Simplicial Sets

Simplicial sets are an extension to simplicial complexes.

Model Category

A model category is a context in which we can do homotopy theory or some generalization thereof; two model categories are ‘the same’ for this purpose if they are Quillen equivalent. (nlab) simplical sets

A model category (sometimes called a Quillen model category) is a context for doing homotopy theory.

It is a category equipped with three classes of morphisms, each closed under composition:

Quillen Equivalence

We need a mapping that preserves the structure of a topological space: the cycles, the boundaries, the holes and so on. However it does not need to conserve the exact points, faces of the complexes that hold them.

The two shapes on the right are the same from a topology point of view they are a disk with a boundary round it.

quillen equivilance
In order to do this we map the connections between the dimensions in the chain (boundaries), not the dimensions (points, faces, etc.) themselves.

Simplicial Set

X: Δop -> Set is a simpicial set



See page about chain complexes.

Chain Homotopy

Weak Equivalences

Usually equivalences are defined in terms of two functors in opposite directions however a weaker notion of equivalence is defined in terms of a functor going in one direction only.


A Serre fibration arises when we reverse the functor defining the weak equivalence.


In any model category:

Cubical Type Theory

Cubical type theory extends these concepts with the idea of an interval [0,1]. This is discussed further on the page here.

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Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Mathematics++: Selected Topics Beyond the Basic Courses (Student Mathematical Library) Kantor, Ida.


  1. Measure
  2. High Dimensional Geometry
  3. Fourier Analysis
  4. Representations of Finite Groups
  5. Polynomials
  6. Topology

Chapter 6 - Topology. Contains a relatively gentle introduction to homology.


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