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 {\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)).}

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