Bisimplicial set

Template:Short description In higher category theory in mathematics, a bisimplicial set is a simplicial object in the category of simplicial sets, which themselves are simplicial objects in the category of sets. Many concepts from homotopical algebra, which studies simplicial sets, can be transported over to the study of bisimplicial sets, which for example includes Kan fibrations and Kan complexes.

Definition

Bisimplicial sets are simplicial objects in the category of simplicial sets $𝐬𝐒𝐞𝐭$, hence functors ${\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}\to 𝐬𝐒𝐞𝐭$ with the simplex category $\mathrm{\Delta }$. The category of bisimplicial sets is denoted:

$𝐛𝐢𝐬𝐒𝐞𝐭:=𝐅𝐮𝐧({\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}},𝐬𝐒𝐞𝐭)\cong 𝐅𝐮𝐧((\mathrm{\Delta }\times \mathrm{\Delta }{)}^{\mathrm{o}\mathrm{p}},𝐒𝐞𝐭)$

Let ${\mathrm{pr}}_1,{\mathrm{pr}}_2:\mathrm{\Delta }\times \mathrm{\Delta }\to \mathrm{\Delta }$ be the canonical projections, then there are induced functors ${\mathrm{pr}}_1^{\ast },{\mathrm{pr}}_2^{\ast }:𝐬𝐒𝐞𝐭\to 𝐛𝐢𝐬𝐒𝐞𝐭$ by precomposition. For simplicial sets $X$ and $Y$, there is a bisimplicial set $X⊠Y$with:

$X⊠Y={\mathrm{pr}}_1^{\ast }(A)\times {\mathrm{pr}}_1^{\ast }(B),$

$(X⊠Y{)}_{m,n}=X_m\times Y_n.$

Let $\delta :\mathrm{\Delta }\to \mathrm{\Delta }\times \mathrm{\Delta }$ be the diagonal functor, then there is an induced functor ${\delta }^{\ast }=\mathrm{diag}:𝐛𝐢𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭$ by precomposition. For a bisimplicial set $Z$, there is a simplicial set ${\delta }^{\ast }(Z)$ with:

${\delta }^{\ast }(Z{)}_n=Z_{n,n}.$

Adjoints

The diagonal ${\delta }^{\ast }=\mathrm{diag}:𝐛𝐢𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭$ has a left adjoint ${\delta }_{!}:𝐬𝐒𝐞𝐭\to 𝐛𝐢𝐬𝐒𝐞𝐭$ with ${\delta }_{!}⊣{\delta }^{\ast }$ and a right adjoint ${\delta }_{\ast }:𝐬𝐒𝐞𝐭\to 𝐛𝐢𝐬𝐒𝐞𝐭$ with ${\delta }^{\ast }⊣{\delta }_{\ast }$.

Let $K$ be a simplicial set. The functor $K⊠-:𝐬𝐒𝐞𝐭\to 𝐛𝐢𝐬𝐒𝐞𝐭$ has a right adjoint:

${}^K-:𝐛𝐢𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭,({}^{K}X{)}_n:=\mathrm{Hom}(K⊠{\mathrm{\Delta }}^n,X)=\underset{{\mathrm{\Delta }}^m\to K}{\underleftarrow{\lim }}{X}_{m,n}.$

The functor $-⊠K:𝐬𝐒𝐞𝐭\to 𝐛𝐢𝐬𝐒𝐞𝐭$ has a right adjoint:

${-}^K:𝐛𝐢𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭,(X^K{)}_m:=\mathrm{Hom}({\mathrm{\Delta }}^n⊠K,X)=\underset{{\mathrm{\Delta }}^n\to K}{\underleftarrow{\lim }}{X}_{m,n}.$

Model structures

Model structures from the category of simplicial sets, with the most important being the Joyal and Kan–Quillen model structure, can be transported over to the category of bisimplicial sets using the injective and projective model structure. But it is more useful to instead take the analog replacements of the morphisms $\partial {\mathrm{\Delta }}^n\to {\mathrm{\Delta }}^n$ and ${\mathrm{\Lambda }}_k^n\to {\mathrm{\Delta }}^n$, which are:

$\partial {\mathrm{\Delta }}^m⊠{\mathrm{\Delta }}^n\cup {\mathrm{\Delta }}^m⊠\partial {\mathrm{\Delta }}^n\to {\mathrm{\Delta }}^m⊠{\mathrm{\Delta }}^n,$

${\mathrm{\Lambda }}_k^m⊠{\mathrm{\Delta }}^n\cup {\mathrm{\Delta }}^m⊠\partial {\mathrm{\Delta }}^n\to {\mathrm{\Delta }}^m⊠{\mathrm{\Delta }}^n,$

$\partial {\mathrm{\Delta }}^m⊠{\mathrm{\Delta }}^n\cup {\mathrm{\Delta }}^m⊠{\mathrm{\Lambda }}_k^n\to {\mathrm{\Delta }}^m⊠{\mathrm{\Delta }}^n$

and which lead from Kan fibrations to bifibrations, left/right fibrations to left/right bifibrations, anodyne extensions to bi-anodyne extensions, left/right anodyne extensions to left/right bi-anodyne extensions and Kan complexes to Kan bicomplexes.

Properties

• The diagonal functor ${\delta }^{\ast }=\mathrm{diag}:𝐛𝐢𝐬𝐒𝐞𝐭\to 𝐬𝐒𝐞𝐭$ send left/right bi-anodyne extensions to left/right anodyne extensions.
• The diagonal functor ${\delta }_{!}:𝐬𝐒𝐞𝐭\to 𝐛𝐢𝐬𝐒𝐞𝐭$ send left/right anodyne extensions to left/right bi-anodyne extensions.
• For simplicial sets $X$ and $Y$, one has an isomorphism of slice categories:

  $(\mathrm{\Delta }\times \mathrm{\Delta })/(A⊠B)\cong \mathrm{\Delta }/A\times \mathrm{\Delta }/B,$

  ${\delta }^{\ast }(A⊠B)\cong A\times B.$

Literature

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