Twisted Diagonal (simplicial Sets)

In higher category theory in mathematics, the twisted diagonal of a simplicial set (for ∞-categories also called the twisted arrow ∞-category) is a construction, which generalizes the twisted diagonal of a category to which it corresponds under the nerve construction. Since the twisted diagonal of a category is the category of elements of the Hom functor, the twisted diagonal of an ∞-category can be used to define the Hom functor of an ∞-category.

Twisted diagonal with the join operation

For a simplicial set $A$ define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:Cisinski 2019, 5.6.1.

: $\mathbf(A)_ =\operatorname((\Delta^m)^\mathrm*\Delta^n,A),$
: $\operatorname(A) =\delta^*(\mathbf(A)).$

The canonical morphisms $(\Delta^m)^\mathrm\rightarrow(\Delta^m)^\mathrm*\Delta^n\leftarrow\Delta^n$ induce canonical morphisms $\mathbf(A)\rightarrow A^\mathrm\boxtimes A$ and $\operatorname(A)\rightarrow A^\mathrm\times A$.

Twisted diagonal with the diamond operation

For a simplicial set $A$ define a bisimplicial set and a simplicial set with the diamond operation by:

: $\mathbf_\diamond(A)_ =\operatorname((\Delta^m)^\mathrm\diamond\Delta^n,A),$
: $\operatorname_\diamond(A) =\delta^*(\mathbf_\diamond(A)).$

The canonical morphisms $(\Delta^m)^\mathrm\rightarrow(\Delta^m)^\mathrm\diamond\Delta^n\leftarrow\Delta^n$ induce canonical morphisms $\mathbf_\diamond(A)\rightarrow A^\mathrm\boxtimes A$ and $S_\diamond(A)\rightarrow A^\mathrm\times A$. The weak categorical equivalence $\gamma_\colon (\Delta^m)^\mathrm\diamond\Delta^n\rightarrow(\Delta^m)^\mathrm*\Delta^n$ induces canonical morphisms $\mathbf(A)\rightarrow\mathbf_\diamond(A)$ and $\operatorname(A)\rightarrow\operatorname_\diamond(A)$.

Properties

* Under the nerve, the twisted diagonal of categories corresponds to the twisted diagonal of simplicial sets. Let $\mathcal$ be a small category, then:Kerodon
Proposition 8.1.1.10.
/ref>
*: $N\operatorname(\mathcal) =\operatorname(N\mathcal).$
* For an ∞-category $A$, the canonical map $\operatorname(A)\rightarrow A^\mathrm\times A$ is a left fibration. Therefore, the twisted diagonal $\operatorname(A)$ is also an ∞-category.
* For a Kan complex $A$, the canonical map $\operatorname(A)\rightarrow A^\mathrm\times A$ is a Kan fibration. Therefore, the twisted diagonal $\operatorname(A)$ is also a Kan complex.
* For an ∞-category $A$, the canonical map $\mathbf_\diamond(A)\rightarrow A^\mathrm\boxtimes A$ is a left bifibration and the canonical map $\operatorname_\diamond(A)\rightarrow A^\mathrm\times A$ is a left fibration. Therefore, the simplicial set $\operatorname_\diamond(A)$ is also an ∞-category.
* For an ∞-category $A$, the canonical morphism $\operatorname(A)\rightarrow\operatorname_\diamond(A)$ is a fiberwise equivalence of left fibrations over $A^\mathrm\times A$.
* A functor $u\colon A\rightarrow B$ between ∞-categories $A$ and $B$ is fully faithful if and only if the induced map:
*: $\operatorname(A)\rightarrow(A^\mathrm\times A)\times_\operatorname(B)$ is a fiberwise equivalence over $A^\mathrm\times A$.
* For a functor $u\colon A\rightarrow B$ between ∞-categories $A$ and $B$, the induced maps:
*: $\operatorname(A)\rightarrow(A^\mathrm\times B)\times_\operatorname(B),$
*: $\operatorname(A)\rightarrow(B^\mathrm\times A)\times_\operatorname(B),$

: are cofinal.Cisinski 2019, Proposition 5.6.9.

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

The Twisted Arrow Construction
on Kerodon

Higher category theory
Simplicial sets