## Covarient and Contravarient Functors

There are two ways that a given morphism can preserve structure:

This is the covarient case where the start of the arrow in C maps to the start of the arrow in D and the end of the arrow in C maps to the end of the arrow in D. | |

This is the contravarient case where the start of the arrow in C maps to the end of the arrow in D and the end of the arrow in C maps to the start of the arrow in D. |

Unless otherwise stated we usually assume that a morphism/functor is covarient so we can draw this as on the left. | |

For the contravarient case we could invent some sort of special symbol for a contravarient morphism/functor. However is usually easier to reverse the internal arrows in either C or D. On the left we have reversed C by making it C^{op}. So now we can use an ordinary arrow between C and D. |

The way that covarient and contravarient functors arise, in this case, is as follows.

Let 'A', 'X' and 'Y' be elements of a class 'C'. let 'f' be a functor from 'X' to 'Y'. Assume that both 'A' and 'f' are fixed:

covarient | contravarient |
---|---|

- If 'f' and 'h' are known then we can derive 'g' (by composition)
- If 'f' and 'g' are known then we cannot derive 'h' (see concepts of section and retraction).
So given some fixed 'f' the 'g' depends on 'h' which I have drawn below as an arrow between arrows. |
- If 'f' and 'g' are known then we can derive 'h' (by composition)
- If 'f' and 'h' are known then we cannot derive 'g' (see concepts of section and retraction).
So given some fixed 'f' the 'h' depends on 'g' which I have drawn below as an arrow between arrows. |

This arrow goes in the same direction to f, hence it is covarient. | This arrow goes in the opposite direction to f, hence it is contravarient. |

We now need to combine these ideas with the concept of hom sets as follows.