Simplicial Sets

A simplicial set is a functor from deltaop to set.

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

simplicial set diagram

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.


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. diagram

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.



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: diagram

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.

triangle 1 diagram

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.

triangle 2 diagram
Or we can glue more simplicies. For instance we can glue (A0,A1) to (C1,C0) diagram

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.

glueing diagram


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.

subset structure diagram

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.


Face Maps
di : Xn -> Xn-1 | 0≤i≤n

Degeneracy Maps
sj : Xn -> Xn+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, 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 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
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

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


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



See page about chain complexes.



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


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

3 {0,0,0,0} {0,0,0,0}


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

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


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

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


{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.


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


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.


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

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


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