I often forget the conceptual background of these well known universal constructions. This bothers me especially when I need to give a new mathematical instrument the conceptual meaning as the extended or restricted version of limit or colimit. Here I write of limit/colimit that include definitions and some concrete examples of instantiating the concept, in order for reminding me of least rigoristic idea that activate the lost memory.
Definition 1. Let are categories and functors and are given as: . The comma category is a category composed of a triple as an object and the morphism is a pair of morphisms and (we write these only as ) that commutes the following diagram:
Note that this definition generalizes the case when a functor is an object of the category , since can be regarded as the functor . In this sense, when both the functors are objects , then the corresponding comma category is exactly the hom-set written as:
Definition 2. Let be categories ( is called index category, usually taken to be small or finite). The diagonal functor is the functor that maps for each object the constant functor , and for each morphism, the morphism function that assigns each index constantly to the given morphism as in the diagram:
Definition 3. Let be a functor. If exists, the initial object of comma category , regarding the functors as to fit in the sequence , is called the colimit (of ), denoted .
Definition 4. Let be a functor. If exists, the terminal object of comma category , regarding the functors as to fit in the sequence , is called the limit (of ), denoted .
Example 1. Coproduct is the colimit of , regarding the corresponding functor as indexed objects. We usually denote for the coproduct of given indexed objects. In the following diagram, the colimit as the universal arrow is .
Example 2. Let be a category of two objects and two morphisms . Then an object of the category is exactly a pair of morphisms, also denote in . We call the colimit of the coequalizer (of ). An object of comma category can be represented by a commutative diagram:
Hence the coequalizer of is the pair such that for any , there is unique morphism in that commutes the following diagram:
Example 2.1. When is the category of abelian group or the category of -module for a certain ring , and suppose , then the coequalizer of is the quotient .
Example 3. Let denotes an index category , then an object of is again of the same shape, namely . Any element suffices the commutative diagram:
hence the colimit of , called pushout of , is a pair that for any pair , there exists unique morphism such that the following diagram commutes: