Re: Exercise from "Categories for the Working Mathematician"

On Mar 30, 10:43 pm, s...@xxxxxxxxxxxxx wrote:
Hi. I am trying to learn category theory on my own and think I may be
confused, so I would be grateful for any help.

On Page 92 of the second edition of Saunders Mac Lane's "Categories
for the Working Mathematician", exercise 3 asks the reader to show
that if <G, F, phi>: X --> A is an adjunction with G full and every
unit eta_x monic, then every eta_x is also epi.

Exercise 1 on the same page asks the reader to show, among other
things, that "G is full" is self-dual, the dual of "<G, F, phi>: X -->
A is an adjunction" is "<F, G, phi^(-1)>: A --> X is an adjunction",
and the dual of "eta is the unit of <G, F, phi>" is "eta is the counit
of <F, G, phi^(-1)>".

Two pages previously there is a theorem which states that for an
adjunction <F, G, phi>, G is full iff every counit is a split monic.

From these facts it seems to follow that, under the hypotheses of the

original exercise, every counit eta_x: (FGx)* --> x* (where * denotes
dual objects) of the dual adjunction <F, G, phi^(-1)> is monic, and
therefore the unit eta_x: x --> FGx of the original adjunction is epi.
But I have not made any use of the fact that every eta_x is monic,
which leads me to suspect that there is something wrong with the above
argument. Can anyone tell me what?

this is a naming convention confusion i think

you have two categories A and B
with these two functors
F: A -> B
G: B -> A

you are given an original full adjunction

phi: Hom (F-, -) -> Hom (-, G-)
B A

and the dual adjunctions defined by flipping arrows

-1
phi : Hom (-, G-) -> Hom (F-, -)
A B

each of these have their own units

eta: A -> A : a |-> FGa
phi

eta: B -> B : b |-> GFb
-1
phi

and counits

epsilon: B -> B : b |-> GFb
phi

epsilon: A -> A : a |-> FGa
-1
phi

now phi_eta is the same as (phi^-1)_epsilon
and phi_epsilon is the same as (phi^-1)_eta

but phi_eta and phi_epsilon are still distinct
as they occur in different categories

now your problem says G is full

therefore

every epsilon is monic
phi

this does not prove anything yet of

eta
phi

because you are not given that F is full
-1
and so can say nothing of phi 's counit

you are also given though

every eta is monic
phi
-1
which does give information on phi 's counit

the adjunction has the following properties

(epsilon F) o (F eta) = id
F

(G epsilon) o (eta G) = id
G

since G is surjective we have a natural isomorphism among dual
adjoints
and

eta is epic
phi

-+-+-

i may have over explained here
but i just wanted to show the necessity
of distinguishing the four "units" by name
and showing that only a pairing was equivalent

the main point of the proof
is to map the units on A to B with F
and look at the adjoint situation

since G is full we get a natural isomorphism
that provides us to switch monic for epic in B
which we can bring back to the unit in A

this is one of those
follow the arrows and twist
category exercises
that require careful naming
( which i may have erred on above - check! )

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galathaea: prankster, fablist, magician, liar

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