Delta set

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

:$d\_i^n\; \backslash colon\; S\_\{n+1\}\; \backslash rightarrow\; S\_n$

for each $n\backslash geq\; 0$ and $i=0,1,\backslash dots,n+1$, that satisfy

:$d\_i^\{n\}\backslash circ\; d\_j^\{n+1\}=d\_\{j-1\}^\{n\}\backslash circ\; d\_i^\{n+1\}$

whenever . Often, the superscript of $\{d\_i^n\}$ is omitted for brevity.

This definition generalizes the notion of a simplicial complex, where the $S\_n$ are the sets of n-simplices, and the $d\_i$ are the associated face maps, each mapping the $i$-th face of a simplex in $S\_\{n+1\}$ to a simplex in $S\_\{n\}$. The composition rule ensures that the faces in $S\_\{n+1\}$ of a simplex in $S\_\{n+2\}$ share their neighboring faces in $S\_n$, i.e. that the simplexes are well-formed. Δ-set is not as general as a simplicial set, since it lacks "degeneracies".

File:Delta maps.svg

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

:$\backslash \{\; f\_n\backslash colon\; S\_n\; \backslash rightarrow\; T\_n\; \backslash \}\_\{n=0\}^\{\backslash infty\}$

such that

:$f\_n\; \backslash circ\; d\_i\; =\; d\_i\; \backslash circ\; f\_\{n+1\}$

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, associating a geometrically defined space (a standard n-simplex) with each abstract simplex in Δ-set, and then "gluing" the spaces together using inclusion relations between the spaces to define an equivalence relation:

:$|S|\; =\; \backslash left(\; \backslash coprod\_\{n=0\}^\{\backslash infty\}\; S\_n\; \backslash times\; \backslash Delta^n\; \backslash right)/\_\{\backslash sim\}$

where we declare $\{\backslash sim\}$ as

: $(\backslash sigma,\backslash iota\; ^i\; t)\; \backslash sim\; (d\_i\; \backslash sigma,\; t)\; \backslash quad\; \backslash text\{\; for\; all\; \}\; \backslash sigma\; \backslash in\; S\_n,\; t\; \backslash in\; \backslash Delta^\{n-1\}.$

Here, $\backslash Delta^n$ denotes a standard n-simplex as a space, and

:$\backslash iota^i\; \backslash colon\; \backslash Delta^\{n-1\}\; \backslash rightarrow\; \backslash Delta^n$

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

: $|S|\_0\; \backslash subset\; |S|\_1\; \backslash subset\; \backslash cdots\; \backslash subset\; |S|,$

where

:$|S|\_N\; =\; \backslash left(\; \backslash coprod\_\{n=0\}^\{N\}\; S\_n\; \backslash times\; \backslash Delta^n\; \backslash right)/\_\{\backslash sim\}$

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 $(\backslash Z\; S,\; \backslash partial\; )$, whose n-th group is the free abelian group

: $(\backslash Z\; S)\_n\; =\; \backslash Z\; \backslash langle\; S\_n\; \backslash rangle,$

generated by the set $S\_n$, and whose n-th differential is defined by

: $\backslash partial\_n\; =\; d\_0\; -\; d\_1\; +\; d\_2\; -\; \backslash cdots\; +\; (-1)^n\; d\_n.$

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 $\backslash mathrm\{sing\}(X)$ as follows. A singular n-simplex in X is a continuous map

: $\backslash sigma\; \backslash colon\; \backslash Delta^n\; \backslash rightarrow\; X.$

Define

: $\backslash mathrm\{sing\}\_n^\{\}(X)$

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

:$d\_i\; \backslash colon\; \backslash mathrm\{sing\}\_\{i+1\}(X)\; \backslash rightarrow\; \backslash mathrm\{sing\}\_i(X)$

by

:$d\_i(\backslash sigma)\; =\; \backslash sigma\; \backslash circ\; d^i,$

where again $d^i$ is the $i$-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 $S^1$, and use it to compute the homology of this space. Thinking of $S^1$ as an interval with the endpoints identified, define

:$S\_0\; =\; \backslash \{v\backslash \},\; \backslash quad\; S\_1\; =\; \backslash \{e\backslash \},$

with $S\_n\; =\; \backslash varnothing$ for all $n\; \backslash ge\; 2$. The only possible maps $d\_0,d\_1\backslash colon\; S\_1\; \backslash rightarrow\; S\_0,$ are

