Delta set

Formally, a Δ-set is a sequence of sets ${\displaystyle \{S_{n}\}_{n=0}^{\infty }}$ together with maps

${\displaystyle d_{i}\colon S_{n+1}\rightarrow S_{n}}$

with ${\displaystyle i=0,1,\ldots ,n+1}$ for ${\displaystyle n\geq 1}$ that satisfy

${\displaystyle d_{i}\circ d_{j}=d_{j-1}\circ d_{i}}$

whenever ${\displaystyle i<j}$.

This definition generalizes the notion of a simplicial complex, where the ${\displaystyle S_{n}}$ are the sets of n-simplices, and the ${\displaystyle d_{i}}$ are the face maps. It is not as general as a simplicial set, since it lacks "degeneracies."

Given Δ-sets S and T, a map of Δ-sets is a collection of set-maps

${\displaystyle \{f_{n}\colon S_{n}\rightarrow T_{n}\}_{n=0}^{\infty }}$

such that

${\displaystyle f_{n}\circ d_{i}=d_{i}\circ f_{n+1}}$

whenever both sides of the equation are defined. With this notion, we can define the category of Δ-sets, whose objects are Δ-sets and whose morphisms are maps of Δ-sets.

Each Δ-set has a corresponding geometric realization, defined as

${\displaystyle |S|=\left(\coprod _{n=0}^{\infty }S_{n}\times \Delta ^{n}\right)/_{\sim }}$

where we declare that

${\displaystyle (\sigma ,d^{i}t)\sim (d_{i}\sigma ,t)\quad {\text{ for all }}\sigma \in S_{n},t\in \Delta ^{n-1}.}$

Here, ${\displaystyle \Delta ^{n}}$ denotes the standard n-simplex, and

${\displaystyle d^{i}\colon \Delta ^{n-1}\rightarrow \Delta ^{n}}$

is the inclusion of the i-th face. The geometric realization is a topological space with the quotient topology.

The geometric realization of a Δ-set S has a natural filtration

${\displaystyle |S|_{0}\subset |S|_{1}\subset \cdots \subset |S|,}$

where

${\displaystyle |S|_{N}=\left(\coprod _{n=0}^{N}S_{n}\times \Delta ^{n}\right)/_{\sim }}$

is a "restricted" geometric realization.

The geometric realization of a Δ-set described above defines a covariant functor from the category of Δ-sets to the category of topological spaces. Geometric realization takes a Δ-set to a topological space, and carries maps of Δ-sets to induced continuous maps between geometric realizations.

If S is a Δ-set, there is an associated free abelian chain complex, denoted ${\displaystyle (\mathbb {Z} S,\partial )}$, whose n-th group is the free abelian group

${\displaystyle (\mathbb {Z} S)_{n}=\mathbb {Z} \langle S_{n}\rangle ,}$

generated by the set ${\displaystyle S_{n}}$, and whose n-th differential is defined by

${\displaystyle \partial _{n}=d_{0}-d_{1}+d_{2}-\cdots +(-1)^{n}d_{n}.}$

This defines a covariant functor from the category of Δ-sets to the category of chain complexes of abelian groups. A Δ-set is carried to the chain complex just described, and a map of Δ-sets is carried to a map of chain complexes, which is defined by extending the map of Δ-sets in the standard way using the universal property of free abelian groups.

Given any topological space X, one can construct a Δ-set ${\displaystyle \mathrm {sing} (X)}$ as follows. A singular n-simplex in X is a continuous map

${\displaystyle \sigma \colon \Delta ^{n}\rightarrow X.}$

Define

${\displaystyle \mathrm {sing} _{n}^{}(X)}$

to be the collection of all singular n-simplicies in X, and define

${\displaystyle d_{i}\colon \mathrm {sing} _{i+1}(X)\rightarrow \mathrm {sing} _{i}(X)}$

by

${\displaystyle d_{i}(\sigma )=\sigma \circ d^{i},}$

where again ${\displaystyle d^{i}}$ is the ${\displaystyle i}$-th face map. One can check that this is in fact a Δ-set. This defines a covariant functor from the category of topological spaces to the category of Δ-sets. A topological space is carried to the Δ-set just described, and a continuous map of spaces is carried to a map of Δ-sets, which is given by composing the map with the singular n-simplices.

This example illustrates the constructions described above. We can create a Δ-set S whose geometric realization is the unit circle ${\displaystyle S^{1}}$, and use it to compute the homology of this space. Thinking of ${\displaystyle S^{1}}$ as an interval with the endpoints identified, define

${\displaystyle S_{0}=\{v\},\quad S_{1}=\{e\},}$

with ${\displaystyle S_{n}=\varnothing }$ for all ${\displaystyle n\geq 2}$. The only possible maps ${\displaystyle d_{0},d_{1}\colon S_{1}\rightarrow S_{0},}$ are

${\displaystyle d_{0}(e)=d_{1}(e)=v.\quad }$

It is simple to check that this is a Δ-set, and that ${\displaystyle |S|\cong S^{1}}$. Now, the associated chain complex ${\displaystyle (\mathbb {Z} S,\partial )}$ is

${\displaystyle 0\longrightarrow \mathbb {Z} \langle e\rangle {\stackrel {\partial _{1}}{\longrightarrow }}\mathbb {Z} \langle v\rangle \longrightarrow 0,}$

where

${\displaystyle \partial _{1}(e)=d_{0}(e)-d_{1}(e)=v-v=0.}$

In fact, ${\displaystyle \partial _{n}=0}$ for all n. The homology of this chain complex is also simple to compute:

${\displaystyle H_{0}(\mathbb {Z} S)={\frac {\ker \partial _{0}}{\mathrm {im} \partial _{1}}}=\mathbb {Z} \langle v\rangle \cong \mathbb {Z} ,}$

${\displaystyle H_{1}(\mathbb {Z} S)={\frac {\ker \partial _{1}}{\mathrm {im} \partial _{2}}}=\mathbb {Z} \langle e\rangle \cong \mathbb {Z} .}$

All other homology groups are clearly trivial.

One advantage of using Δ-sets in this way is that the resulting chain complex is generally much simpler than the singular chain complex. For reasonably simple spaces, all of the groups will be finitely generated, whereas the singular chain groups are, in general, not even countably generated.

One drawback of this method is that one must prove that the geometric realization of the Δ-set is actually homeomorphic to the topological space in question. This can become a computational challenge as the Δ-set increases in complexity.

• Simplicial complexes
• Simplicial sets
• Singular homology

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