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 formula_1 define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:
 formula_2
 formula_3
The canonical morphisms formula_4 induce canonical morphisms formula_5 and formula_6.
Twisted diagonal with the diamond operation.
For a simplicial set formula_1 define a bisimplicial set and a simplicial set with the diamond operation by:
 formula_8
 formula_9
The canonical morphisms formula_10 induce canonical morphisms formula_11 and formula_12. The weak categorical equivalence formula_13 induces canonical morphisms formula_14 and formula_15.
 are cofinal.