Adjoints in Category theory

From: Jose Capco (nospam_at_nospam.org)
Date: 02/23/05 

Date: Wed, 23 Feb 2005 16:28:35 +0100

Dear NG,

I'm not very new to this, but I would like to hear opinions. I started
first learning category theory in basic algebra (actually there was
never a real course of category theory in my graduate coursework, and
while doing postgraduate studies I had to learn them all on my own) and
there I learned about the definition of adjoints without the usage of
"units" and "counits"
Basically
Functors between categories F:A -> B, G: B -> A are called adjoint
pairs (F,G) (F being the left adjoint and G being the right adjoint)
If for any A-object 'a' and B-object b we have

A(a,Gb) (thats the A-morphisms between a and Gb) is isomorphic (in our
algebra we dealt with the abelian categories) to B(Fa,b)

Now here there isn't mention of unit and counit.. but then as my
mathematical understanding grew a bit mature, I learned about another
definition with the units and counits (I will not state it here).. is
there a natural way to relate the equivalence of these definition? and
if there is any importance to learn the definition using units and
counits (ie. are these special transformation really important?), which
is much longer... (I've to admit that my supervisor explained this to me
one sleepy day, but I was not very attentive!). I would appreciate any
suggestion

Sincerely,
Jose Capco

Quantcast