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In mathematics, a Δ-set ''S'', often called a semi-simplicial set, is a
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
object that is useful in the construction and
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle ...
of
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s, and also in the computation of related algebraic invariants of such spaces. A Δ-set is somewhat more general than a
simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
, yet not quite as general as a
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
.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 ,1with the two endpoints identified, we can define a triangulation with a single vertex 0, and a single edge looping between 0 and 0.


Definition and related data

Formally, a Δ-set is a sequence of sets \_^ together with maps :d_i \colon S_ \rightarrow S_n with i = 0,1,\ldots,n+1 for n\ge 1 that satisfy : d_i \circ d_j = d_ \circ d_i whenever i < j. 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 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 : \_^ such that : f_n \circ d_i = d_i \circ f_ whenever both sides of the equation are defined. With this notion, we can define the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...
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 :, S, = \left( \coprod_^ S_n \times \Delta^n \right)/_ where we declare that : (\sigma,d^i t) \sim (d_i \sigma, t) \quad \text \sigma \in S_n, t \in \Delta^. Here, \Delta^n denotes the standard ''n''-simplex, and : d^i \colon \Delta^ \rightarrow \Delta^n is the inclusion of the ''i''-th face. The geometric realization is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
with the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
. The geometric realization of a Δ-set ''S'' has a natural
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
: , S, _0 \subset , S, _1 \subset \cdots \subset , S, , where :, S, _N = \left( \coprod_^ S_n \times \Delta^n \right)/_ is a "restricted" geometric realization.


Related functors

The geometric realization of a Δ-set described above defines a covariant
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
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 In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
, denoted (\Z S, \partial ), whose ''n''-th group is the
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a su ...
: (\Z S)_n = \Z \langle S_n \rangle, generated by the set S_n, and whose ''n''-th differential is defined by : \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 In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...
of free abelian groups. Given any topological space ''X'', one can construct a Δ-set \mathrm(X) as follows. A singular ''n''-simplex in ''X'' is a continuous map : \sigma \colon \Delta^n \rightarrow X. Define : \mathrm_n^(X) to be the collection of all singular ''n''-simplicies in ''X'', and define :d_i \colon \mathrm_(X) \rightarrow \mathrm_i(X) by : d_i(\sigma) = \sigma \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.


Examples

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 = \, \quad S_1 = \, with S_n = \varnothing for all n \ge 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 (\Z S,\partial) is : 0 \longrightarrow \Z \langle e \rangle \stackrel \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(\Z S) = \frac = \mathbb \langle v \rangle \cong \Z, : H_1(\Z S) = \frac = \mathbb \langle e \rangle \cong \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 \partial_0/\text \partial_1 = \mathbb. Let \ 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 \partial_2 = \langle e_0^1+e_1^1-e_2^1 \rangle, and hence H_1(T^2) = \text\partial_1 / \text \partial_2 = \mathbb^3/\mathbb = \mathbb^2. Since there are no 3-simplices,  H_2(T^2)=\text \partial_2. We have that \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 \text \partial_2 is infinite cyclic generated by e_0^2-e_1^2. So H_2(T^2)=\mathbb. Clearly H_n(T^2)=0 for n\ge 3. Thus, H_n(T^2) = \begin \mathbb & n=0,2\\ \mathbb^2 & n=1\\ 0 & n\ge 3. \end 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.


Abstract nonsense

We now inspect the relation between Δ-sets and
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
s. Consider the
simplex category In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving maps. It is used to define simplicial and cosimplicial objects. Formal definit ...
\Delta, whose objects are the finite totally ordered sets := \ and whose morphisms are monotone maps. A
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
is defined to be a
presheaf In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
on \Delta, i.e. a (contravariant)
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...
S: \Delta^ \to \text. On the other hand, consider the subcategory \hat of \Delta whose morphisms are only the ''strict'' monotone maps. Note that the morphisms in \hat are precisely the injections in \Delta, and one can prove that these are generated by the monotone maps of the form \delta^i: \to +1/math> which "skip" the element i \in +1/math>. From this we see that a
presheaf In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
S: \hat^ \to \text on \hat is determined by a sequence of sets \_^\infty (where we denote S( by S_n for simplicity) together with maps d_i: S_ \to S_ 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^ in \hat, one concludes that : d_i \circ d_j = d_ \circ d_i whenever i < j. Thus, a
presheaf In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
on \hat 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 transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a na ...
s when we view S and T as (contravariant) functors. In this sense, Δ-sets ''are'' presheaves on \hat while simplicial sets are presheaves on \Delta. From this persepective, it is now easy to see that every
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
is a Δ-set. Indeed, notice there is an inclusion \hat \hookrightarrow \Delta ; so that every
simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
S: \Delta^ \to \text naturally gives rise to a Δ-set, namely the composite \hat^ \hookrightarrow \Delta^ \xrightarrow \text.


Pros and cons

One advantage of using Δ-sets in this way is that the resulting
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
is generally much simpler than the
singular chain complex In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
. 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.


See also

*
Simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...
es *
Simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...
s *
Singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...


References

* * * {{DEFAULTSORT:Delta Set Topology Algebraic topology Simplicial sets