Simplicial set Guide, Meaning , Facts, Information and Description

In mathematics, a simplicial set is a sequence of sets

together with face maps

and degeneracy maps

for each and every . These maps must obey certain identities:

1. if
2. if
3. if or
4. if
5. if

When it's understood which we're working with the first subscript is usually omitted. That is, is written and as . When each is a group we say that is a simplicial group.

Table of contents

1 Examples 1.1 The Standard Simplicial Set 1.2 Singular complex for a space 1.3 Braids 2 Homotopy 3 Geometric realization 4 Categorical definition 5 References

Examples

The Standard Simplicial Set

Let , the -simplex. The face maps are

given by

The degeneracy maps are

given by

Singular complex for a space

See singular homology

Given a space we can define . For each , define the degeneracy map where is the

Braids

''See braid theory, braid group

Homotopy

If is a simplicial group we may define its homotopy groups. Let

Then we have a sequence

where the maps between the sets are .

Define the th homotopy group of as the th homology group of this chain. That is

Geometric realization

Give the definition of the geometric realization.

Note that the homotopy group defined for this is actually the homotopy group of the realization.

Categorical definition

Using the language of category theory, a simplicial set is a simplicial object in , that is, a contravariant functor from the simplicial category to .

Alternatively, a simplicial set can be viewed as a presheaf on .

These definitions arise from the relationship of the conditions imposed on the face and degeneracy maps to the category of finite, totally ordered sets. There is an equivalence of categories between it and the subcategory with objects

{0, 1, ..., n}

with the usual order ≤. This is the usual definition of Δ.

References

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