Join (simplicial sets)

Template:Short description Template:Other usesIn higher category theory in mathematics, the join of simplicial sets is an operation making the category of simplicial sets into a monoidal category. In particular, it takes two simplicial sets to construct another simplicial set. It is closely related to the diamond operation and used in the construction of the twisted diagonal. Under the nerve construction, it corresponds to the join of categories and under the geometric realization, it corresponds to the join of topological spaces.

Definition

Visualization of the join $X\ast Y$ with the blue part representing $X$ and the green part representing $Y$.

For natural numbers $m,p,q\in \mathbb{N}$, one has the identity:

$\mathrm{Hom}([m],[p+q+1])=\prod _{i+j+1=n}\mathrm{Hom}([i],[p])\times \mathrm{Hom}([j],[q]),$

which can be extended by colimits to a functor $-\ast -:𝐬𝐒𝐞𝐭\times 𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭$, which together with the empty simplicial set as unit element makes the category of simplicial sets $𝐬𝐒𝐞𝐭$ into a monoidal category. For simplicial set $X$ and $Y$, their join $X\ast Y$ is the simplicial set:

$(X\ast Y{)}_n=\prod _{i+j+1=n}{X}_i\times Y_j.$

A $n$-simplex $\sigma :{\mathrm{\Delta }}^n\to X\ast Y$ therefore either factors over $X$ or $Y$ or splits into a $p$-simplex ${\sigma }_{-}:{\mathrm{\Delta }}^p\to X$ and a $q$-simplex ${\sigma }_{+}:{\mathrm{\Delta }}^q\to Y$ with $n=p+q+1$ and $\sigma ={\sigma }_{-}\ast {\sigma }_{+}$.

One has canonical morphisms $X,Y\to X\ast Y$, which combine into a canonical morphism $X+Y\to X\ast Y$ through the universal property of the coproduct. One also has a canonical morphism $X\ast Y\to {\mathrm{\Delta }}^0\ast {\mathrm{\Delta }}^0\cong {\mathrm{\Delta }}^1$ of terminal maps, for which the fiber of $0$ is $X$ and the fiber of $1$ is $Y$.

For a simplicial set $X$, one further defines its left cone and right cone as:

$X^{◃}:={\mathrm{\Delta }}^0\ast X,$

$X^{▹}:=X\ast {\mathrm{\Delta }}^0.$

Right adjoint

Let $Y$ be a simplicial set. The functor $Y\ast -:𝐬𝐒𝐞𝐭\to Y\mathrm{\setminus }𝐬𝐒𝐞𝐭,X\mapsto (Y\mapsto Y\ast X)$ has a right adjoint $Y\mathrm{\setminus }𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭,(t:Y\to W)\mapsto t\mathrm{\setminus }W$ (alternatively denoted $Y\mathrm{\setminus }W$) and the functor $-\ast Y:𝐬𝐒𝐞𝐭\to Y\mathrm{\setminus }𝐬𝐒𝐞𝐭,X\mapsto (Y\mapsto X\ast Y)$ also has a right adjoint $Y\mathrm{\setminus }𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭,(t:Y\to W)\mapsto W/t$ (alternatively denoted $W/Y$). A special case is $Y={\mathrm{\Delta }}^0$ the terminal simplicial set, since ${𝐬𝐒𝐞𝐭}_{\ast }={\mathrm{\Delta }}^0\mathrm{\setminus }𝐬𝐒𝐞𝐭$ is the category of pointed simplicial sets.

Let $𝒞$ be a category and $X\in \mathrm{Ob}𝒞$ be an object. Let $[0]$ be the terminal category (with the notation taken from the terminal object of the simplex category), then there is an associated functor $t:[0]\to 𝒞,0\mapsto X$, which with the nerve induces a morphism $Nt:{\mathrm{\Delta }}^0\to N𝒞$. For every simplicial set $A$, one has by additionally using the adjunction between the join of categories and slice categories:

$\begin{array}{cc}𝐬𝐒𝐞𝐭(A,N𝒞/Nt)& \cong {𝐬𝐒𝐞𝐭}_{\ast }({\mathrm{\Delta }}^0\to A\ast {\mathrm{\Delta }}^0,Nt)\cong {𝐂𝐚𝐭}_{\ast }([0]\to \tau (A)\star [0],t)\\ & \cong 𝐂𝐚𝐭(\tau (A),𝒞/X)\cong 𝐬𝐒𝐞𝐭(A,N(𝒞/X)).\end{array}$

Hence according to the Yoneda lemma, one has (with the alternative notation, which here better underlines the result):

$N𝒞/NX\cong N(𝒞/X).$

Examples

One has:

$\partial {\mathrm{\Delta }}^m\ast {\mathrm{\Delta }}^n\cup {\mathrm{\Delta }}^m\ast \partial {\mathrm{\Delta }}^n\cong \partial {\mathrm{\Delta }}^{m+n+1},$

${\mathrm{\Lambda }}_k^m\ast {\mathrm{\Delta }}^n\cup {\mathrm{\Delta }}^m\ast \partial {\mathrm{\Delta }}^n\cong {\mathrm{\Lambda }}_k^{m+n+1},$

$\partial {\mathrm{\Delta }}^m\ast {\mathrm{\Delta }}^n\cup {\mathrm{\Delta }}^m\ast {\mathrm{\Lambda }}_k^n\cong {\mathrm{\Lambda }}_{m+k+1}^{m+n+1}.$

Properties

• For simplicial sets $X$ and $Y$, there is a unique morphism ${\gamma }_{X,Y}:X\diamond Y\to X\ast Y$ into the diamond operation compatible with the maps $X+Y\to X\ast Y,X\diamond Y$ and $X\ast Y,X\diamond Y\to {\mathrm{\Delta }}^1$. It is a weak categorical equivalence, hence a weak equivalence of the Joyal model structure.
• For a simplicial set $X$, the functors $X\ast -,-\ast X:𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭$ preserve weak categorical equivalences.
• For ∞-categories $X$ and $Y$, the simplicial set $X\ast Y$ is also an ∞-category.
• The join is associative. For simplicial sets $X$, $Y$ and $Z$, one has:

  $(X\ast Y)\ast Z\cong X\ast (Y\ast Z).$

• The join reverses under the opposite simplicial set. For simplicial sets $X$ and $Y$, one has:

  $(X\ast Y{)}^{\mathrm{o}\mathrm{p}}\cong Y^{\mathrm{o}\mathrm{p}}\ast X^{\mathrm{o}\mathrm{p}}.$

• For a morphism $t:Y\to W$, one has (as adjoint of the previous result):

  $(W/t{)}^{\mathrm{o}\mathrm{p}}\cong t^{\mathrm{o}\mathrm{p}}\mathrm{\setminus }{W}^{\mathrm{o}\mathrm{p}}.$

• For a morphism $z:Y\ast X\to W$, its precomposition with the canonical inclusion $x:X\to Y\ast X\to W$ and $y:Y\to W/x$ corresponding to $z$ under the adjunction $𝐬𝐒𝐞𝐭(Y,W/x)\cong X\mathrm{\setminus }𝐬𝐒𝐞𝐭(X\to Y\ast X,x)$, one has $W/z\cong (W/x)/y$ or in alternative notation:

$W/(Y\ast X)\cong (W/X)/Y.$

Proof: For every simplicial set $A$, one has:

$\begin{array}{cc}𝐬𝐒𝐞𝐭(A,W/z)& \cong (Y\ast X)\mathrm{\setminus }𝐬𝐒𝐞𝐭((Y\ast X)\to A\ast (Y\ast X),z)\cong X\mathrm{\setminus }𝐬𝐒𝐞𝐭(X\to (A\ast Y)\ast X,x)\\ & \cong 𝐬𝐒𝐞𝐭(A\ast Y,W/x)\cong Y\mathrm{\setminus }𝐬𝐒𝐞𝐭(Y\to A\ast Y,y)\cong 𝐬𝐒𝐞𝐭(A,(W/x)/y),\end{array}$

so the claim follows from the Yoneda lemma.

• 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

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External links

• join of simplicial sets and join of quasi-categories at the nLab
• Joins of Simplicial Sets on Kerodon