Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

12Proposition <strong>2.</strong>20. Let C be a category, let (Z, X, Y, d, d 0 , d 1 , s, t) be a contractibleequalizer and let F : C → D be a functor. Thenis a contractible equalizer in D.F Z F d F XF sF d 0F tF d 1 F YProof. Since functors preserve composition, the statement is proved.Proposition <strong>2.</strong>2<strong>1.</strong> Assume thatZ d Xis an equalizer and there exists t : Y → X such thatd 0d 1 Y□t ◦ d 0 = Id Xd 1 ◦ t ◦ d 1 = d 0 ◦ t ◦ d 1Then there exists s : X → Z such that (Z, X, Y, d, d 0 , d 1 , s, t) is a contractible equalizer.Proof. Since d 1 ◦ t ◦ d 1 = d 0 ◦ t ◦ d 1 and (Z, d) = Equ (d 0 , d 1 ), there exists s : X → Zsuch thatt ◦ d 1 = d ◦ s.Let us computed ◦ s ◦ d = t ◦ d 1 ◦ d = t ◦ d 0 ◦ d = dand since d is mono we gets ◦ d = Id Z .□Definition <strong>2.</strong>2<strong>2.</strong> Let F : C → D be a functor. An F -contractible equalizer pair isa parallel pairin C such that there exists a contractible equalizerin D.XD d F Xsd 0d 1All the previous <strong>results</strong> can be considered in the opposite category so that theygive the dual notion, namely contractible coequalizers.Definition <strong>2.</strong>23. Let C be a category.(C, X, Y, c, d 0 , d 1 , u, v) whereF d 0tF d 1 Y F YA contractible coequalizer is a eightupleXd 0vd 1 YcuC