Simplicial Set

A simplicial set is a functor from delta^op to set.

Where delta is a (weakly) ordered set as described on the page here.

When denoting faces in a complex, order is important for finding things like boundaries.

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 multiple times.

This is an example of a degenerate face.

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

Example Graph

Each edge has two maps into node, s (source) and t (target).

The map in the other direction is the degeneracy map. That is each node maps to a self loop.

So both the nodes and edges can be represented as 2 dimensional faces.

Example Triangle

Things get more complicated when we move up from lines to surfaces.

There are 3 ways to get from a triangle to an edge (by deleting each node) and 2 ways to get from a edge to a triangle (degenerate map from each end of the edge).

A face, such as a triangle, contains its boundary, such as the edges and nodes.

If we take all the permutations of 3 numbers in (non strict) order we have 1 for each node, 1 for the whole triangle but 2 for each edge.

Gluing

Here we glue two faces of simplicies to form a simplicial set.

So here we have glued (a0) to (b1) and (a1) to (b0) to form a loop.

So if we apply M twice we get back to where we started.

We can express this as a group presentation like this:

[M | M*M=1]

In the next example there are 3 gluings

(a0) to (b1)

(b0) to (c1)

(c0) to (a1)

to form a loop.

So if we apply R 3 times we get back to where we started.

We can express this as a group presentation like this:

[R | R*R*R=1]

We can take the product of these two examples:

Gluing in 2 dimensions

So here we have glued (A1,A2) to (B2,B1).

So, order is not preserved over the whole simplicial set, just over an individual simplex. The whole simplical set is not a total order as there are faces (for example A0 and B0) which cant be compared.

We can glue any dimension face (point, line, triangle... ) as long as the two faces being glued have the same dimension.

We can then go on to glue more faces, for instace, in addition to gluing (A1,A2) to (B2,B1) as above we could additionally glue (A0,A1) to (B1,B2).

Note that, in this case, this only leaves one remaining face. This is known as a horn clause, this means that the remaining face is also equal. I t also means that the two simplicies are also equal or opposite.

Or we can glue more simplicies. For instance we can glue (A0,A1) to (C1,C0)

By doing this glueing we define an equivalence relation over the top of the original ordered set.

To do this we can either construct the quotient space, that is S\~ where S is the original ordered set and ~ is the equivalence relation. This effectively makes them the same point (which may not work with the ordering) or we use a setoid to represent ~ over S.

see

Adjunction space Wiki
Quotient space Wiki
Setoid page here
Setoid related categories page here

Subset Structure

Above we have the set of all functions into the complex which is a homset.

We can treat this homset as a set (see Yoneda) that is ignore the structure below it so that it is a set category including functions (the structure above it).

A structure we can take from the homset is the subset structure.

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

Degeneracy Maps s_j :

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

Face Maps
d_i : X_n -> X_n-1 | 0≤i≤n

Degeneracy Maps
s_j : X_n -> X_n+1 | 0≤j≤n

Simplicial Identities

The following is based on a category of ordered sets, Order preserving arrows can be decomposed into a sequence of arrows that insert or merge single elements one at a time. More about ordered sets on page here.

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

Identity	An example of one case in a triangle {0,1,2}
d_i d_j = d_j-1 d_i if i < j
d_i s_j = s _j-1 d_i if i < j
d_j s_j = id = d_j+1 s_j
d_i s_j = s_j d_i-1 if i > j+1
s_i s_j = s _j+1 s _i if i ≤ j

Here are all the possible 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
d_i d_j = d_j-1 d_i if i < j	since i < j these are the only combinations allowed	d_j	d_i d_j	d_i	d_j-1 d_i
	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
d_i s_j = s _j-1 d_i if i < j	since i < j these are the only combinations allowed	s_j	d_i s_j	d_i	s _j-1 d_i
	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
d_j s_j = id = d_j+1 s_j		s_j	d_j s_j	s_j	d_j+1 s_j
	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
d_i s_j = s_j d_i-1 if i > j+1	since i > j+1 these are the only combinations allowed	s_j	d_i s_j	d_i-1	s_j d_i-1
d_i s_j = s_j d_i-1 if i > j+1	i=2,j=0	{0,0,1,2}	{0,0,2}	{0,2}	{0,0,2}
		1st stage	result	1st stage	result
s_i s_j = s _j+1 s _i if i ≤ j	since i ≤ j these are the only combinations allowed	s_j	s_i s_j	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)

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).

