The Ultimate Simplicial set - American History Information Guide and Reference

In mathematics, a simplicial set $Γ *$ is a sequence of sets

$\Gamma_0, \Gamma_1, \dots$

together with face maps

$s_{n,i}: \Gamma_{n-1} \to \Gamma_{n}$

and degeneracy maps

$d_{n,i}: \Gamma_{n+1} \to \Gamma_n$

for each $0 \geq i \geq n$ and every $n$ . These maps must obey certain identities:

$d_{n,i} \circ d_{n,j} = d_{n,j-1} \circ d_{n,i}$ if $i < j$
$d_{n,i} \circ s_{n,j} = s_{n,j-1} \circ d_{n,i}$ if $i < j$
$d_{n,i} \circ s_{n,j} = \mathrm{id}$ if $i = j$ or $i = j + 1$
$d_{n,i} \circ s_{n,j} = s_{n,j} \circ d_{n,i-1}$ if $i > j + 1$
$s_{n,i} \circ s_{n,j} = s_{n,j+1} \circ s_{n,i}$ if $i \geq j$

When it's understood which $Γ n$ we're working with the first subscript is usually omitted. That is, $d n, i$ is written $d i$ and $s n, i$ as $s i$ . When each $Γ n$ is a group we say that $Γ *$ is a simplicial group.

Contents

1 Examples

1.1 The Standard Simplicial Set
1.2 Singular complex for a space
1.3 Braids

2 Homotopy

3 Geometric realization

4 Categorical definition

5 References

Examples

The Standard Simplicial Set

Let $\Gamma_n = \Delta_n = \{(x_0, \dots, x_n) \in \mathbb{R}^{n+1}: 0\leq x_i \leq 1, \sum x_i = 1 \}$ , the $n$ -simplex. The face maps are

$s_i: \Delta_{n-1} \to \Delta_{n}$

given by

$s_i (x_0, ..., x_{n-1}) = (x_0, \dots, x_{i-1}, 0, x_{i}, \dots, x_{n-1}).$

The degeneracy maps are

$d_i: \Delta_{n+1} \to \Delta_n$

given by

$d_i(x_0, \dots, x_{n+1}) = (x_0, \dots, x_{i-1}, x_i + x_{i+1}, x_{i+2}, \dots, x_{n+1}).$

Singular complex for a space

See singular homology

Given a space $X$ we can define $\Gamma_n = \{ f: \Delta_n \to X \}$ . For each $0 \leq i \leq n$ , define the degeneracy map $d i (f)(x) = f (δ i (x))$ where $δ i$ is the

Braids

See braid theory, braid group

Homotopy

If $Γ *$ is a simplicial group we may define its homotopy groups. Let

$C_q = \cap_{i=1}^n ker( d_{q,i} ).$

Then we have a sequence

$\cdots \to C_{q+1} \to C_{q} \to C_{q-1} \to \cdots$

where the maps between the sets are $d q,0$ .

Define the $q$ th homotopy group of $Γ *$ as the $q$ th homology group of this chain. That is

$\pi_q ( \Gamma_* ) := \frac{ker(d_{q-1,0})}{im(d_{q,0})}.$

Geometric realization

Give the definition of the geometric realization.

Note that the homotopy group defined for this is actually the homotopy group of the realization.

Categorical definition

Using the language of category theory, a simplicial set is a simplicial object in $\mathbf{Set}$ , that is, a contravariant functor from the simplicial category $Δ$ to $\mathbf{Set}$ .

Alternatively, a simplicial set can be viewed as a presheaf on $Δ$ .

These definitions arise from the relationship of the conditions imposed on the face and degeneracy maps to the category of finite, totally ordered sets. There is an equivalence of categories between it and the subcategory with objects

{0, 1, ..., n}

with the usual order ≤. This is the usual definition of Δ.

References

Categories: Algebraic topology

Your American History Reference Guide! - Simplicial set

Simplicial set