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 . 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 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 dataEdit

Formally, a Δ-set is a sequence of sets   together with maps

 

with   for   that satisfy

 

whenever  .

This definition generalizes the notion of a simplicial complex, where the   are the sets of n-simplices, and the   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,   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 functorsEdit

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  , whose n-th group is the free abelian group

 

generated by the set  , 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   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   is the  -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 exampleEdit

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 of this space. Thinking of   as an interval with the endpoints identified, define

 

with   for all  . The only possible maps   are

 

It is simple to check that this is a Δ-set, and that  . Now, the associated chain complex   is

 

where

 

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

 
 

All other homology groups are clearly trivial.

Pros and consEdit

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 alsoEdit

ReferencesEdit

  • Friedman, Greg (2012). "Survey Article: An elementary illustrated introduction to simplicial sets". Rocky Mountain Journal of Mathematics. 42 (2): 353–423. arXiv:0809.4221. doi:10.1216/rmj-2012-42-2-353. MR 2915498.
  • Ranicki, Andrew A. (1993). Algebraic L-theory and Topological Manifolds (PDF). Cambridge Tracts in Mathematics. 102. Cambridge Univ. Press. ISBN 978-0-521-42024-2.
  • Ranicki, Andrew; Weiss, Michael (2012). "On The Algebraic L-theory of Δ-sets". Pure and Applied Mathematics Quarterly. 8 (2): 423–450. arXiv:math.AT/0701833. doi:10.4310/pamq.2012.v8.n2.a3. MR 2900173.
  • Rourke, Colin P.; Sanderson, Brian J. (1971). "Δ-Sets I: Homotopy Theory". Quarterly Journal of Mathematics. 22 (3): 321–338. Bibcode:1971QJMat..22..321R. doi:10.1093/qmath/22.3.321.