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 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.

Definition and related data

Formally, a Δ-set together with maps

, that satisfy

is omitted for brevity.

are the sets of n share their, i.e. that the simplexes are well-formed. Δ-set 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

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:

as

denotes a standard n-simplex as a space, and

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

where

is a "restricted" geometric realization.

Related functors

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, whose n-th group is the free abelian group

, and whose n-th differential is defined by

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 as follows. A singular n-simplex in X is a continuous map

Define

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

by

-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, and use it to compute the homology as an interval with the endpoints identified, define

are

is

where

for all n. The homology of this chain complex is also simple to compute:

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.

which is 0 if and only if.

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.

Relation with simplical sets

We now inspect the relation between Δ-sets and simplicial sets. Consider the simplex category and whose morphisms are monotone maps. A simplicial set is defined to be a presheaf, i.e. a (contravariant) functor whose morphisms are only the strict. From this we see that a presheaf, one concludes that

. Thus, a presheaf determines the data of a Δ-set and, conversely, all Δ-sets arise in this way. between Δ-sets correspond to natural transformations as (contravariant) functors. In this sense, Δ-sets are.

From this perspective, it is now easy to see that every simplicial set; so that every simplicial set.

Pros 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.

See also

• Simplicial complexes

• Simplicial sets

• Singular homology

References