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Wednesday, March 9, 2016

Limits and colimits

 Lecture topic

Definition: Given categories A,B and functors F,G:AB, a natural transformation η:FG is a collection of elements ηXHomB(F(X),G(X)) for all XObj(A) such that the diagram
commutes, whenever fHomA(X,Y).

Definition: For XObj(A), define the constant category X_ to be the category with Obj(X_)={X} and HomX_(X,X)={idX}. For any other category B, this may also be viewed as a natural transformation X_:BA with X_(Y)=X and X_(f)=idX for any object Y and any morphism f of B.

Definition:
Let A be a small category and F:AB a functor. The colimit colim(F) of F is an object colim(F)Obj(B) and a natural transformation ι:Fcolim(F)_ that is initial among all such natural transformations. We write ιX:F(X)colim(F) and have ι(f)=idcolim(F) for any morphism f of A.

In other words, whenever ZObj(B) and η:FZ_ is a natural transformation, there is a unique map ζ:colim(F)Z such that the following diagram commutes:
Definition: Let A be a small category and F:AB a functor. The limit lim(F) of F is an object lim(F)Obj(B) and a natural transformation π:lim(F)_F that is final among all such natural transformations. We write πX:lim(F)F(X) and have π(f)=idlim(F) for any morphism f of A.

In other words, whenever ZObj(B) and ϵ:Z_F is a natural transformation, there is a unique map θ:Zlim(F) such that the following diagram commutes:
Examples of colimits are initial objects, coproducts, cokernels, pushouts, direct limits. Examples of limits are final objects, products, kernels, pullbacks, inverse limits.

 Remark: Hom commutes with limits and tensor commutes with colimits. That is:
Hom(A,lim(Bi))=lim(Hom(A,Bi))(colim(Ai))B=colim(AiB)
References: May (A Concise course in Algebraic Topology, Chapter 2.6), Aluffi (Algebra: Chapter 0, Chapter VIII.1)



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