Join (simplicial Sets)

In 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

For natural numbers $m,p,q\in\mathbb$, one has the identity:Cisinski 2019, 3.4.12.

: \operatorname( +q+1
=\prod_\operatorname( \times\operatorname( ,

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

: $(X*Y)_n =\prod_X_i\times Y_j.$

A $n$-simplex $\sigma\colon \Delta^n\rightarrow X*Y$ therefore either factors over $X$ or $Y$ or splits into a $p$-simplex $\sigma_-\colon \Delta^p\rightarrow X$ and a $q$-simplex $\sigma_+\colon \Delta^q\rightarrow Y$ with $n=p+q+1$ and $\sigma=\sigma_-*\sigma_+$.

One has canonical morphisms $X,Y\rightarrow X*Y$, which combine into a canonical morphism $X+Y\rightarrow X*Y$ through the universal property of the coproduct. One also has a canonical morphism $X*Y\rightarrow\Delta^0*\Delta^0 \cong\Delta^1$ 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:

: $X^\triangleleft :=\Delta^0*X,$
: $X^\triangleright :=X*\Delta^0.$

Right adjoint

Let $Y$ be a simplicial set. The functor $Y*-\colon \mathbf\rightarrow Y\backslash\mathbf, X\mapsto(Y\mapsto Y*X)$ has a right adjoint $Y\backslash\mathbf\rightarrow\mathbf, (t\colon Y\rightarrow W)\mapsto t\backslash W$ (alternatively denoted $Y\backslash W$) and the functor $-*Y\colon \mathbf\rightarrow Y\backslash\mathbf, X\mapsto(Y\mapsto X*Y)$ also has a right adjoint $Y\backslash\mathbf\rightarrow\mathbf, (t\colon Y\rightarrow W)\mapsto W/t$ (alternatively denoted $W/Y$).Lurie 2009, Proposition 1.2.9.2Cisinski 2019, 3.4.14. A special case is $Y=\Delta^0$ the terminal simplicial set, since $\mathbf_* =\Delta^0\backslash\mathbf$ is the category of pointed simplicial sets.

Let $\mathcal$ be a category and $X\in\operatorname\mathcal$ be an object. Let /math> be the terminal category (with the notation taken from the terminal object of the simplex category), then there is an associated functor t\colon
rightarrow\mathcal,
0\mapsto X, which with the nerve induces a morphism $Nt\colon \Delta^0\rightarrow N\mathcal$. For every simplicial set $A$, one has by additionally using the adjunction between the join of categories and slice categories:

:
\begin
\mathbf(A,N\mathcal/Nt)
&\cong\mathbf_*(\Delta^0\rightarrow A*\Delta^0,Nt)
\cong\mathbf_*( rightarrow\tau(A)\star t) \\
&\cong\mathbf(\tau(A),\mathcal/X)
\cong\mathbf(A,N(\mathcal/X)).
\end

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

: $N\mathcal/NX \cong N(\mathcal/X).$

Examples

One has:

: $\partial\Delta^m*\Delta^n\cup\Delta^m*\partial\Delta^n \cong\partial\Delta^,$
: $\Lambda_k^m*\Delta^n\cup\Delta^m*\partial\Delta^n \cong\Lambda_k^,$
: $\partial\Delta^m*\Delta^n\cup\Delta^m*\Lambda_k^n \cong\Lambda_^.$

Properties

* For simplicial sets $X$ and $Y$, there is a unique morphism $\gamma_\colon X\diamond Y\rightarrow X*Y$ into the diamond operation compatible with the maps $X+Y\rightarrow X*Y,X\diamond Y$ and $X*Y,X\diamond Y\rightarrow\Delta^1$.Cisinski 2019, Proposition 4.2.2. It is a weak categorical equivalence, hence a weak equivalence of the Joyal model structure.Lurie 2009, Proposition 4.2.1.2.Cisinksi 2019, Proposition 4.2.3.
* For a simplicial set $X$, the functors $X*-,-*X\colon\mathbf\rightarrow\mathbf$ preserve weak categorical equivalences.
* For ∞-categories $X$ and $Y$, the simplicial set $X*Y$ is also an ∞-category.
* The join is associative. For simplicial sets $X$, $Y$ and $Z$, one has:
*: $(X*Y)*Z \cong X*(Y*Z).$
* The join reverses under the opposite simplicial set. For simplicial sets $X$ and $Y$, one has:Joyal 2008, p. 244Cisinski 2019, Remark 3.4.15.
*: $(X*Y)^\mathrm \cong Y^\mathrm*X^\mathrm.$

* For a morphism $t\colon Y\rightarrow W$, one has (as adjoint of the previous result):
*: $(W/t)^\mathrm \cong t^\mathrm\backslash W^\mathrm.$

* For morphisms $z\colon Y*X\rightarrow W$, its precomposition with the canonical inclusion $x\colon X\rightarrow Y*X\rightarrow W$ and $y\colon Y\rightarrow W/x$, one has $W/z\cong(W/x)/y$ or in alternative notation:

:: $W/(Y*X) \cong(W/X)/Y.$
: For every simplicial set $A$, one has:
:: $\begin \mathbf(A,W/z) &\cong(Y*X)\backslash\mathbf((Y*X)\rightarrow A*(Y*X),z) \cong X\backslash\mathbf(X\rightarrow(A*Y)*X,x) \\ &\cong\mathbf(A*Y,W/x) \cong Y\backslash\mathbf(Y\rightarrow A*Y,y) \cong\mathbf(A,(W/x)/y), \end$
: so the claim follows from the Yoneda lemma.

* 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 book , last=Cisinski , first=Denis-Charles , author-link=Denis-Charles Cisinski , url=https://cisinski.app.uni-regensburg.de/CatLR.pdf , title=Higher Categories and Homotopical Algebra , date=2019-06-30 , publisher=Cambridge University Press , isbn=978-1108473200 , location= , language=en , authorlink=

References

External links

* join of simplicial sets and join of quasi-categories at the ''n''Lab

Joins of Simplicial Sets
on Kerodon

Higher category theory
Simplicial sets