Bisimplicial set

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.

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. =\mathbf{Fun}(\Delta^\mathrm{op},\mathbf{sSet}) \cong\mathbf{Fun}((\Delta\times\Delta)^\mathrm{op},\mathbf{Set}) Let \operatorname{pr}_1,\operatorname{pr}_2\colon \Delta\times\Delta\rightarrow\Delta be the canonical projections, then there are induced functors \operatorname{pr}_1^*,\operatorname{pr}_2^*\colon \mathbf{sSet}\rightarrow\mathbf{bisSet} by precomposition. For simplicial sets X and Y , there is a bisimplicial set X\boxtimes Y with: X\boxtimes Y =\operatorname{pr}_1^*(A)\times\operatorname{pr}_1^*(B), (X\boxtimes Y)_{m,n} =X_m\times Y_n. Let \delta\colon \Delta\rightarrow\Delta\times\Delta be the diagonal functor, then there is an induced functor \delta^*=\operatorname{diag}\colon \mathbf{bisSet}\rightarrow\mathbf{sSet} by precomposition. For a bisimplicial set Z , there is a simplicial set \delta^*(Z) with: \delta^*(Z)_n =Z_{n,n}. The diagonal \delta^*=\operatorname{diag}\colon \mathbf{bisSet}\rightarrow\mathbf{sSet} has a left adjoint \delta_!\colon \mathbf{sSet}\rightarrow\mathbf{bisSet} with \delta_!\dashv\delta^* and a right adjoint \delta_*\colon \mathbf{sSet}\rightarrow\mathbf{bisSet} with \delta^*\dashv\delta_* . Let K be a simplicial set. The functor K\boxtimes-\colon \mathbf{sSet}\rightarrow\mathbf{bisSet} has a right adjoint: {}^K-\colon \mathbf{bisSet}\rightarrow\mathbf{sSet}, ({}^KX)_n =\operatorname{Hom}(K\boxtimes\Delta^n,X) =\varprojlim_{\Delta^m\rightarrow K}X_{m,n}. The functor -\boxtimes K\colon \mathbf{sSet}\rightarrow\mathbf{bisSet} has a right adjoint: -^K\colon \mathbf{bisSet}\rightarrow\mathbf{sSet}, (X^K)_m =\operatorname{Hom}(\Delta^n\boxtimes K,X) =\varprojlim_{\Delta^n\rightarrow K}X_{m,n}. The diagonal functor \delta^*=\operatorname{diag}\colon \mathbf{bisSet}\rightarrow\mathbf{sSet} send left/right bi-anodyne extensions to left/right anodyne extensions. The diagonal functor \delta_!\colon \mathbf{sSet}\rightarrow\mathbf{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: (\Delta\times\Delta)/(A\boxtimes B) \cong\Delta/A\times\Delta/B, \delta^*(A\boxtimes B) \cong A\times B.

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