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Delta set

Abstraction useful in the construction and triangulation of topological spaces

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Source updated June 13, 2026

In mathematics, a Δ-set, often called a Δ-complex or a semi-simplicial set, is a combinatorial object that is useful in the construction and triangulation of topological spaces, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a simplicial complex, yet not quite as sophisticated as a simplicial set. Simplicial sets have additional structure, so that every simplicial set is also a semi-simplicial set.

As an example, suppose we want to triangulate the 1-dimensional circle $S^{1}$ . To do so with a simplicial complex, we need at least three vertices, and edges connecting them. But delta-sets allow for a simpler triangulation: thinking of $S^{1}$ as the interval [0,1] with the two endpoints identified, we can define a triangulation with a single vertex 0, and a single edge looping between 0 and 0.

03Examples

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

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

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

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

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

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

where

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

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

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

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

All other homology groups are clearly trivial.

The following example is from section 2.1 of Hatcher's Algebraic Topology. 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 $\partial _{1}$ is 0 because there is only one vertex, so $H_{0}(T^{2})={\text{ker }}\partial _{0}/{\text{ im }}\partial _{1}=\mathbb {Z}$ . Let $\{e_{0}^{1},e_{1}^{1},e_{0}^{1}+e_{1}^{1}-e_{2}^{1}\}$ be a basis for $\Delta _{1}(T^{2})$ . Then $\partial _{2}(e_{0}^{2})=e_{0}^{1}+e_{1}^{1}-e_{2}^{1}=\partial _{2}(e_{1}^{2})$ , so ${\text{im }}\partial _{2}=\langle e_{0}^{1}+e_{1}^{1}-e_{2}^{1}\rangle$ , and hence $H_{1}(T^{2})={\text{ker }}\partial _{1}/{\text{ im }}\partial _{2}=\mathbb {Z} ^{3}/\mathbb {Z} =\mathbb {Z} ^{2}.$

Since there are no 3-simplices, $H_{2}(T^{2})={\text{ker }}\partial _{2}$ . We have that $\partial _{2}(pe_{0}^{2}+qe_{1}^{2})=(p+q)(e_{0}^{1}+e_{1}^{1}-e_{2}^{1})$ which is 0 if and only if $p=-q$ . Hence ${\text{ker }}\partial _{2}$ is infinite cyclic generated by $e_{0}^{2}-e_{1}^{2}$ .

So $H_{2}(T^{2})=\mathbb {Z}$ . Clearly $H_{n}(T^{2})=0$ for $n\geq 3.$

Thus, $H_{n}(T^{2})={\begin{cases}\mathbb {Z} &n=0,2\\\mathbb {Z} ^{2}&n=1\\0&n\geq 3.\end{cases}}$

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.

04Relation with simplical sets

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

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

whenever $i<j$ . Thus, a presheaf on ${\hat {\Delta }}$ determines the data of a Δ-set and, conversely, all Δ-sets arise in this way. Moreover, Δ-maps $f:S\to T$ between Δ-sets correspond to natural transformations when we view $S$ and $T$ as (contravariant) functors. In this sense, Δ-sets are presheaves on ${\hat {\Delta }}$ while simplicial sets are presheaves on $\Delta$ .

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

05Pros and cons

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.

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Sources and credits

This article is adapted from the Wikipedia article “Delta set”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Circle structures.svg by Homo morphisms, CC0
Delta maps.svg by Homo morphisms, CC0
Delta structures.pdf by Homo morphisms, CC0

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Delta set

03Examples

04Relation with simplical sets

05Pros and cons

Sources and credits

Related topics

Simplicial complex

Simplicial set

Singular homology

Delta set

01Definition and related data

02Related functors

03Examples

04Relation with simplical sets

05Pros and cons

Sources and credits

Related topics

Simplicial complex

Simplicial set

Singular homology