Twisted diagonal (simplicial sets)

Template:Short description Template:Other uses 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:

$𝐓𝐰(A{)}_{m,n}=\mathrm{Hom}(({\mathrm{\Delta }}^m{)}^{\mathrm{o}\mathrm{p}}\ast {\mathrm{\Delta }}^n,A),$

$\mathrm{Tw}(A)={\delta }^{\ast }(𝐓𝐰(A)).$

(${\delta }^{\ast }:𝐛𝐢𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭$ is the functor obtained by precomposition with the diagonal $\delta :\mathrm{\Delta }\to \mathrm{\Delta }\times \mathrm{\Delta }$, hence ${\delta }^{\ast }(A{)}_n=A_{n,n}$.) The canonical morphisms $({\mathrm{\Delta }}^m{)}^{\mathrm{o}\mathrm{p}}\to ({\mathrm{\Delta }}^m{)}^{\mathrm{o}\mathrm{p}}\ast {\mathrm{\Delta }}^n\leftarrow {\mathrm{\Delta }}^n$ induce canonical morphisms $𝐓𝐰(A)\to A^{\mathrm{o}\mathrm{p}}⊠A$ and $\mathrm{Tw}(A)\to A^{\mathrm{o}\mathrm{p}}\times A$.

Twisted diagonal with the diamond operation

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

${𝐓𝐰}_{\diamond }(A{)}_{m,n}=\mathrm{Hom}(({\mathrm{\Delta }}^m{)}^{\mathrm{o}\mathrm{p}}\diamond {\mathrm{\Delta }}^n,A),$

${\mathrm{Tw}}_{\diamond }(A)={\delta }^{\ast }({𝐓𝐰}_{\diamond }(A)).$

The canonical morphisms $({\mathrm{\Delta }}^m{)}^{\mathrm{o}\mathrm{p}}\to ({\mathrm{\Delta }}^m{)}^{\mathrm{o}\mathrm{p}}\diamond {\mathrm{\Delta }}^n\leftarrow {\mathrm{\Delta }}^n$ induce canonical morphisms ${𝐓𝐰}_{\diamond }(A)\to A^{\mathrm{o}\mathrm{p}}⊠A$ and ${\mathrm{Tw}}_{\diamond }(A)\to A^{\mathrm{o}\mathrm{p}}\times A$. The weak categorical equivalence ${\gamma }_{({\mathrm{\Delta }}^m{)}^{\mathrm{o}\mathrm{p}},{\mathrm{\Delta }}^n}:({\mathrm{\Delta }}^m{)}^{\mathrm{o}\mathrm{p}}\diamond {\mathrm{\Delta }}^n\to ({\mathrm{\Delta }}^m{)}^{\mathrm{o}\mathrm{p}}\ast {\mathrm{\Delta }}^n$ induces canonical morphisms $𝐓𝐰(A)\to {𝐓𝐰}_{\diamond }(A)$ and $\mathrm{Tw}(A)\to {\mathrm{Tw}}_{\diamond }(A)$.

Properties

• Under the nerve, the twisted diagonal of categories corresponds to the twisted diagonal of simplicial sets. Let $𝒞$ be a small category, then:

  $N\mathrm{Tw}(𝒞)=\mathrm{Tw}(N𝒞).$

• For an ∞-category $A$, the canonical map $\mathrm{Tw}(A)\to A^{\mathrm{o}\mathrm{p}}\times A$ is a left fibration. Therefore, the twisted diagonal $\mathrm{Tw}(A)$ is also an ∞-category.
• For a Kan complex $A$, the canonical map $\mathrm{Tw}(A)\to A^{\mathrm{o}\mathrm{p}}\times A$ is a Kan fibration. Therefore, the twisted diagonal $\mathrm{Tw}(A)$ is also a Kan complex.
• For an ∞-category $A$, the canonical map ${𝐓𝐰}_{\diamond }(A)\to A^{\mathrm{o}\mathrm{p}}⊠A$ is a left bifibration and the canonical map ${\mathrm{Tw}}_{\diamond }(A)\to A^{\mathrm{o}\mathrm{p}}\times A$ is a left fibration. Therefore, the simplicial set ${\mathrm{Tw}}_{\diamond }(A)$ is also an ∞-category.
• For an ∞-category $A$, the canonical morphism $\mathrm{Tw}(A)\to {\mathrm{Tw}}_{\diamond }(A)$ is a fiberwise equivalence of left fibrations over $A^{\mathrm{o}\mathrm{p}}\times A$.
• A functor $u:A\to B$ between ∞-categories $A$ and $B$ is fully faithful if and only if the induced map:

  $\mathrm{Tw}(A)\to (A^{\mathrm{o}\mathrm{p}}\times A){\times }_{B^{\mathrm{o}\mathrm{p}}\times B}\mathrm{Tw}(B)$

is a fiberwise equivalence over $A^{\mathrm{o}\mathrm{p}}\times A$.

• For a functor $u:A\to B$ between ∞-categories $A$ and $B$, the induced maps:

  $\mathrm{Tw}(A)\to (A^{\mathrm{o}\mathrm{p}}\times B){\times }_{B^{\mathrm{o}\mathrm{p}}\times B}\mathrm{Tw}(B),$

  $\mathrm{Tw}(A)\to (B^{\mathrm{o}\mathrm{p}}\times A){\times }_{B^{\mathrm{o}\mathrm{p}}\times B}\mathrm{Tw}(B),$

are cofinal.

Literature

• Template:Cite book
• Template:Cite web - 5.2.1 Twisted Arrow ∞-Categories

External links

• The Twisted Arrow Construction on Kerodon
• twisted arrow (∞,1)-category at the nLab