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 {\displaystyle A} define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:

T w ( A ) m , n = Hom ⁡ ( ( Δ m ) o p ∗ Δ n , A ) , {\displaystyle \mathbf {Tw} (A)_{m,n}=\operatorname {Hom} ((\Delta ^{m})^{\mathrm {op} }*\Delta ^{n},A),}

Tw ⁡ ( A ) = δ ∗ ( T w ( A ) ) . {\displaystyle \operatorname {Tw} (A)=\delta ^{*}(\mathbf {Tw} (A)).}

(δ ∗ : b i s S e t → s S e t {\displaystyle \delta ^{*}\colon \mathbf {bisSet} \rightarrow \mathbf {sSet} } is the functor obtained by precomposition with the diagonal δ : Δ → Δ × Δ {\displaystyle \delta \colon \Delta \rightarrow \Delta \times \Delta }, hence δ ∗ ( A ) n = A n , n {\displaystyle \delta ^{*}(A)_{n}=A_{n,n}}.) The canonical morphisms ( Δ m ) o p → ( Δ m ) o p ∗ Δ n ← Δ n {\displaystyle (\Delta ^{m})^{\mathrm {op} }\rightarrow (\Delta ^{m})^{\mathrm {op} }*\Delta ^{n}\leftarrow \Delta ^{n}} induce canonical morphisms T w ( A ) → A o p ⊠ A {\displaystyle \mathbf {Tw} (A)\rightarrow A^{\mathrm {op} }\boxtimes A} and Tw ⁡ ( A ) → A o p × A {\displaystyle \operatorname {Tw} (A)\rightarrow A^{\mathrm {op} }\times A}.

Twisted diagonal with the diamond operation

For a simplicial set A {\displaystyle A} define a bisimplicial set and a simplicial set with the opposite simplicial set and the diamond operation by:

T w ⋄ ( A ) m , n = Hom ⁡ ( ( Δ m ) o p ⋄ Δ n , A ) , {\displaystyle \mathbf {Tw} _{\diamond }(A)_{m,n}=\operatorname {Hom} ((\Delta ^{m})^{\mathrm {op} }\diamond \Delta ^{n},A),}

Tw ⋄ ⁡ ( A ) = δ ∗ ( T w ⋄ ( A ) ) . {\displaystyle \operatorname {Tw} _{\diamond }(A)=\delta ^{*}(\mathbf {Tw} _{\diamond }(A)).}

The canonical morphisms ( Δ m ) o p → ( Δ m ) o p ⋄ Δ n ← Δ n {\displaystyle (\Delta ^{m})^{\mathrm {op} }\rightarrow (\Delta ^{m})^{\mathrm {op} }\diamond \Delta ^{n}\leftarrow \Delta ^{n}} induce canonical morphisms T w ⋄ ( A ) → A o p ⊠ A {\displaystyle \mathbf {Tw} _{\diamond }(A)\rightarrow A^{\mathrm {op} }\boxtimes A} and Tw ⋄ ⁡ ( A ) → A o p × A {\displaystyle \operatorname {Tw} _{\diamond }(A)\rightarrow A^{\mathrm {op} }\times A}. The weak categorical equivalence γ ( Δ m ) o p , Δ n : ( Δ m ) o p ⋄ Δ n → ( Δ m ) o p ∗ Δ n {\displaystyle \gamma _{(\Delta ^{m})^{\mathrm {op} },\Delta ^{n}}\colon (\Delta ^{m})^{\mathrm {op} }\diamond \Delta ^{n}\rightarrow (\Delta ^{m})^{\mathrm {op} }*\Delta ^{n}} induces canonical morphisms T w ( A ) → T w ⋄ ( A ) {\displaystyle \mathbf {Tw} (A)\rightarrow \mathbf {Tw} _{\diamond }(A)} and Tw ⁡ ( A ) → Tw ⋄ ⁡ ( A ) {\displaystyle \operatorname {Tw} (A)\rightarrow \operatorname {Tw} _{\diamond }(A)}.

Properties

  • Under the nerve, the twisted diagonal of categories corresponds to the twisted diagonal of simplicial sets. Let C {\displaystyle {\mathcal {C}}} be a small category, then: N Tw ⁡ ( C ) = Tw ⁡ ( N C ) . {\displaystyle N\operatorname {Tw} ({\mathcal {C}})=\operatorname {Tw} (N{\mathcal {C}}).}
  • For an ∞-category A {\displaystyle A}, the canonical map Tw ⁡ ( A ) → A o p × A {\displaystyle \operatorname {Tw} (A)\rightarrow A^{\mathrm {op} }\times A} is a left fibration. Therefore, the twisted diagonal Tw ⁡ ( A ) {\displaystyle \operatorname {Tw} (A)} is also an ∞-category.
  • For a Kan complex A {\displaystyle A}, the canonical map Tw ⁡ ( A ) → A o p × A {\displaystyle \operatorname {Tw} (A)\rightarrow A^{\mathrm {op} }\times A} is a Kan fibration. Therefore, the twisted diagonal Tw ⁡ ( A ) {\displaystyle \operatorname {Tw} (A)} is also a Kan complex.
  • For an ∞-category A {\displaystyle A}, the canonical map T w ⋄ ( A ) → A o p ⊠ A {\displaystyle \mathbf {Tw} _{\diamond }(A)\rightarrow A^{\mathrm {op} }\boxtimes A} is a left bifibration and the canonical map Tw ⋄ ⁡ ( A ) → A o p × A {\displaystyle \operatorname {Tw} _{\diamond }(A)\rightarrow A^{\mathrm {op} }\times A} is a left fibration. Therefore, the simplicial set Tw ⋄ ⁡ ( A ) {\displaystyle \operatorname {Tw} _{\diamond }(A)} is also an ∞-category.
  • For an ∞-category A {\displaystyle A}, the canonical morphism Tw ⁡ ( A ) → Tw ⋄ ⁡ ( A ) {\displaystyle \operatorname {Tw} (A)\rightarrow \operatorname {Tw} _{\diamond }(A)} is a fiberwise equivalence of left fibrations over A o p × A {\displaystyle A^{\mathrm {op} }\times A}.
  • A functor u : A → B {\displaystyle u\colon A\rightarrow B} between ∞-categories A {\displaystyle A} and B {\displaystyle B} is fully faithful if and only if the induced map: Tw ⁡ ( A ) → ( A o p × A ) × B o p × B Tw ⁡ ( B ) {\displaystyle \operatorname {Tw} (A)\rightarrow (A^{\mathrm {op} }\times A)\times _{B^{\mathrm {op} }\times B}\operatorname {Tw} (B)}

is a fiberwise equivalence over A o p × A {\displaystyle A^{\mathrm {op} }\times A}.

  • For a functor u : A → B {\displaystyle u\colon A\rightarrow B} between ∞-categories A {\displaystyle A} and B {\displaystyle B}, the induced maps: Tw ⁡ ( A ) → ( A o p × B ) × B o p × B Tw ⁡ ( B ) , {\displaystyle \operatorname {Tw} (A)\rightarrow (A^{\mathrm {op} }\times B)\times _{B^{\mathrm {op} }\times B}\operatorname {Tw} (B),} Tw ⁡ ( A ) → ( B o p × A ) × B o p × B Tw ⁡ ( B ) , {\displaystyle \operatorname {Tw} (A)\rightarrow (B^{\mathrm {op} }\times A)\times _{B^{\mathrm {op} }\times B}\operatorname {Tw} (B),}

are cofinal.

Literature

External links

  • on Kerodon
  • twisted arrow (∞,1)-category at the nLab