Diamond operation

In higher category theory in mathematics, the diamond operation of simplicial sets is an operation taking two simplicial sets to construct another simplicial set. It is closely related to the join of simplicial sets and used in an alternative construction of the twisted diagonal.

Definition

File:Join.svg

For simplicial set $X$ and $Y$ , their diamond $X\diamond Y$ is the pushout of the diagram:Lurie 2009, Definition 4.2.1.1Cisinksi 2019, 4.2.1.

: $X\times Y\times\Delta^1\leftarrow X\times Y\times\partial\Delta^1\rightarrow X+Y.$

One has a canonical map $X\diamond Y\rightarrow\Delta^0\diamond\Delta^0
\cong\Delta^1$ for which the fiber of $0$ is $X$ and the fiber of $1$ is $Y$ .

Right adjoints

Let $Y$ be a simplicial set. The functor $Y\diamond -\colon
\mathbf{sSet}\rightarrow Y\backslash\mathbf{sSet},
X\mapsto(Y\mapsto X\diamond Y)$ has a right adjoint $Y\backslash\mathbf{sSet}\rightarrow\mathbf{sSet},
(t\colon Y\rightarrow W)\mapsto t\backslash\backslash W$ (alternatively denoted $Y\backslash\backslash W$ ) and the functor $-\diamond Y\colon
\mathbf{sSet}\rightarrow Y\backslash\mathbf{sSet},
X\mapsto(Y\mapsto X\diamond Y)$ has a right adjoint $Y\backslash\mathbf{sSet}\rightarrow\mathbf{sSet},
(t\colon Y\rightarrow W)\mapsto W//t$ (alternatively denoted $W//Y$ ).Lurie 2009, after Corollary 4.2.1.4.Cisinski 2019, 4.2.1. A special case is $Y=\Delta^0$ the terminal simplicial set, since $\mathbf{sSet}_*
=\Delta^0\backslash\mathbf{sSet}$ is the category of pointed simplicial sets.

Properties

For simplicial sets $X$ and $Y$ , there is a unique morphism $\gamma_{X,Y}\colon$

X\diamond Y\rightarrow X*Y from the join of simplicial sets 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 $X\diamond -,-\diamond X\colon\mathbf{sSet}\rightarrow\mathbf{sSet}$ preserve weak categorical equivalences.Lurie 2009, Corollary 4.2.1.3.Cisinski 2019, Proposition 4.2.4.

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}}
{{cite book |last1=Lurie |first1=Jacob |author-link=Jacob Lurie |title=Higher Topos Theory |title-link=Higher Topos Theory |publisher=Princeton University Press |year=2009 |isbn=978-0-691-14049-0 |series=Annals of Mathematics Studies |volume=170 |mr=2522659 |arxiv=math.CT/0608040}}
{{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

Category:Higher category theory

Category:Simplicial sets