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 $A$ 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.

: $\mathbf{Tw}(A)_{m,n}
=\operatorname{Hom}((\Delta^m)^\mathrm{op}*\Delta^n,A),$

: $\operatorname{Tw}(A)
=\delta^*(\mathbf{Tw}(A)).$

The canonical morphisms $(\Delta^m)^\mathrm{op}\rightarrow(\Delta^m)^\mathrm{op}*\Delta^n\leftarrow\Delta^n$ induce canonical morphisms $\mathbf{Tw}(A)\rightarrow A^\mathrm{op}\boxtimes A$ and $\operatorname{Tw}(A)\rightarrow A^\mathrm{op}\times A$ .

Twisted diagonal with the diamond operation

For a simplicial set $A$ define a bisimplicial set and a simplicial set with the diamond operation by:Cisinski 2019, 5.6.10.

: $\mathbf{Tw}_\diamond(A)_{m,n}
=\operatorname{Hom}((\Delta^m)^\mathrm{op}\diamond\Delta^n,A),$

: $\operatorname{Tw}_\diamond(A)
=\delta^*(\mathbf{Tw}_\diamond(A)).$

The canonical morphisms $(\Delta^m)^\mathrm{op}\rightarrow(\Delta^m)^\mathrm{op}\diamond\Delta^n\leftarrow\Delta^n$ induce canonical morphisms $\mathbf{Tw}_\diamond(A)\rightarrow A^\mathrm{op}\boxtimes A$ and $S_\diamond(A)\rightarrow A^\mathrm{op}\times A$ . The weak categorical equivalence $\gamma_{(\Delta^m)^\mathrm{op},\Delta^n}\colon
(\Delta^m)^\mathrm{op}\diamond\Delta^n\rightarrow(\Delta^m)^\mathrm{op}*\Delta^n$ induces canonical morphisms $\mathbf{Tw}(A)\rightarrow\mathbf{Tw}_\diamond(A)$ and $\operatorname{Tw}(A)\rightarrow\operatorname{Tw}_\diamond(A)$ .

Properties

Under the nerve, the twisted diagonal of categories corresponds to the twisted diagonal of simplicial sets. Let $\mathcal{C}$ be a small category, then:Kerodon, [https://kerodon.net/tag/03JN Proposition 8.1.1.10.]
: $N\operatorname{Tw}(\mathcal{C})$

=\operatorname{Tw}(N\mathcal{C}).

For an ∞-category $A$ , the canonical map $\operatorname{Tw}(A)\rightarrow A^\mathrm{op}\times A$ is a left fibration. Therefore, the twisted diagonal $\operatorname{Tw}(A)$ is also an ∞-category.Cisinski 2019, Proposition 5.6.2.[https://kerodon.net/tag/03JQ Kerodon, Proposition 8.1.1.11.][https://kerodon.net/tag/03JR Kerodon, Corollary 8.1.1.12.]
For a Kan complex $A$ , the canonical map $\operatorname{Tw}(A)\rightarrow A^\mathrm{op}\times A$ is a Kan fibration. Therefore, the twisted diagonal $\operatorname{Tw}(A)$ is also a Kan complex.[https://kerodon.net/tag/048H Kerodon, Corollary 8.1.1.13.]
For an ∞-category $A$ , the canonical map $\mathbf{Tw}_\diamond(A)\rightarrow A^\mathrm{op}\boxtimes A$ is a left bifibration and the canonical map $\operatorname{Tw}_\diamond(A)\rightarrow A^\mathrm{op}\times A$ is a left fibration. Therefore, the simplicial set $\operatorname{Tw}_\diamond(A)$ is also an ∞-category.Cisinski 2019, Proposition 5.6.12.
For an ∞-category $A$ , the canonical morphism $\operatorname{Tw}(A)\rightarrow\operatorname{Tw}_\diamond(A)$ is a fiberwise equivalence of left fibrations over $A^\mathrm{op}\times A$ .Cisinski 2019, Corollary 5.6.14.
A functor $u\colon A\rightarrow B$ between ∞-categories $A$ and $B$ is fully faithful if and only if the induced map:
: $\operatorname{Tw}(A)\rightarrow(A^\mathrm{op}\times A)\times_{B^\mathrm{op}\times B}\operatorname{Tw}(B)$ is a fiberwise equivalence over $A^\mathrm{op}\times A$ .Cisinski 2019, Corollary 5.6.6.
For a functor $u\colon A\rightarrow B$ between ∞-categories $A$ and $B$ , the induced maps:
: $\operatorname{Tw}(A)\rightarrow(A^\mathrm{op}\times B)\times_{B^\mathrm{op}\times B}\operatorname{Tw}(B),$
: $\operatorname{Tw}(A)\rightarrow(B^\mathrm{op}\times A)\times_{B^\mathrm{op}\times B}\operatorname{Tw}(B),$

: are cofinal.Cisinski 2019, Proposition 5.6.9.

Literature

{{cite book |last=Cisinski |first=Denis-Charles |author-link=Denis-Charles Cisinski |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |title=Higher Categories and Homotopical Algebra |date=2019-06-30 |publisher=Cambridge University Press |isbn=978-1108473200 |location= |language=en |authorlink=}}

References

External links

[https://kerodon.net/tag/03JF The Twisted Arrow Construction] on Kerodon

Category:Higher category theory

Category:Simplicial sets