Simplicial Sets

Simplicial sets are an extension to simplicial complexes (see page here).

To get to simplicial sets from simplicial complexes it is necessary to order (or partially order) the vertices.

In addition to the face maps in simplicial complexes we add degeneracy maps in the opposite direction these maps are order preserving.

Simplicial sets have an additional property that they are closed under products and this is important in Cubical TT for example.

Degeneracy

In homology (as portrayed on the page here) the emphasis is on the relationship between adjacent dimensions. For instance, if we take the boundary, we map from a given dimension to the dimension immediately below it (face maps). This situation can be modeled by simplicial complexes already described on the page here.

In homotopy (as portrayed on the page here) we may have an n-dimensional path in a dimension several levels higher. This situation may be better modeled by simplicial sets as described here.

For a 2-dimensional face in 2-dimensional space all the vertices have different indices.

For a 0-dimensional point in 2-dimensional space we can use the same index.

This is a degenerate face.

In cubical type theory degeneracy maps corresponds to weakening in Logic.

Simplicial Set Structure

In order to handle these degenerate faces better we add degenerate maps going in the reverse direction to the face maps.

We have maps going in both directions:

Face Maps di :
Takes an n-simplex and gives its i-th (n-1) face.

Degeneracy Maps sj :

Takes an n-simplex and gives its j-th degenerate (n+1) simplex inside it.

diagram

Face Maps
di : Xn -> Xn-1 | 0≤i≤n

Degeneracy Maps
sj : Xn -> Xn+1 | 0≤j≤n

Simplicial Identities

Note: when combining maps below the map on the right is done first. For example, di dj means do dj then do di.

Identity An example of one case in a triangle {0,1,2}
di dj = dj-1 di if i < j
di sj = s j-1 di if i < j
dj sj = id = dj+1 sj
di sj = sj di-1 if i > j+1
si sj = s j+1 s i if i ≤ j

Here are all the posible cases of the identities for the example of a triangle {0,1,2}.

 
example {0,1,2}
    left hand side right hand side
    1st stage result 1st stage result
di dj = dj-1 di if i < j since i < j these are the only combinations allowed dj di dj di dj-1 di
i=0,j=1 {0,2} d0d1={2} {1,2} d0d0={2}
i=0,j=2 {0,1} d0d2={1} {1,2} d1d0={1}
i=1,j=2 {0,1} d1d2={0} {0,2} d1d1={0}
    1st stage result 1st stage result
di sj = s j-1 di if i < j since i < j these are the only combinations allowed sj di sj di s j-1 di
i=0,j=1 {0,1,1,2} d0s1={1,1,2} {1,2} s0d0={1,1,2}
i=0,j=2 {0,1,2,2} d0s2={1,2,2} {1,2} s1d0={1,2,2}
i=1,j=2 {0,1,2,2} d1s2={0,2,2} {0,2} s1d1={0,2,2}
    1st stage result 1st stage result
dj sj = id = dj+1 sj   sj dj sj sj dj+1 sj
i=0 {0,0,1,2} {0,1,2} {0,0,1,2} {0,1,2}
i=1 {0,1,1,2} {0,1,2} {0,1,1,2} {0,1,2}
i=2 {0,1,2,2} {0,1,2} {0,1,2,2} {0,1,2}
    1st stage result 1st stage result
di sj = sj di-1 if i > j+1 since i > j+1 these are the only combinations allowed sj di sj di-1 sj di-1
i=2,j=0 {0,0,1,2} {0,0,2} {0,2} {0,0,2}
    1st stage result 1st stage result
si sj = s j+1 s i if i ≤ j since i ≤ j these are the only combinations allowed sj si sj s i s j+1 s i
i=0,j=0 {0,0,1,2} {0,0,0,1,2} {0,0,1,2} {0,0,0,1,2}
i=0,j=1 {0,1,1,2} {0,0,1,1,2} {0,0,1,2} {0,0,1,1,2}
i=0,j=2 {0,1,2,2} {0,0,1,2,2} {0,0,1,2} {0,0,1,2,2}
i=1,j=1 {0,1,1,2} {0,1,1,1,2} {0,1,1,2} {0,1,1,1,2}
i=1,j=2 {0,1,2,2} {0,1,1,2,2} {0,1,1,2} {0,1,1,2,2}
i=2,j=2 {0,1,2,2} {0,1,2,2,2} {0,1,2,2} {0,1,2,2,2}

Simplicial Set from Delta Set

On the delta set page each type of simplex was indexed, for example:

  • The 4 vertices (in red)
  • The 6 edges (in green)
  • The 4 triangles (in blue)
indexed representation
This representation holds all these indexes so that we don't have to keep creating them and they can be used consistently.

Face Maps

So the tetrahedron indexes its 4 triangles, each triangle indexes its 3 edges and each edge indexes its 2 vertices.

Each face table therefore indexes into the next. To show this I have drawn some of the arrows (I could not draw all the arrows as that would have made the diagram too messy).

indexed table
diagram

The delta category Δ has:

  • Objects: The simplicies.
  • Morphisms: Order preserving injective maps. They embed a face into a larger simplex.

The vertices of the simplicies are (partially) ordered by labeling them with natural numbers.

Notation: [M] stands for a simplex with vertices 0 to M. For example:
[2] := {0,1,2}

A morphism [M]->[N] is only defined if M≤N

diagram

Δop has the same objects as Δ but the morphisms are reversed.

The morphisms are now the same as the face maps above.

We can then make this into a sheaf Δop->set (as described on page here). This separates out the structure inherent in the maps between simplicies of different dimensions and the structure which comes from gluing them together. Perhaps we can think of the simplex structure as the local structure and the gluing provides the global structure.

