⎯ΘωΘ⟶ · @yog
203 followers · 852 posts · Server mathstodon.xyz#First we #reformulate #ordinary #colimits in the #language of #tensor #products, in a #way that #suggests #more #general #weighted colimits.
#Assume for the #moment that the #receiving category C has all #coproducts and #coequalizers. As is #well #known, it #follows that C #has #all colimits; the #proof is we can #write #down a #formula for the colimit of F:J→C: as a #coequalizer of a #pair:
∑j,k∈Ob(J)hom(j,k)×F(j) [[→ →]] ∑k∈Ob(J)F(k) [[→]] colimJF
#pair #coequalizer #formula #down #write #proof #all #has #follows #known #well #coequalizers #coproducts #receiving #moment #Assume #weighted #general #more #suggests #way #products #tensor #language #colimits #ordinary #reformulate #first