**Exercise 2.1.17**

Fix a topological space . Write for the poset of open subsets of under inclusion. Let be a functor from to the (functor) category of presheaves on . For , we define to be the functor with and . We wish to intuit a whole chain of adjoint functors as follows:

with .

Let’s tackle first. will be a ‘evaluation on the whole space’ functor. Given an , let – the set that sends the entire space to. Given a natural transformation , we let , the component of at .

The claim is that is right-adjoint to . So, for and , we need . So, we demonstrate the rule. Given – that is, , we need to define a natural transformation . has components . Since , we have . Since , there is an associated map via the (contravariant) functor . So, define . For the other way, let be a natural transformation. It has components , or . We want to define , or . This time, simply let .

will be the ‘evaluation at the empty set’ functor, and the construction proceeds exactly dual to that of .

will be a functor takes a set to the ‘least interesting’ presheaf with that set in its image. Particularly, given an , let and for all other open sets , and for arrows is the unique empty function , and all other arrows are taken to .

This will be left-adjoint to . So, for and , we have . Given a natural transformation , we need to define a map i.e. . has components . When we have , so define . Given a map , or , we need to define a natural transformation . It will have components . So, define , and as the unique empty function for all other open sets .

will be dual to in a certain sense.
For , we made a presheaf that assigns on the terminal object of (), and for everything assigns the initial object of .
So for , we will assign on the *initial* object of () and a *terminal* object of on all other open sets.
To be precise, given , let and for all other open sets .
For arrows is the unique map , and all other arrows are sent to .

This is right-adjoint to . That is, for and , we need . Given a natural transformation , we need to define a map , that is . has components . So let . Given a map , or , we need to define a natural transformation . Simply let , and let be the unique function for all other open sets .