Bisimplicial set, diagonal, and the realisation lemma

Intuition [Beginner]

A bisimplicial set is a grid of simplicial sets indexed by another simplicial coordinate. Imagine a two-dimensional array where each row is a simplicial set, and the rows themselves are organised like the simplices of another simplicial set. This gives a simplicial-object-in-simplicial-sets, or equivalently, a functor from ${\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}$ to sets.

The diagonal of a bisimplicial set picks out the diagonal entries: the simplices at position $(n,n)$. This produces an ordinary simplicial set. The realisation lemma says that if two bisimplicial sets are equivalent row-by-row (each corresponding pair of rows is weakly equivalent), then their diagonals are also weakly equivalent.

This sounds abstract, but it has a concrete purpose: it lets you prove that two constructions produce the same homotopy type by checking them level by level, which is often much easier than a direct comparison. For example, the homotopy colimit of a diagram of spaces can be computed as the diagonal of a bisimplicial set built from the diagram.

Visual [Beginner]

A grid of squares. Along the horizontal axis, simplicial degree $p$. Along the vertical axis, simplicial degree $q$. Each cell at position $(p,q)$ is a set. The diagonal entries $(0,0),(1,1),(2,2),\dots$ are highlighted. An arrow from the grid to a single column (a simplicial set) labelled "diag."

The diagonal extracts a single simplicial set from the two-dimensional array, preserving homotopical information.

Worked example [Beginner]

The diagonal of the nerve of a category. Let $\mathcal{C}$ be a small category. The nerve $N(\mathcal{C})$ is a simplicial set. Consider the bisimplicial set $X$ with $X_{p,q}=N(\mathcal{C}{)}_p$ (constant in the $q$-direction). Then $\mathrm{diag}(X{)}_n=X_{n,n}=N(\mathcal{C}{)}_n$, so the diagonal recovers the original nerve.

A more interesting example: the bisimplicial set $X$ with $X_{p,q}=\mathrm{Hom}([p]\times [q],\mathcal{C})$ (functors from the product category $[p]\times [q]$ to $\mathcal{C}$). The diagonal $\mathrm{diag}(X)$ is the nerve of the twisted arrow category of $\mathcal{C}$, which encodes the higher associativity data of composition.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Bisimplicial set). A bisimplicial set is a functor $X:{\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}}\to \mathbf{Set}$. Equivalently, $X$ is a simplicial object in the category of simplicial sets: a functor ${\Delta }^{\mathrm{op}}\to \mathbf{sSet}$. The category of bisimplicial sets is $\mathbf{ssSet}=\mathbf{Fun}({\Delta }^{\mathrm{op}}\times {\Delta }^{\mathrm{op}},\mathbf{Set})$.

For a bisimplicial set $X$, we write $X_{p,q}$ for the set at position $(p,q)$. The face and degeneracy operators come in two families: horizontal operators $d_i^h,s_i^h$ acting on the first index $p$, and vertical operators $d_i^v,s_i^v$ acting on the second index $q$.

Definition (Diagonal). The diagonal of a bisimplicial set $X$ is the simplicial set $\mathrm{diag}(X)$ defined by $\mathrm{diag}(X{)}_n=X_{n,n}$, with face operators $d_i=d_i^h\circ d_i^v$ and degeneracy operators $s_i=s_i^h\circ s_i^v$.

Definition (Realisation). The realisation of a bisimplicial set $X$ is the geometric realisation of its diagonal: $\Vert X\Vert =|\mathrm{diag}(X)|$. Equivalently, $\Vert X\Vert ={⨆}_{p,q}{X}_{p,q}\times |{\Delta }^p|\times |{\Delta }^q|$ modulo the face and degeneracy relations in both directions.

Key theorem with proof [Intermediate+]

Theorem (Realisation lemma). Let $f:X\to Y$ be a map of bisimplicial sets that is a weak homotopy equivalence in each vertical degree: for each fixed $p$, the map $f_{p,} : X_{p,} \to Y_{p,}$ofsimplicialsetsisaweakequivalence.Thentheinducedmap$\mathrm{diag}(f) : \mathrm{diag}(X) \to \mathrm{diag}(Y)$ is a weak equivalence of simplicial sets.*

Proof sketch. The key ingredient is the skeletal filtration. For a bisimplicial set $X$, the diagonal $\mathrm{diag}(X)$ has a filtration by the horizontal degree: ${\mathrm{filt}}_n(\mathrm{diag}(X))$ consists of those simplices coming from $X_{p,\ast }$ with $p\le n$. This gives a sequence of pushouts building $\mathrm{diag}(X)$ from $\mathrm{diag}({\mathrm{sk}}_0^{h}X)$, and the associated graded pieces are wedges of realisations of the vertical simplicial sets. The map $f$ induces weak equivalences on each associated graded piece (since $f_{p,\ast }$ is a weak equivalence for each $p$), and by induction on the skeletal filtration and the five lemma applied to the long exact sequences of homotopy groups, $\mathrm{diag}(f)$ is a weak equivalence. $\square$

Bridge. The realisation lemma is the bisimplicial refinement of the skeletal filtration of 03.12.24; the horizontal skeleta play the role of CW skeleta while the vertical simplicial sets provide the "cells," and the level-wise equivalence condition parallels the CW-approximation guarantee of 03.12.26 that weak equivalences on skeleta lift to the full space. The diagonal construction itself is a two-dimensional generalisation of the nerve construction for categories, connecting back to the classifying-space ideas used in the simplicial group machinery of 03.12.39.

