Twisted diagonal (simplicial sets)

In higher category theory in mathematics, the twisted diagonal of a simplicial set 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 define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:
The canonical morphisms induce canonical morphisms and.

For a simplicial set define a bisimplicial set and a simplicial set with the opposite simplicial set and the diamond operation by:
The canonical morphisms induce canonical morphisms and. The weak categorical equivalence induces canonical morphisms and.

Under the nerve, the twisted diagonal of categories corresponds to the twisted diagonal of simplicial sets. Let be a small category, then:
:
For an ∞-category, the canonical map is a left fibration. Therefore, the twisted diagonal is also an ∞-category.
For a Kan complex, the canonical map is a Kan fibration. Therefore, the twisted diagonal is also a Kan complex.
For an ∞-category, the canonical map is a left bifibration and the canonical map is a left fibration. Therefore, the simplicial set is also an ∞-category.
For an ∞-category, the canonical morphism is a fiberwise equivalence of left fibrations over.
A functor between ∞-categories and is fully faithful if and only if the induced map:
:
For a functor between ∞-categories and, the induced maps:
:
: