Modeling Maths

For a given mathematical structure, say natural numbers, there a 2 sorts of relationships that may hold:

• Things that are true in all instances (such as axioms or identities).
• Things that apply only to the current problem under investigation (such as some equations)

There are correspondences between homotopy and other mathematical structures.

Try to represent the above in homotopy terms in the form of simplicial sets.

Equations might be loops in simplicial sets.

• Simplicies have a substructure that is fixed.
• Complexes may be glued together in different ways depending on what we are doing.

How do these two things interact?

See fibrations

Fibration and Co-fibration

Homotopy has the concept of:

• a fibration which has the lifting property.
• a co-fibration which has the extension property -extension is dual to lift.

Fibration
(lifting property)

Co-fibration
(Extension Property)
Homotopy

Fibration
(see page here)

Co-fibration
(see page here)

Combinatorics
(simplicial sets)

Kan fibration
(see page here)
Kan extension
(see page here)

Kan fibrations are combinatorial analogs of Serre fibrations of topological spaces.

Further Info

see pages on this site: