operatorname(\mathcal)\sqcup\operatorname(\mathcal);$
: $\operatorname_(X,Y) :=\begin \operatorname_(X,Y); & X,Y\in\operatorname(\mathcal) \\ \operatorname_(X,Y); & X,Y\in\operatorname(\mathcal) \\ \; & X\in\operatorname(\mathcal), Y\in\operatorname(\mathcal) \\ \emptyset; & X\in\operatorname(\mathcal), Y\in\operatorname(\mathcal) \end.$

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

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

:
\mathcal^\triangleleft
:= star\mathcal,

:
\mathcal^\triangleright
:=\mathcal\star

Right adjoints

Let $\mathcal$ be a small category. The functor $\mathcal\star-\colon \mathbf\rightarrow \mathcal\backslash\mathbf, \mathcal\mapsto(\mathcal\mapsto\mathcal\star\mathcal)$ has a right adjoint $\mathcal\backslash\mathbf\rightarrow\mathbf, (F\colon\mathcal\rightarrow\mathcal)\mapsto F\backslash\mathcal$ (alternatively denoted $\mathcal\backslash\mathcal$) and the functor $-\star\mathcal\colon \mathbf\rightarrow \mathcal\backslash\mathbf, \mathcal\mapsto(\mathcal\mapsto\mathcal\star\mathcal)$ also has a right adjoint $\mathcal\backslash\mathbf\rightarrow\mathbf, (F\colon\mathcal\rightarrow\mathcal)\mapsto\mathcal/F$ (alternatively denoted $\mathcal/\mathcal$). A special case is \mathcal= /math> the terminal small category, since \mathbf_*
= backslash\mathbf is the category of pointed small categories.

Properties

* The join is associative. For small categories $\mathcal$, $\mathcal$ and $\mathcal$, one has:
*: $(\mathcal\star\mathcal)\star\mathcal \cong\mathcal\star(\mathcal\star\mathcal).$
* The join reverses under the dual category. For small categories $\mathcal$ and $\mathcal$, one has:Kerodon
Warning 4.3.2.8.
/ref>
*: $(\mathcal\star\mathcal)^\mathrm \cong\mathcal^\mathrm\star\mathcal^\mathrm.$

* Under the nerve, the join of categories becomes the join of simplicial sets. For small categories $\mathcal$ and $\mathcal$, one has:Joyal 2008, Corollary 3.3.Kerodon
Example 4.3.3.14.
/ref>
*: $N(\mathcal\star\mathcal) \cong N\mathcal*N\mathcal.$

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 ''n''Lab

Joins of Categories
on Kerodon

References

Category theory