Diagram (category theory)

In category theory, a special part of mathematics, we learn about something called a diagram. It is like a way to organize many pieces together, just like how we might collect and sort different types of toys. But instead of just collecting things, category theory uses special links called morphisms to connect these pieces.

In regular math, when we talk about a family of sets, we are just grouping sets together using a list or an index. This is similar to having a list of books where each book has a number to show its place in the list.

In category theory, a diagram is like that family of sets, but with an extra step. It not only groups objects (which are like the sets) but also includes the links (morphisms) between them. These links help us see how the objects relate to each other. So, a diagram is really a fun way to see both the pieces and how they fit together in a big picture.

Definition

In category theory, a special kind of math, a diagram is a way to organize different pieces and how they connect. Think of it as a map showing how things are linked together. The map follows a pattern, called the index category, which guides how the pieces fit.

We often use small or simple patterns for these diagrams, making them easier to understand. These diagrams help mathematicians study how different pieces in math connect and change.

Examples

In category theory, a diagram is a group of objects and how they connect. One common type is the constant diagram. This diagram sends every object to the same one and every connection to the same connection.

When the diagram has just objects with no connections, it is like a family of objects. Depending on what we do with this diagram, we can get products or coproducts. Adding connections between objects helps us build more complex constructions.

Cones and limits

A cone with a point called N of a diagram D : J → C is a special kind of connection from a simple diagram to D. This simple diagram, called the constant diagram, sends every part of J to the same object N in C.

The limit of a diagram D is a special cone that connects to D in the most general way. If limits exist for all diagrams of a certain type in a category C, we get a special mapping called a functor:

lim : C^J → C

This functor takes each diagram to its limit.

Similarly, the colimit of a diagram D is another special cone, but this time coming from D. If colimits exist for all diagrams of a certain type, we have a functor:

colim : C^J → C

This functor takes each diagram to its colimit.

Commutative diagrams

Main article: Commutative diagram

Diagrams in math are sometimes shown using special drawings called commutative diagrams. These drawings are useful when the index category is small and simple. In these drawings, each object is shown as a dot, and important links between objects are shown as arrows. The idea is that there is only one way to go from one object to another, which makes the diagram "commute."

But not all diagrams can be shown this way. For example, a diagram with one object linking to itself might not follow the commute rule. Some diagrams are too big or complicated to draw fully, so people use simpler drawings with dots and arrows to give a general idea.