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:^[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 $\operatorname {Tw} (A)\rightarrow A^{\mathrm {op} }\times A$ .^[1]

Twisted diagonal with the diamond operation

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

\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:^[3]
$N\operatorname {Tw} ({\mathcal {C}})=\operatorname {Tw} (N{\mathcal {C}}).$
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.^[4]^[5]^[6]
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.^[7]
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.^[8]
For an ∞-category $A$ , the canonical morphism $\operatorname {Tw} (A)\rightarrow \operatorname {Tw} _{\diamond }(A)$ is a fiberwise equivalence of left fibrations over $A^{\mathrm {op} }\times A$ .^[9]
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 $A^{\mathrm {op} }\times A$ .^[10]
For a functor $u\colon A\rightarrow B$ between ∞-categories $A$ and $B$ , the induced maps:
$\operatorname {Tw} (A)\rightarrow (A^{\mathrm {op} }\times B)\times _{B^{\mathrm {op} }\times B}\operatorname {Tw} (B),$

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