Lecture topic
Definition: Given categories A,B and functors F,G:A→B, a natural transformation η:F→G is a collection of elements ηX∈HomB(F(X),G(X)) for all X∈Obj(A) such that the diagram
commutes, whenever f∈HomA(X,Y).Definition: For X∈Obj(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_:B→A 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:A→B a functor. The colimit colim(F) of F is an object colim(F)∈Obj(B) and a natural transformation ι:F→colim(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 Z∈Obj(B) and η:F→Z_ is a natural transformation, there is a unique map ζ:colim(F)→Z such that the following diagram commutes:
In other words, whenever Z∈Obj(B) and ϵ:Z_→F is a natural transformation, there is a unique map θ:Z→lim(F) such that the following diagram commutes:
Remark: Hom commutes with limits and tensor commutes with colimits. That is:
Hom(A,lim(Bi))=lim(Hom(A,Bi))(colim(Ai))⊗B=colim(Ai⊗B)
References: May (A Concise course in Algebraic Topology, Chapter 2.6), Aluffi (Algebra: Chapter 0, Chapter VIII.1)
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