Exercises [Intermediate+]

Advanced results [Master]

Bousfield-Kan homotopy colimit. The diagonal of the bisimplicial bar construction $B(\ast ,\mathcal{C},F)$ is the Bousfield-Kan formula for the homotopy colimit of a diagram $F:\mathcal{C}\to \mathbf{sSet}$. The realisation lemma ensures that weakly equivalent diagrams produce weakly equivalent homotopy colimits.

Artin-Mazur codiagonal. The codiagonal (or "total decoherence") $\nabla :\mathbf{ssSet}\to \mathbf{sSet}$ is the left adjoint to the functor that sends a simplicial set $K$ to the constant bisimplicial set at $K$. The codiagonal can be computed as a coend $\nabla (X)={\int }^{[p]}{X}_{p,\ast }\times {\Delta }^p$. There is a natural weak equivalence $|\mathrm{diag}(X)|\simeq |\nabla (X)|$, so the diagonal and codiagonal give the same homotopy type.

Synthesis. The bisimplicial framework extends the simplicial-set machinery of 03.12.24 to two dimensions, the diagonal functor condenses the two-dimensional array back to a one-dimensional simplicial set while preserving homotopical information, the realisation lemma provides the level-wise comparison tool that makes the CW-approximation of 03.12.26 work inductively, and the Bousfield-Kan homotopy colimit connects to the Puppe sequences of 03.12.27 and 03.12.28 by ensuring that homotopy colimits of fibre sequences remain fibre sequences. The entire construction underpins the Dwyer-Kan simplicial localisation of categories and the theory of $\infty$-categories as bisimplicial sets satisfying Segal conditions.

Full proof set [Master]

Proposition (Spectral sequence for the diagonal). For a bisimplicial set $X$ that is level-wise fibrant (each $X_{p,}$isaKancomplex),thereisahomotopyspectralsequence$E^2_{p,q} = \pi_p^h \pi_q^v(X) \Rightarrow \pi_{p+q}(|\mathrm{diag}(X)|)$,where$\pi_q^v(X)$denotestheverticalhomotopygroups(computing$\pi_q(X_{p,})$foreach$p$)and$\pi_p^h$ denotes the horizontal homotopy groups of the resulting simplicial group.

Proof. Filter $\mathrm{diag}(X)$ by horizontal skeleton: ${\mathrm{filt}}_n(\mathrm{diag}(X))$ is the diagonal of the $n$-skeleton of $X$ in the horizontal direction. The successive quotients are: ${\mathrm{filt}}_n/{\mathrm{filt}}_{n-1}={\bigvee }_{\sigma \in N{X}_{n,\ast }}{\Sigma }^n(|\sigma |)$ where $N{X}_{n,\ast }$ is the set of non-degenerate horizontal $n$-simplices and $|\sigma |$ is the geometric realisation of the vertical simplicial set at position $\sigma$. The spectral sequence of this filtration has $E^1$ page given by the homotopy groups of these successive quotients, which reorganise into $E_{p,q}^1={\pi }_q(X_{p,\ast })$ (the vertical homotopy groups). The $d^1$ differential is induced by the horizontal face maps, giving $E_{p,q}^2=H_p({\pi }_q^v(X);d^1)$. Convergence to ${\pi }_{p+q}(|\mathrm{diag}(X)|)$ follows from the skeletal convergence theorem for homotopy spectral sequences of filtered spaces. $\square$

Connections [Master]

Simplicial sets 03.12.24 provide both coordinates of a bisimplicial set; the face and degeneracy operators in each direction satisfy the same simplicial identities.

The realisation lemma is a level-wise version of the CW-approximation theorem 03.12.26, and it ensures that the diagonal construction preserves weak equivalences constructed skeleton by skeleton.

The simplicial group and W-bar construction 03.12.39 uses bisimplicial techniques in its proof: the bar construction $W(G)$ is defined as a bisimplicial set whose diagonal gives the classifying space $BG$.

Bibliography [Master]

@book{quillen1967,
  author = {Quillen, Daniel},
  title = {Homotopical Algebra},
  series = {Lecture Notes in Mathematics},
  volume = {43},
  publisher = {Springer},
  year = {1967}
}

@book{goerss-jardine1999,
  author = {Goerss, Paul G. and Jardine, John F.},
  title = {Simplicial Homotopy Theory},
  publisher = {Birkh{\"a}user},
  year = {1999}
}

@book{bousfield-kan,
  author = {Bousfield, Aldridge K. and Kan, Daniel M.},
  title = {Homotopy Limits, Completions and Localizations},
  series = {Lecture Notes in Mathematics},
  volume = {304},
  publisher = {Springer},
  year = {1972}
}

@article{dwyer-kan1980,
  author = {Dwyer, William G. and Kan, Daniel M.},
  title = {Simplicial localizations of categories},
  journal = {J. Pure Appl. Algebra},
  volume = {17},
  pages = {267--284},
  year = {1980}
}

Historical & philosophical context [Master]

Bisimplicial sets were introduced in the context of Quillen's 1967 "Homotopical Algebra" ^{[Quillen 1967]}, which used them to construct model structures on categories of diagrams and to prove theorems about homotopy colimits. The realisation lemma appears implicitly in Quillen's work and was made explicit by Bousfield and Kan in their 1972 monograph ^{[Bousfield-Kan 1972]}.

The philosophical significance is that bisimplicial sets allow "two-dimensional" homotopy theory: constructions that proceed simultaneously in two simplicial directions. The diagonal condenses these two dimensions into one while preserving homotopical information, providing a powerful tool for comparing constructions. The Bousfield-Kan homotopy colimit formula, built from bisimplicial sets, is the standard tool for computing homotopy colimits in algebraic topology and underlies the Dwyer-Kan theory of simplicial localisations, which connects category theory to homotopy theory.