2024-08-20

Category examples

  1. Let be a set. We can form a category where and if and otherwise.
  2. We make a category as follows:

which we denote by . This is a special case of the following:
3. Let be a poset. We form the category whose set of objects is and we have a morphism if . This is a category because if then , so we can compose morphisms. The identity morphism comes from .
4. Let be a topological space. Then , the open subsets of ordered by inclusion is a poset. If is a base of the topology, then we also get a subposet.
5. Let be a topological manifold. Then there is an open cover where each is homeomorphic to an open subset of .
6. Let be a ring. Then has a base given by the distinguished affine open sets , where .

Definition (Opposite category)

Given a category , we can take its opposite category , with the same objects as but and the composition law reversed.

Example

If is a set, . If is a poset, .

Definition (Subcategory)

Given a category , the objects of a subcategory is a subcollection of the objects of , with morphisms a subcollection of the morphisms in compatible to make a category.

Definition (Full subcategory)

A subcategory is full if it has all the morphisms of the ambient category.

Definition (Connected and filtered)

A category is

  • connected if for every pair , there exists and maps and :
  • filtered if it is connected and for every pair of maps , there exists such that (pairs of maps can be coequalised):
  • coconneted or cofiltered if is connected or filtered respectively

Example: posets

The following are equivalent for a poset :

  1. is (co)filtered.
  2. is (co)connected.
  3. (Directed) For all , there exists a (lower) upper bound for and .

Example

Let be a topological space and be a basis. Let . Then there is a subposet (under inclusion)

This is cofiltered: if , then .

If and . If , then

Definition (Functor)

A functor :

  • sends objects of to : , and
  • morphisms compatible with maps in .

Examples

  1. A forgetful functor .
  2. , (forget the -algebra structure).
  3. For a ring homomorphism , we get forgetful functors
  4. forgets the -action. Tannakian formalism.
  5. , .
  6. , .
  7. , . If , we get square matrices.
  8. , .
  9. Let be a ring homomorphism, i.e., is an -algebra. Define
  10. If is a category and is an object, we can define two functors
    These are the Yoneda embeddings.
  11. Let be a set and a category. Then a functor is the same as specifying an object for all .
  12. In general, for a poset , a functor is the same as specifying an object for all and a morphism for all such that
  13. A functor is equivalent to a sequence
  14. Let and be topological spaces. We get a presheaf
    is a poset, so we need to specify what the restrictions are: the restrictions are literally the restrictions of functions.
  15. Let be a topological space. Then a functor (or we can map out of for a basis) consists of the following data:
    1. For each , an object of .
    2. For each inclusion , a restriction .
    3. For each triple , we require that (compatibility of restrictions).
  16. Let be a ring. is a basis for . We define
    Note that we need to be careful because we have for example:

But there is indeed a canonical isomorphism . Alternatively, we can define

which does not depend on the choice of , and take

Now our objects are given to us canonically.

Tradeoff between good objects vs good maps.

More generally, for an -module, we can define

The above are actually the structure sheaf on , and the sheaf associated to a module.

Definition (Faithful, full, essentially surjective)

A functor is:

  • faithful or full if the map is injective or surjective respectively;
  • essentially surjective if every object of is isomorphic to an object where belongs to .

Fully faithful means faithful and full. Fully faithful functors are injective on objects up to isomorphism.

Definition (Natural transformation)

A natural transformation between two functors is a collection of morphisms for all objects such that for every morphism in , the following diagram commutes:

A natural transformation is a natural isomorphism if the morphisms are isomorphisms for all objects .

Example

Let be a ring and an -module homomorphism. Then for each , there is a uniquely induced map (given by the universal properties) that makes the following diagram commute:

Further, if and , then the universal properties guarantee that the following diagram also commutes:

where the vertical maps are the restrictions and : these maps exist because

and so has more denominators than . This defines a natural transformation between functors .

Definition (Equivalence)

A functor is an equivalence if there exist a functor such that there exist natural isomorphisms and .

Definition (Presheaves)

For a category , the category of presheaves of sets on is

Here, stands for functors.