EX125. fully faithful functor reflects limit and colimit. скачать в хорошем качестве

EX125. fully faithful functor reflects limit and colimit.

Informally we recall (define): 1) Def (diagram); Def (diagram f type J: with J a category given); 2) Def (cone over a diagram); 3) Def (limit/ colimit) as universal cone; 4) Def (informally: a functor reflects limits): formal definition see remark; Exs: 4) Ex1: epimorphism is reflected by faithful functor; 5) Ex2: limits and colimits are reflected by fully-faithful functors; Remark: 1) I didn't give formal definition of "F reflects limit": Def (reflect) Let F: C→D be a functor and J: I→C a diagram. We say that F reflects limits of J if whenever we have a cone η : const^I_x → J over J in C such that F(η) is a limit of F J in D, then was already a limit of J in C. Notice this explains why in the pf of Ex2, we start with a cone. Link: https://www.math3ma.com/blog/a-diagra... https://www.math3ma.com/blog/limits-a... https://ncatlab.org/nlab/show/reflect...