:$d\_0(e)\; =\; d\_1(e)\; =\; v.\; \backslash quad$

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

:$0\; \backslash longrightarrow\; \backslash Z\; \backslash langle\; e\; \backslash rangle\; \backslash stackrel\{\backslash partial\_1\}\{\backslash longrightarrow\}\; \backslash Z\; \backslash langle\; v\; \backslash rangle\; \backslash longrightarrow\; 0,$File:Delta_structures.pdf

where

:$\backslash partial\_1(e)\; =\; d\_0(e)\; -\; d\_1(e)\; =\; v\; -\; v\; =\; 0.$

In fact, $\backslash partial\_n\; =\; 0$ for all n. The homology of this chain complex is also simple to compute:

:$H\_0(\backslash Z\; S)\; =\; \backslash frac\{\backslash ker\; \backslash partial\_0\}\{\backslash mathrm\{im\}\; \backslash partial\_1\}\; =\; \backslash mathbb\{Z\}\; \backslash langle\; v\; \backslash rangle\; \backslash cong\; \backslash Z,$

:$H\_1(\backslash Z\; S)\; =\; \backslash frac\{\backslash ker\; \backslash partial\_1\}\{\backslash mathrm\{im\}\; \backslash partial\_2\}\; =\; \backslash mathbb\{Z\}\; \backslash langle\; e\; \backslash rangle\; \backslash cong\; \backslash Z.$

All other homology groups are clearly trivial.

The following example is from section 2.1 of Hatcher's Algebraic Topology.{{Cite book |last=Hatcher |first=Allen |url=https://www.worldcat.org/oclc/45420394 |title=Algebraic topology |date=2002 |publisher=Cambridge University Press |isbn=0-521-79160-X |location=Cambridge |oclc=45420394}} Consider the Δ-set structure given to the torus in the figure, which has one vertex, three edges, and two 2-simplices.

The boundary map $\backslash partial\_1$ is 0 because there is only one vertex, so $H\_0(T^2)\; =\; \backslash text\{ker\; \}\; \backslash partial\_0/\backslash text\{\; im\; \}\; \backslash partial\_1\; =\; \backslash mathbb\{Z\}$. Let $\backslash \{e\_0^1,\; e\_1^1,e\_0^1+e\_1^1-e\_2^1\backslash \}$ be a basis for $\backslash Delta\_1(T^2)$. Then $\backslash partial\_2(e\_0^2)\; =\; e\_0^1+e\_1^1-e\_2^1=\; \backslash partial\_2(e\_1^2)$, so $\backslash text\{im\; \}\; \backslash partial\_2\; =\; \backslash langle\; e\_0^1+e\_1^1-e\_2^1\; \backslash rangle$, and hence $H\_1(T^2)\; =\; \backslash text\{ker\; \}\backslash partial\_1\; /\; \backslash text\{\; im\; \}\; \backslash partial\_2\; =\; \backslash mathbb\{Z\}^3/\backslash mathbb\{Z\}\; =\; \backslash mathbb\{Z\}^2.$

Since there are no 3-simplices,  $H\_2(T^2)=\backslash text\{ker\; \}\; \backslash partial\_2$. We have that $\backslash partial\_2(p\; e\_0^2\; +\; q\; e\_1^2)\; =\; (p+q)\; (e\_0^1+e\_1^1-e\_2^1)$ which is 0 if and only if $p=-q$.  Hence $\backslash text\{ker\; \}\; \backslash partial\_2$ is infinite cyclic generated by $e\_0^2-e\_1^2$.

So $H\_2(T^2)=\backslash mathbb\{Z\}$. Clearly $H\_n(T^2)=0$ for $n\backslash ge\; 3.$

Thus, $H\_n(T^2)\; =\; \backslash begin\{cases\}$

\mathbb{Z} & n=0,2\\

\mathbb{Z}^2 & n=1\\

0 & n\ge 3.

\end{cases}

It is worth highlighting that the minimum number of simplices needed to endow $T^2$

with the structure of a simplicial complex is 7 vertices,  21 edges, and 14 2-simplices, for a total of 42 simplices. This would make the above calculations, which only used 6 simplices, much harder for someone to do by hand.

This is a non-example. Consider a line segment. This is a 1-dimensional Δ-set and a 1-dimensional simplicial set. However, if we view the line segment as a 2-dimensional simplicial set, in which the 2-simplex is viewed as degenerate, then the line segment is not a Δ-set, as we do not allow for such degeneracies.

