Limits and Colimits

Cones and Limits

Definition

Def Diagram Suppose is a category and is a small category. A diagram of type in is a functor . We write the objects in the index category lower case, and the values of the functor in the form etc.

Cone

A cone over a diagram with summit or apex is a natural transformation whose domain is the constant functor at . The components are called the legs of the cone. Explicitly:

• The data of a cone over with summit is a collection of morphisms , indexed by the objects .
• A family of morphisms defines a cone over if and only if, for each morphism in , the following triangle commutes in :

Proposition

Prop Cones form a category for by its morphisms are defined by morphisms in : such that the following diagram commutes in for all :

Limit & Colimit

A limit for a diagram is a terminal object in , written as A finite limit is a limit for a diagram on a finite index category . A colimit is an initial object in the category of cones under .

Continuity

Definition

Def Continuity A functor is said to preserve limits of type if, whenever is a limit for a diagram , the cone is then a limit for the diagram . Briefly, A functor that preserves all limits is said to be continuous.

Proposition

Representable functors are continuous.

Contravariant Functor

A functor of the form is called a contravariant functor on . Explicitly, such a functor takes to and . e.g. A typical example of a contravariant functor is a representable functor of the form,

Proposition

Contravariant representable functors map all colimits to limits.

Definition

Def Complete A complete category is a category in which all small limits exist. That is, a category is complete if every diagram (where is small) has a limit in . e.g. has all small limits and colimits, thus complete.

Prop A category is complete if and only if it has equalizers (of all pairs of morphisms) and all (small) products.

Inverse Limit

Inverse System

Let be a category, a poset. Suppose is a family of objects with morphisms for with the following properties:

• for all .
• for all .

Then the pair is called an inverse system in .

Inverse Limit

Suppose is an inverse system of objects and morphisms in a category . The inverse limit of this system is an object in with morphisms , called projections, such that the following diagram commutes for all , and for all : The inverse limit is denoted by .