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

Δ^op 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 Δ^op->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.

X: Δ^op -> Set is a simplicial set

where:

Δ is a finite totally ordered set {0,1,2...,n}

See page about chain complexes.

Notation


Δ	The delta category whose objects are the simplicies.
Δ^op	The delta category with morphisms reversed.
Δⁿ	an n-simplex which may be degenerate
\|Δⁿ\|	Ordered n-simplex = [0, ... ,n]
X	A simplicial complex.
X_n	The set of all n simplicies of X
d_i(x)	an order preserving function which removes the i^th element from a simplex
s_i(x)	an order preserving function which duplicates the i^th element in a simplex
D_i(x)	Like d_i(x) but in Δ^op so it increases the dimension rather than decrementing it
S_i(x)	Like s_i(x) but in Δ^op 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.

Δ⁰

Δ¹

Δ²

...

{0}

{0}
{1}

{0}
{1}
{2}

{0,0}

{0,0}
{0,1}
{1,1}

{0,0}
{0,1}
{0,2}
{1,1}
{1,2}
{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}

{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:

(X × Y)_n = X_n × Y_n = {(x,y) xX_n, yY_n}
if (x,y)(X × Y)_n then d_i(x,y) = (d_i x, d_i y)
if (x,y)(X × Y)_n then s_i(x,y) = (s_i x, s_i y)

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)

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.

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}
Δ^op	ordered sets	Order preserving maps example: {0,1,2,3,4} -> {0,1,2} may remove points (choose a face).

Objects

Morphisms

ordered sets

example: {0,1,2}

Order preserving maps

may be inclusions (add points), example:

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

Δ^op

ordered sets

Order preserving maps

example:

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

may remove points (choose a face).

Chain Homotopy

Wiki
Youtube

Weak Equivalences

This is discussed on the page here.
Wiki

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:

A fibration that is also a weak equivalence is called a trivial (or acyclic) fibration.
A cofibration that is also a weak equivalence is called a trivial (or acyclic) cofibration.

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.

Simplicial Sets

Degeneracy

Example Graph

Example Triangle

Gluing

Gluing in 2 dimensions

Subset Structure

Simplicial Set Structure

Simplicial Identities

Simplicial Set from Delta Set

Face Maps

Simplicial Set

Notation

Combinatorics

Products

Example - product of two lines Δ¹×Δ¹

Cubical Set

Model Category

Quillen Equivalence

Simplicial Set in Category Theory

Chain Homotopy

Weak Equivalences

Fibrations

Cofibrations

Cubical Type Theory

Code

EuclideanSpace

Where Are You

Other Sites

Computer Code Implementation

Algebraic Topology

Other Sites

Simplicial Sets

Degeneracy

Example Graph

Example Triangle

Gluing

Gluing in 2 dimensions

Subset Structure

Simplicial Set Structure

Simplicial Identities

Simplicial Set from Delta Set

Face Maps

Simplicial Set

Notation

Combinatorics

Products

Example - product of two lines Δ1×Δ1

Cubical Set

Model Category

Quillen Equivalence

Simplicial Set in Category Theory

Chain Homotopy

Weak Equivalences

Fibrations

Cofibrations

Cubical Type Theory

Code

EuclideanSpace

Where Are You

Other Sites

Computer Code Implementation

Algebraic Topology

Other Sites

Example - product of two lines Δ¹×Δ¹