Concrete and Non-Concrete Categories (Informally)

The categories that naturally come to mind when trying to think up examples of a category (such as sets with functions, groups with homomorphisms, topological spaces with continuous maps, …) are sometimes called concrete categories.  These are categories where the objects are sets, usually with some additional structure (group structure, a topology, etc.), and the morphisms are well-defined functions between those sets that preserve the structure.  For example, consider Top, the category of topological spaces with continuous maps.

In the case of Top it’s easy to check that the identity map is continuous, that composition of continuous functions yields another continuous function, and function composition in general is associative.  With a continuous map, say , we’re taking a point and assigning it to a single point (so we have a well-defined function between two sets) in such a way that if we have an open subset and look at the preimage of each point in , we get an open set in .  It is in this sense that the structure of the topological space is preserved: preimages of open sets are open sets.  In the case of Grp, our morphisms are group homomorphisms, which by their very definition respect the group structure.

There are, of course, categories which are not concrete.  In the last post we saw a very simple category with three objects.  These objects aren’t sets, they’re just “things” that we decided to label A, B, and C.  The morphisms between these objects aren’t functions in any sense of the word: they don’t associate inputs with outputs, there aren’t even any “inputs” or “outputs” to speak of!  Our morphisms are literally just arrows that start at one object and end at another.  (For this reason some people simply refer to the morphisms of a category as arrows.)

The above is about as far from a concrete category as you can get, but consider the following example from Awodey.  Let the objects of the category be sets.  For any sets and a morphism from to is any subset of (that is, any relation on and ).  The identity relation is taken to be .  For composition of two relations, we say that is the composition of and where is in the set iff there is a such that and are in the original relations.  Here are objects are sets, but the arrows are not functions.  The morphisms are collections of ordered pairs with and .  This is not a function as we may “associate” one with several items in .  For example, we may have a morphism with .

Category Theory – Basic Definitions

There are certain ideas that are common across many areas of mathematics.  Regardless of whether we’re talking about vector spaces, topological spaces, groups, or simply sets, we have some notion of a mapping between entities in that space.  In the case of sets we may require nothing more than that the maps be well-defined, whereas with groups or vector spaces we may also require that the map preserves some structure.  For example, when discussing maps between vector spaces we may require linearity (e.g., a linear transformation); for groups we may require that the map respects the group operation (homomorphisms).  In either case, we may place some other requirements on the maps that do not require the structure of the space.  In set-theoretic terms, we may concern ourselves with maps that are injective (one-to-one; for each output, there is only one input); that are surjective (there is an input that will give each possible output), or both (bijective).  The language and ideas of category theory allow us to generalize this common notion and also to explore consequences of the requirements we place on these maps.

A category is a class of objects and a class of morphisms between those objects that satisfy certain properties.  Before discussing those properties it should be noted that though in many instances morphisms will be actual maps or functions, they don’t necessarily have to be.  We may in fact think of a category simply as a directed graph, where the vertices are objects and the edges are morphisms.  Here there is no structure or mapping quality of the morphisms, they are just “arrows” that start at one object and end at another.

Collectively, the class of objects of the category is referred to as , and the class of morphisms as .  To refer to the collection of morphisms whose source is the object and whose target is the object , we write (or just if the category is clear from the context).  The morphisms of must satisfy the following.

1. For each there is an identity morphism from that object to itself.  That is, there is a with the property that for any and such that and .
2. Composition of morphisms yields a morphism:  if and , then there is a morphism, .
3. Composition of morphisms is associative:  if , , and are morphisms, then .

For example, the following diagram could be taken as the definition of a category .

In this category we have three objects, , , and .   In addition to the identity morphisms, we have an , a , and their composition.