We now inspect the relation between Δ-sets and simplicial sets. Consider the simplex category $\backslash Delta$, whose objects are the finite totally ordered sets $[n]\; :=\; \backslash \{0,\; 1,\backslash cdots\; ,\; n\backslash \}$ and whose morphisms are monotone maps. A simplicial set is defined to be a presheaf on $\backslash Delta$, i.e. a (contravariant) functor $S:\; \backslash Delta^\{\backslash text\{op\}\}\; \backslash to\; \backslash text\{Set\}$. On the other hand, consider the subcategory $\backslash hat\{\backslash Delta\}$ of $\backslash Delta$ whose morphisms are only the strict monotone maps. Note that the morphisms in $\backslash hat\{\backslash Delta\}$ are precisely the injections in $\backslash Delta$, and one can prove that these are generated by the monotone maps of the form $\backslash delta^i:\; [n]\; \backslash to\; [n+1]$ which "skip" the element $i\; \backslash in\; [n+1]$. From this we see that a presheaf $S:\; \backslash hat\{\backslash Delta\}^\{\backslash text\{op\}\}\; \backslash to\; \backslash text\{Set\}$ on $\backslash hat\{\backslash Delta\}$ is determined by a sequence of sets $\backslash \{S\_n\backslash \}\_\{n\; =\; 0\}^\backslash infty$ (where we denote $S([n])$ by $S\_n$ for simplicity) together with maps $d\_i:\; S\_\{n+1\}\; \backslash to\; S\_\{n\}$ for $i\; =\; 0,1,\backslash ldots,n+1$ (where we denote $S(\backslash delta^i)$ by $d\_i$ for simplicity as well). In fact, after checking that $\backslash delta^j\; \backslash circ\; \backslash delta^i\; =\; \backslash delta^i\; \backslash circ\; \backslash delta^\{j-1\}$ in $\backslash hat\{\backslash Delta\}$, one concludes that

:$d\_i\; \backslash circ\; d\_j\; =\; d\_\{j-1\}\; \backslash circ\; d\_i$

whenever . Thus, a presheaf on $\backslash hat\{\backslash Delta\}$ determines the data of a Δ-set and, conversely, all Δ-sets arise in this way.{{cite journal

| last = Friedman | first = Greg

| arxiv = 0809.4221

| doi = 10.1216/RMJ-2012-42-2-353

| issue = 2

| journal = The Rocky Mountain Journal of Mathematics

| mr = 2915498

| pages = 353–423

| title = Survey article: An elementary illustrated introduction to simplicial sets

| volume = 42

| year = 2012}} Moreover, Δ-maps $f:\; S\; \backslash to\; T$ between Δ-sets correspond to natural transformations when we view $S$ and $T$ as (contravariant) functors. In this sense, Δ-sets are presheaves on $\backslash hat\{\backslash Delta\}$ while simplicial sets are presheaves on $\backslash Delta$.

From this perspective, it is now easy to see that every simplicial set is a Δ-set. Indeed, notice there is an inclusion $\backslash hat\{\backslash Delta\}\; \backslash hookrightarrow\; \backslash Delta$; so that every simplicial set $S:\; \backslash Delta^\{\backslash text\{op\}\}\; \backslash to\; \backslash text\{Set\}$ naturally gives rise to a Δ-set, namely the composite $\backslash hat\{\backslash Delta\}^\{\backslash text\{op\}\}\; \backslash hookrightarrow\; \backslash Delta^\{\backslash text\{op\}\}\; \backslash xrightarrow\{S\}\; \backslash text\{Set\}$.

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

{{Reflist}}

• {{Cite book

|last= Ranicki

|first= Andrew A.

|author-link = Andrew Ranicki

|title= Algebraic L-theory and Topological Manifolds

|publisher= Cambridge Univ. Press

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|series= Cambridge Tracts in Mathematics

|volume= 102

|isbn= 978-0-521-42024-2

|url=http://www.maths.ed.ac.uk/~aar/books/topman.pdf

}}

• {{cite journal

| last1=Ranicki | first1=Andrew |author1-link = Andrew Ranicki

| last2=Weiss | first2=Michael |author2-link = Michael Weiss (mathematician)

| title=On The Algebraic L-theory of Δ-sets | journal=Pure and Applied Mathematics Quarterly | volume=8 | issue=2 | year=2012 | doi=10.4310/pamq.2012.v8.n2.a3 | pages=423–450 |arxiv=math.AT/0701833| mr=2900173}}

• {{Cite journal

|last1= Rourke

|first1= Colin P.

|author-link1=Colin P. Rourke

|last2= Sanderson

|first2= Brian J.

|title= Δ-Sets I: Homotopy Theory

|journal= Quarterly Journal of Mathematics

|year= 1971

|volume= 22

|pages=321–338

|doi= 10.1093/qmath/22.3.321

|issue=3

|bibcode= 1971QJMat..22..321R

}}

{{DEFAULTSORT:Delta Set}}

Category:Topology

Category:Algebraic topology

Category:Simplicial sets