Twisted diagonal (simplicial sets)

For other uses, see Twisted diagonal (disambiguation).

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 ${\displaystyle A}$ define a bisimplicial set and a simplicial set with the opposite simplicial set and the join of simplicial sets by:^[1]

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

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

The canonical morphisms ${\displaystyle (\Delta ^{m})^{\mathrm {op} }\rightarrow (\Delta ^{m})^{\mathrm {op} }*\Delta ^{n}\leftarrow \Delta ^{n}}$ induce canonical morphisms ${\displaystyle \mathbf {Tw} (A)\rightarrow A^{\mathrm {op} }\boxtimes A}$ and ${\displaystyle \operatorname {Tw} (A)\rightarrow A^{\mathrm {op} }\times A}$.^[1]

Twisted diagonal with the diamond operation

For a simplicial set ${\displaystyle A}$ define a bisimplicial set and a simplicial set with the diamond operation by:^[2]

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

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

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

Properties

• Under the nerve, the twisted diagonal of categories corresponds to the twisted diagonal of simplicial sets. Let ${\displaystyle {\mathcal {C}}}$ be a small category, then:^[3]

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

• For an ∞-category ${\displaystyle A}$, the canonical map ${\displaystyle \operatorname {Tw} (A)\rightarrow A^{\mathrm {op} }\times A}$ is a left fibration. Therefore, the twisted diagonal ${\displaystyle \operatorname {Tw} (A)}$ is also an ∞-category.^[4][5][6]
• For a Kan complex ${\displaystyle A}$, the canonical map ${\displaystyle \operatorname {Tw} (A)\rightarrow A^{\mathrm {op} }\times A}$ is a Kan fibration. Therefore, the twisted diagonal ${\displaystyle \operatorname {Tw} (A)}$ is also a Kan complex.^[7]
• For an ∞-category ${\displaystyle A}$, the canonical map ${\displaystyle \mathbf {Tw} _{\diamond }(A)\rightarrow A^{\mathrm {op} }\boxtimes A}$ is a left bifibration and the canonical map ${\displaystyle \operatorname {Tw} _{\diamond }(A)\rightarrow A^{\mathrm {op} }\times A}$ is a left fibration. Therefore, the simplicial set ${\displaystyle \operatorname {Tw} _{\diamond }(A)}$ is also an ∞-category.^[8]
• For an ∞-category ${\displaystyle A}$, the canonical morphism ${\displaystyle \operatorname {Tw} (A)\rightarrow \operatorname {Tw} _{\diamond }(A)}$ is a fiberwise equivalence of left fibrations over ${\displaystyle A^{\mathrm {op} }\times A}$.^[9]
• A functor ${\displaystyle u\colon A\rightarrow B}$ between ∞-categories ${\displaystyle A}$ and ${\displaystyle B}$ is fully faithful if and only if the induced map:

  ${\displaystyle \operatorname {Tw} (A)\rightarrow (A^{\mathrm {op} }\times A)\times _{B^{\mathrm {op} }\times B}\operatorname {Tw} (B)}$ is a fiberwise equivalence over ${\displaystyle A^{\mathrm {op} }\times A}$.^[10]

• For a functor ${\displaystyle u\colon A\rightarrow B}$ between ∞-categories ${\displaystyle A}$ and ${\displaystyle B}$, the induced maps:

  ${\displaystyle \operatorname {Tw} (A)\rightarrow (A^{\mathrm {op} }\times B)\times _{B^{\mathrm {op} }\times B}\operatorname {Tw} (B),}$

  ${\displaystyle \operatorname {Tw} (A)\rightarrow (B^{\mathrm {op} }\times A)\times _{B^{\mathrm {op} }\times B}\operatorname {Tw} (B),}$

are cofinal.^[11]

Literature

• Cisinski, Denis-Charles (2019-06-30). Higher Categories and Homotopical Algebra (PDF). Cambridge University Press. ISBN 978-1108473200.

References

1. Cisinski 2019, 5.6.1.
2. Cisinski 2019, 5.6.10.
3. Kerodon, Proposition 8.1.1.10.
4. Cisinski 2019, Proposition 5.6.2.
5. Kerodon, Proposition 8.1.1.11.
6. Kerodon, Corollary 8.1.1.12.
7. Kerodon, Corollary 8.1.1.13.
8. Cisinski 2019, Proposition 5.6.12.
9. Cisinski 2019, Corollary 5.6.14.
10. Cisinski 2019, Corollary 5.6.6.
11. Cisinski 2019, Proposition 5.6.9.

External links

• The Twisted Arrow Construction on Kerodon