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.