§ Left and right adjoints to inverse image
§ The story in set
- Suppose is a morphism of sets.
- Then there is an associated morphism , the inverse image.
- We get two associated morphisms, , which perform universal and existential quantification "relative to ".
- The idea is this: think of as being fibered over by . Then gives the set of such that the fiber of lies entirely in . That is, .
- In pictures, Suppose the
@ mark the subset of , while the
- is outside the subset. We draw as being fibered over .
- @ @
- - @
- - @
| | |
v v v
b1 b2 b3
- Then, will give us , because it's only whose entire fiber lies in .
- Dually, will give us , because some portion of the fiber lies in .
§ The story in general
- Suppose we have a presheaf category . Take a morphism
§ The story in slice categories
- If we have in
Set, then we have , which sends a morphism to .
- This also motivates the presheaves story, as .
- Recall that any morphism can be equally seen as a morphism . This is the mapping between slice and exponential.
- We can think of as a collection . This is the fibrational viewpoint.
- Then the functor .