higher category theory In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. H ...

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the diamond operation of

simplicial sets In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

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

For simplicial set

X

and

Y

, their ''diamond''

X\diamond Y

is the pushout of the diagram: :

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

X

and the fiber of

1

Y

Right adjoints

Let

Y

be a simplicial set. The functor

Y\diamond -\colon
\mathbf\rightarrow Y\backslash\mathbf,
X\mapsto(Y\mapsto X\diamond Y)

has a

right adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

Y\backslash\mathbf\rightarrow\mathbf,
(t\colon Y\rightarrow W)\mapsto t\backslash\backslash W

(alternatively denoted

Y\backslash\backslash W

) and the functor

-\diamond Y\colon
\mathbf\rightarrow Y\backslash\mathbf,
X\mapsto(Y\mapsto X\diamond Y)

has a right adjoint

Y\backslash\mathbf\rightarrow\mathbf,
(t\colon Y\rightarrow W)\mapsto W//t

(alternatively denoted

W//Y

). A special case is

Y=\Delta^0

the terminal simplicial set, since

\mathbf_*
=\Delta^0\backslash\mathbf

is the category of pointed simplicial sets.

Properties

* For simplicial sets

X

and

Y

, there is a unique morphism

\gamma_\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 In higher category theory in mathematics, the Joyal model structure is a special model structure on the category of simplicial sets. It consists of three classes of morphisms between simplicial sets called ''fibrations'', ''cofibrations'' and ''wea ...

.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\rightarrow\mathbf

preserve weak categorical equivalences.Cisinski 2019, Proposition 4.2.4.

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 Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, isbn=978-1108473200 , location= , language=en , authorlink=

References

Higher category theory Simplicial sets