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 formula_1, hence functors formula_2 with the simplex category formula_3. The category of bisimplicial sets is denoted:
 formula_4
Let formula_5 be the canonical projections, then there are induced functors formula_6 by precomposition. For simplicial sets formula_7 and formula_8, there is a bisimplicial set formula_9with:
 formula_10
 formula_11
Let formula_12 be the diagonal functor, then there is an induced functor formula_13 by precomposition. For a bisimplicial set formula_14, there is a simplicial set formula_15 with:
 formula_16
Adjoints.
The diagonal formula_13 has a left adjoint formula_18 with formula_19 and a right adjoint formula_20 with formula_21.
Let formula_22 be a simplicial set. The functor formula_23 has a right adjoint:
 formula_24
The functor formula_25 has a right adjoint:
 formula_26
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 formula_27 and formula_28, which are:
 formula_29
 formula_30
 formula_31
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".