Delta set

In mathematics, a Δ-set "S", often called 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 general as a simplicial set.
As an example, suppose we want to triangulate the 1-dimensional circle formula_1. To do so with a simplicial complex, we need at least two vertices (e.g. one at the top and one at the bottom), and two edges connecting them. But delta-sets allow for a simpler triangulation: thinking of formula_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.
Definition and related data.
Formally, a Δ-set is a sequence of sets formula_3 together with maps
with formula_5 for formula_6 that satisfy
whenever formula_8.
This definition generalizes the notion of a simplicial complex, where the formula_9 are the sets of "n"-simplices, and the formula_10 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
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
where we declare that
Here, formula_15 denotes the standard "n"-simplex, 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, denoted formula_19, whose "n"-th group is the free abelian group
generated by the set formula_9, 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", one can construct a Δ-set formula_23 as follows. A singular "n"-simplex in "X" is a continuous map
Define
to be the collection of all singular "n"-simplicies in "X", and define
by
where again formula_28 is the formula_29-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.
An example.
This example illustrates the constructions described above. We can create a Δ-set "S" whose geometric realization is the unit circle formula_1, and use it to compute the homology of this space. Thinking of formula_1 as an interval with the endpoints identified, define
with formula_33 for all formula_34. The only possible maps formula_35 are
It is simple to check that this is a Δ-set, and that formula_37. Now, the associated chain complex formula_38 is
where
In fact, formula_41 for all "n". The homology of this chain complex is also simple to compute:
All other homology groups are clearly trivial.
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.