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 define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:Cisinski 2019, 5.6.1.
( 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:Cisinski 2019, 5.6.10.
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:Kerodon, Proposition 8.1.1.10.
For an ∞-category , the canonical map is a left fibration. Therefore, the twisted diagonal is also an ∞-category.Cisinski 2019, Proposition 5.6.2.Kerodon, Proposition 8.1.1.11.Kerodon, Corollary 8.1.1.12. For a Kan complex , the canonical map is a Kan fibration. Therefore, the twisted diagonal is also a Kan complex.Kerodon, Corollary 8.1.1.13. 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.Cisinski 2019, Proposition 5.6.12. For an ∞-category , the canonical morphism is a fiberwise equivalence of left fibrations over .Cisinski 2019, Corollary 5.6.14. A functor between ∞-categories and is fully faithful if and only if the induced map:
is a fiberwise equivalence over .Cisinski 2019, Corollary 5.6.6. For a functor between ∞-categories and , the induced maps:
are cofinal.Cisinski 2019, Proposition 5.6.9.
Literature
- 5.2.1 Twisted Arrow ∞-Categories
References
External links
The Twisted Arrow Construction on Kerodon twisted arrow (∞,1)-category at the nLab