Processing math: 100%

Sunday, May 11, 2014

Injection, full, monic, faithful, conservative in category theory

The concepts named in the subject line are related, even interdefinable, and provide examples of each other.
The purpose of this post is to show some of those interrelations.

XfYa function in Set(x=x)(xf=xf)f is injective iff, x,xX, this logical implication holds xXxxfxxfYxfthe arrow in V between the hom-objects

If we consider XfY as, not (a function in Set), but as (a V-functor between discrete categories enriched in either V=2 or V=Set),
then this condition amounts to saying that the arrow in the third line, in V, is invertible, i.e.,
that (the V-functor f between discrete V-categories) is not merely faithful, as it automatically is in those cases, but also full, i.e.,
that f is a full and faithful, or fully faithful V-functor.

No comments:

Post a Comment

MathJax 2.7.9