Introduction
This page continues from the page about fiber bundles here. Here this is generalised to the notion of a fibration.
Fibrations have additional structure to fiber bundles which allows some structure in B to be 'lifted' to E. (see Wikipedia page)
Lifting
Given a path [0,1] in the base space B and a point e_{0} in the total space E. We can 'lift' this path into the total space E as a path starting at e_{0.} 
Here I am mostly concentrating on discrete structures so we need a looser concept of continuous mappings. 
An inclusion from [0,1] into B can be lifted into E: If we are using CWcomplexes to model our topological spaces then we can use a slightly weaker form of fibration known as a Serre fibration. 
Simple ExampleFibration and cofibration is related to fibre bundles as discussed on the page here. To recap from this page we can start with a surjective mapping between sets and then add extra structure to these sets. In this example we have two sets B the base set and E the total set and a surjective mapping p between them. 
We get a fibre bundle by trying to get as close as we can to the inverse of the function p without making arbitrary choices. 
On this page want to expand the idea of fibre bundles on sets to add more structure. As an example lets go from sets to directed graphs. The function p must now map the arrows as well as the vertices. 
What is the closest we can get to an inverse function? Here is a possibility. This 'lifts' the vertices to sets which may contain multiple vertices. 
Is this valid because two paths join and this is like a tear which is not allowed in topology/homotopy transforms? 
So E is like a product. Here we lift the interval to the red arrow in E. note: we think of this as being continuous even though this example has discrete vertices. 
But E only needs to be like a product locally. 
Can we think of this lifted mapping as being between the sets? 
Fibration
Fibrations of Graphs
theory: topological graph theory
undirected graph:  covering projection 
directed graph:  fibration (weaker form of covering projection) 
Here is an example for directed multigraphs. Each node in the top graph maps to the bottom graph (fibration). The corresponding node always has the same number and colour of incoming arcs (but not necessarily outgoing arcs). This gives some sort of local invariance. 
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.
Next
If we reverse the arrows in the diagram for fibrations we get the diagram for cofobrations:
Cofibration involves the concept of extension
 fibration  lift
 cofibration  extension
Extension is dual to lift.
 Fibrations and Cofibrations are used in model theory.