### Mappings in Set

In order to get some sort of intuitive understanding it may help to compare with the following mappings in set:

- bijections
- surjections
- inclusions

These have some interesting properties (wiki):

- Mappings/functions can be decomposed into surjections and inclusions. So we can define mappings as a sequence of surjections and inclusions.
- A bijection is both a surjection and an inclusion.
- A bijection is invertible one-to-one correspondence (permutation).
- An inclusion is like a one-to-one correspondence with possible additional elements in the codomain.

#### Mappings and Subsets

Inclusions give us a subset structure. The combination of subobjects and functions from and to them gives us some interesting issues.

For example, in some programming languages such, as Java, we can have functions from and to subobjects (that is an object that has been extended):

## CovarianceGiven an object Any function out of |

## ContravarianceAny function in to |

### Mappings in Model Categories

Having looked at the above structures in set we can now look at related structures in model categories.

The weak equivalences are like ‘homotopy equivalences’ that is shapes that can be deformed into each other. They are 'weak' because they are spaces that can be represented by simplicial sets, that is there are no awkward singularities or things like that which may be possible in more general case (see: Warsaw circle - ncatlab).

The fibrations are Serre fibrations and play the role of ‘nice surjections’. For a general discussion about fibrations see the page here.

The cofibrations play the role of ‘nice inclusions’.

### Retractions

A retraction is a surjective map that is continuous. For instance we can continuously reduce a band down to a circle: |

In general a map ƒ: M1 -> M2 is continuous if and only if the preimage of every open set is open. |