In higher category theory in mathematics, the twisted diagonal of a simplicial set (for âÂÂ-categories also called the twisted arrow âÂÂ-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 âÂÂ-category can be used to define the Hom functor of an âÂÂ-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:
( is the functor obtained by precomposition with the diagonal , hence .) The canonical morphisms induce canonical morphisms and .
Twisted diagonal with the diamond operation
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 .
Properties
- Under the nerve, the twisted diagonal of categories corresponds to the twisted diagonal of simplicial sets. Let be a small category, then:
- :
- For an âÂÂ-category , the canonical map is a left fibration. Therefore, the twisted diagonal is also an âÂÂ-category.
- For a Kan complex , the canonical map is a Kan fibration. Therefore, the twisted diagonal is also a Kan complex.
- For an âÂÂ-category , the canonical map is a left bifibration and the canonical map is a left fibration. Therefore, the simplicial set is also an âÂÂ-category.
- For an âÂÂ-category , the canonical morphism is a fiberwise equivalence of left fibrations over .
- A functor between âÂÂ-categories and is fully faithful if and only if the induced map:
- :
is a fiberwise equivalence over .
- For a functor between âÂÂ-categories and , the induced maps:
- :
- :
are cofinal.
Literature
- - 5.2.1 Twisted Arrow âÂÂ-Categories
References
External links