Simplicial Set

X: Δop -> Set is a simplicial set

where:

  tetrahedron

See page about chain complexes.

Notation

   
Δ

The delta category whose objects are the simplicies.

Δop The delta category with morphisms reversed.
Δn an n-simplex which may be degenerate
n| Ordered n-simplex = [0, ... ,n]
X A simplicial complex.
Xn The set of all n simplicies of X
di(x) an order preserving function which removes the ith element from a simplex
si(x) an order preserving function which duplicates the ith element in a simplex
Di(x) Like di(x) but in Δop so it increases the dimension rather than decrementing it
Si(x) Like si(x) but in Δop so it decrements the dimension rather than increasing it

Combinatorics

The degeneracy loosens the coupling between dimension and number of elements.

The following table shows the posibilities in a combinatorial way.

Each entry must be ordered, the element values can stay the same (degeneracy) but they must not decrease.

Degenerate entries (where an element occurs more than once) are shown in red.

n Δ0 Δ1 Δ2 ...
0 {0} {0}
{1}
{0}
{1}
{2}
 
1 {0,0}
{0,0}
{0,1}
{1,1}
{0,0}
{0,1}
{0,2}
{1,1}
{1,2}
{2,2}
 
2 {0,0,0} {0,0,0}
{0,0,1}
{0,1,1}
{1,1,1}
{0,0,0}
{0,0,1}
{0,0,2}
{0,1,1}

{0,1,2}
{0,2,2}
{1,1,1}
{1,1,2}
{1,2,2}
{2,2,2}
 
3 {0,0,0,0} {0,0,0,0}
{0,0,0,1}
{0,0,1,1}
{0,1,1,1}
{0,1,1,1}
{0,0,0,0}
{0,0,0,1}
{0,0,0,2}
{0,0,1,1}
{0,0,1,2}
{0,0,2,2}
{0,1,1,1}
{0,1,1,2}
{0,1,2,2}
{0,2,2,2}
{1,1,1,1}
{1,1,1,2}
{1,1,2,2}
{1,2,2,2}
{2,2,2,2}
 
...        

Products

One of the advantages of simplicial sets, over simplicial complexes, is that they simplify products (products of simplicial complexes on page here).

The rules are:

To take the product of two simplicial sets multiply all the terms which have the same dimension. Duplicates can be removed, also degenerate versions of existing terms can be removed, for example:

{[0,0],[1,1]} = {[s0(0),s0(1)]} = s0{[0],[1]}

so that can cancel out with {[0],[1]}

Example - product of two lines Δ1×Δ1

Multiplying out all the terms at each dimension and canceling out duplicates as above gives:

{0}
{1}
× {0}
{1}
= {0}×{0} = {[0],[0]}
{0}×{1} = {[0],[1]}
{1}×{0} = {[1],[0]}
{1}×{1} = {[1],[1]}
{0,0}
{0,1}
{1,1}
× {0,0}
{0,1}
{1,1}
=

{0,0}×{0,1} = {[0,0],[0,1]}
{1,1}×{0,1} = {[1,1],[0,1]}
{0,1}×{0,0} = {[0,1],[0,0]}
{0,1}×{1,1} = {[0,1],[1,1]}
{0,1}×{0,1} = {[0,1],[0,1]}

{0,0,1}
{0,1,1}
× {0,0,1}
{0,1,1}
= {[0,0,1],[0,1,1]}
{[0,1,1],[0,0,1]}

 

Cubical Set

A cubical version of simplicial sets is described on the page here.

Model Category

A model category is a context in which we can do homotopy theory or some generalization thereof; two model categories are ‘the same’ for this purpose if they are Quillen equivalent. (nlab) simplical sets

A model category (sometimes called a Quillen model category) is a context for doing homotopy theory.

It is a category equipped with three classes of morphisms, each closed under composition:

    related structure in set  
weak equivalences   bijection  
fibrations   nice surjections (Serre fibrations)  
cofibrations   nice inclusions neighborhood retract

Quillen Equivalence

We need a mapping that preserves the structure of a topological space: the cycles, the boundaries, the holes and so on. However it does not need to conserve the exact points, faces of the complexes that hold them.

The two shapes on the right are the same from a topology point of view they are a disk with a boundary round it.

quillen equivilance
In order to do this we map the connections between the dimensions in the chain (boundaries), not the dimensions (points, faces, etc.) themselves.

Simplicial Set in Category Theory

  Objects Morphisms
Δ

ordered sets

example: {0,1,2}

Order preserving maps

may be inclusions (add points), example:

{0,1,2} -> {0,1,2,3,4}

Δop ordered sets

Order preserving maps

example:

{0,1,2,3,4} -> {0,1,2}

may remove points (choose a face).

Chain Homotopy

Weak Equivalences

Usually equivalences are defined in terms of two functors in opposite directions however a weaker notion of equivalence is defined in terms of a functor going in one direction only.

Fibrations

A Serre fibration arises when we reverse the functor defining the weak equivalence.

Cofibrations

In any model category:

Cubical Type Theory

Cubical type theory extends these concepts with the idea of an interval [0,1]. This is discussed further on the page here.

Code

Sage is mathematical software which implements simplicial sets, more about this on page here.


metadata block
see also:

See Sage:

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Mathematics++: Selected Topics Beyond the Basic Courses (Student Mathematical Library) Kantor, Ida.

Chapters:

  1. Measure
  2. High Dimensional Geometry
  3. Fourier Analysis
  4. Representations of Finite Groups
  5. Polynomials
  6. Topology

Chapter 6 - Topology. Contains a relatively gentle introduction to homology.

 

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