Irritating set of examples- I

by ayushkhaitan3437

I am trying to collect explicit examples for concepts and calculations. My hope is that this website becomes a useful repository of examples for anyone looking for them on the internet.

First some words of wisdom from the master himself, Professor Ravi Vakil: “Finally, if you attempt to read this without workign through a significant number of exercises, I will come to your house and pummel you with the EGA until you beg for mercy. As Mark Kisin has said, “You can wave your hands all you want, but it still won’t make you fly.”

We first start with some category theory examples:

1. Can we have two products of the same two objects, say A and B, in the same category? This question is much more general than I am making it out to be. Can we have two distinct universal objects of the same kind in a category (although they may be isomorphic, and even through unique isomorphism)? The only example of the product of objects being isomorphic but not the same is the following: A\times B and B\times A. These aren’t the same objects, but they’re isomorphic through unique isomorphism.

2. Groupoid- In the world of categories, a groupoid is a category in which all morphisms between objects are isomorphisms. An example of a groupoid, which is not a group, is the category \mathfrak{Set} with the following restriction: \text{Hom(A,B)} now only consists of isomorphisms, and not just any morphisms. This example, although true, is not very illustrative. This [link]( provides a much better demonstration of what is going on. Wikipedia says that the way in which a groupoid is different from a group is that the product of some pairs of elements may not be defined. The Overflow link suggests the same thing. You can’t take any any pair of moves that one may make on the current state of the jigsaw puzzle, and just compose them. The most important thing to note here is that the elements of the group do not correspond to objects of the categories. They correspond to morphisms between those objects. This is the most diabolical shift of perspective that one encounters while dealing with categories. Suddenly, morphisms encode much more information than you expect them to.

3. Algebraic Topology example: Consider a category in which points are objects of the category, and the paths between points, upto homotopy, are morphisms. This is a groupoid, as paths between points are invertible. The return path should not wrap around a wayward hole, obviously. One may consider the path as the same, just travelling in the opposite direction. The automorphism group of a point would be the fundamental group of paths centred at that point.

Another category that stems from Algebraic Topology is one in which all objects are topological spaces, and the morphisms between maps are the continuous maps between those spaces. Predictably, the isomorphisms are the homeomorphisms.

4. Subcategory: An example would be one in which objects are sets with cardinaly 1, and morphisms would be the same as those defined in the parent category- \mathfrak{Set}.

5. Covariant functor: Consider the forgetful functor from \mathfrak{Set} to \mathfrak{Vec}_k. The co-domain is bigger than the domain. One could think of this functor as an embedding.

A topological example is the following: one which sends the topological space X, with the choice of a point x_0, to the object \pi(X,x_0). How does this functor map morphisms? It just maps paths in X to their image under the same continous map. How do we know that the image is a path? This is easy to see. We can prove that we ultimately have just a continuous map from [0,1] to that image, and we will be done. Do we have to choose a point in each topological space? Yes. What if we have the following two tuple (X,x_0), (Y,y_o), such that x_0 is not mapped to y_0? Then there is no morhism between these two objects. In other words, the set of morphisms \text{Hom}((X,x_0), (Y,y_o)) consists of only those morphisms which map x_0 to y_0. An illustrative example is the following: f_1: [0,1]\to (\cos (2\pi t),\sin (2\pi t)) and f_2: [0,1]\to (\cos (4\pi t),\sin (4\pi t)). These are two different continuous maps between the same two topological spaces. They both map 0 to the point (1,0) in S^1, but they map a path starting and ending

Side note: Example of two homotopic paths being mapped to homotopic paths under a continuous map. Let f: [0,1]\to S^1 be the continuous map under consideration. Consider any path p in [0,1] which starts and ends at 0. We know that this is homotopic to the constant path at 0 (one may visualize the homotopy as shrinking this path successively toward 0). Then the image of this homotopy is mapped to a path in S^1 that shrinks toward the constant map at (1,0).

6. Contravariant functors: Mapping a vector space to its dual. This example is pretty self-explanatory.

7. Natural Transformation: A natural transformation is a morphism between functors. Abelianization is a common example of a natural transformation. The two functors, both of which are covariant, are id and id^{ab}. The first one maps a group to itself, and the second one maps a group to its commutator. The resultant commutative diagram is easy to see too. The data of the natural transformation is just m:G\to G^{ab} and m:G'\to G'^{ab}.

The double dual of a vector space is another example of a natural transformation. The dual would have worked too, except for the fact that the dual functor is contravariant. Note: one of the functors, in both these natural transformations, is the identity functor.

8. Equivalence of categories- This is exactly what you think it is. Two categories that are not equivalent are \mathfrak{Grp} and \mathfrak{Grp^{ab}}. Too much information is lost while abelianizing the group, which cannot be regained easily.

9. Initial object- The empty set is the initial object in the category \mathfrak{Set}. Why not a singleton? Because the map from the initial object to any object also has to be unique. Moreover, a singleton will not map to an object- namely the empty set. And an initial object should map to all objects.

10. Final object- A singleton will be a good final object in the category \mathfrak{Set}.

11. Zero object- The identity element in the category \mathfrak{Grp} would be such an object.

12. Localization through universal property: Consider \Bbb{Z}, with the multiplicative subset \Bbb{Z}-\langle 7\rangle. The embedding \iota: \Bbb{Z}\to \Bbb{Q} ensures that every integer goes to an invertible element. Trivially, so does every element of \Bbb{Z}-\langle 7\rangle. Hence, there exists a unique map from (\Bbb{Z}-\langle 7\rangle)^{-1}\Bbb{Z} to \Bbb{Q}. We can clearly see that this is overkill. Many more elements than just those of \Bbb{Z}-\langle 7\rangle are mapped to invertible elements. The point is that there may be a ring A such that only elements of \Bbb{Z}-\langle 7\rangle are mapped to invertible elements in A. Hence, in that case too, there will exist a unique map from (\Bbb{Z}-\langle 7\rangle)^{-1}\Bbb{Z} to A. Why do we care about there existing a map from some other object to rings which \Bbb{Z} maps to at all? When we have a morphism \phi:A\to B, and we can say that there exists a map A/S\to B, where S is a set of relations between elements of A, then we’re saying something special about the properties of elements in B (at least the properties of elements mapped to by S).