Category Theory, Graded Category

Graded Category

Once again we are going to convert an existing category C into a larger category G.

Let sequences of objects in C become objects in a new category G. Here G is a graded category with respect to C. If C is groups, G is graded groups, and so on.

At the outset, let's assume there are no restrictions on the sequences; any sequence of objects is fair game. We may place restrictions on the sequences later on.

A morphism m from U to V in G is a sequence of morphisms in C, such that m_i maps U_i into V_i. Morphisms in G are composed by composing the individual morphisms in C. Associativity is inherited. Set m_i to the identity morphism on U_i and find the identity morphism on U in G. We have a valid category.

Free Object

If C is concrete then the objects in G are sequences of sets, which are also sets. A morphism in G is defined by morphisms in C, which are defined by functions obeying certain constraints. thus functions from sequences of sets to sequences of sets determine morphisms in G. In short, G is a concrete category.

Let F be free on the set X, in the category C. The sequence F,F,F,F… is free on X,X,X,X… in the category G.

Conversely, a free object in G is free at every level, and is therefore a sequence of objects that are all equivalent to F.

Limits and Colimits

Start with a graph, points and edges, and imagine a commutative diagram, in C, on that graph. Call this D₁. Then build D₂, and D₃, and so on. These diagrams might stack up, like pancakes on a plate. Now the column above a vertex is an object in G, and the stack of morphisms from one vertex to another is a morphism in G. The result is a commutative diagram D in G.

Conversely, each slice of a commutative diagram D in G implies a commutative diagram D_i in C.

Let H be a new category, derived from D in the category G. This process was described in the previous section. Objects in H are objects in G and morphisms into the graph, such that D commutes. Each slice is an object in C and morphisms into D_i such that the augmented diagram commutes. A morphism in H is a morphism from U to V that makes D∪U∪V commute. And it has to commute at every level.

An object V in H is the limit of D iff each V_i is the limit of D_i. If every diagram on the graph has a limit in C, then every diagram on the same graph, in the category G, has a limit in G.

The same holds for colimits.

Remember that the product is a special case of the limit, where the underlying graph has no edges. If every collection of objects in C has a product, then the same must be true in G. Similarly, colimits in C imply colimits in G.

Chains of Objects

Sometimes graded objects are restricted, so that a morphism must tie each object to the next. And these morphisms may be restricted in other ways, e.g. an embedding.

Let G consist of sequences of objects in C, with morphisms in C connecting each object to the next. A morphism in G is a sequence of morphisms, as described above, but each square must commute. Let's illustrate with X and Y objects of G, and f a morphism from X to Y. You can follow the morphism from X₃ to X₄, then apply f₄ to reach Y₄, or you can follow f₃ from X₃ to Y₃, then use the morphism from Y₃ to Y₄. The result is the same.

For example, let C be the category of groups and group homomorphisms. Now an object in G is a sequence of groups and homomorphisms, where each group is mapped into the next. A morphism in G is a sequence of parallel homomorphisms, like the rungs of an infinite ladder. We can move along a graded sequence, then across to another sequence, or we can move across first, and then along the second sequence. The composite homomorphism is the same either way.

The existence of a limit for each diagram on a given graph in C might not imply a limit in G. Given a stack of commutative diagrams on the given graph, and a limit V_i for each D_i, We can build a sequence V using the objects V_i, but it is not clear that there is a chain of morphisms connecting each V_i to V_i+1, that makes everything commute between the layers. Fortunately, we can often build such a chain in practice.

Return to the example of groups and group homomorphisms, and look at products. At each level, the product is the direct product of the groups. To go from one level to the next, there is a homomorphism from each component group into the next. This implies a homomorphism from the first product into the second. (Run the individual homomorphisms within their components.) Furthermore, the map between products commutes with projections and the maps on component groups. Within the categories of graded groups, graded rings, graded modules, and even graded topological spaces, product is well defined.

Yeah I know, it makes my head spin too.