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 Cofibration
Homotopy has the concept of:
 a fibration which has the lifting property.
 a cofibration which has the extension property extension is dual to lift.
Fibration 
Cofibration (Extension Property) 


Homotopy  Fibration 
Cofibration (see page here) 
Combinatorics 
Kan fibration (see page here) 
Kan extension (see page here) 
Kan fibrations are combinatorial analogs of Serre fibrations of topological spaces.
Homotopy Type Theory
Lawvere Theory
Further Info
see pages on this site: