Re: monomorphisms, epimorphisms, and isomorphisms

Thant Tessman <adm@xxxxxxxxxxxxxxxxxxxx> wrote:
Pierce defines 'monomorphism' like so:

"An arrow f : B -> C in a category C is a monomorphism if, for any pair
of C-arrows g : A -> B and h : A -> B, the equality f o g = f o h
implies that g = h."

First off, is the "C" in "category C" different than the "C" in "f : B
-> C"?

Yes. The non-bold-C is an object in the category bold-C.

Second, why the goofy formulation? Why the need to compose g and h with
f? Is it because in category theory we don't talk about the objects but
only the arrows?

Remember that in category Set, the objects are sets. The objects are
NOT members of sets. The "goofy" definition is because the set-
theoretic definition of an injection, which the "monomorphism" concept
is approximating, has to do with members of sets. In category theory we
don't talk about members of sets, because they may not exist. The sets
themselves are fine things to talk about -- because they're just
objects. Monomorphism, though, is defined for all categories regardless
of whether they have "members", so you can NOT talk about members.

Most of the standard categories (sets, posets, monoids, etc.) all have
members of their objects. Don't let that confuse you; Pierce has
already, by now, given several examples of categories where there is no
concept of "members" of the objects -- e.g., the single-poset-as-
category example.

Anyway, I think I see how the definition of monomorphism implies that
the only time a function can evaluate to the same value is if it is
provided the same argument--what he refers to as an "injective function"
in set theory. That is, unique arguments always produce unique results.

You are either just being sloppy with your words, or are
misunderstanding something fundamental. Arrows are not functions. They
are just arrows; that's all. They come from an object, and go to an
object, and they are related by this 'o' operator. If you can ignore
that the source is confusingly called a "domain", or that the
destination is confusingly called a "codomain", things will be clearer.
In particular, arrows do not take "arguments" or "evaluate" to anything.

Here is 'epimorpism':

"An arrow f : A -> B is an epimorphism if, for any pair of arrows g : B
-> C, and h : B -> C, the equality g o f = h o f implies g = h."

He says that in set theory, this corresponds to surjective functions. a
function is surjective if for each b element of B there is an a element
of A for which f(a) = b. I understand this to mean that every element in
the codomain can be produced by the function f given the right unique
element in the domain.

Right (in set theory; none of this has meaning for arbitrary
categories).

I don't understand how this follows from the definition of epimorphism,
and I don't understand how an epimorphism is different than an
isomorphism.

I found the correspondence to surjective functions in the category Set a
little tricky, as well. A way to think of it is this. If g o f = h o f
implies g = h, then f can't leave g or h any room to differ. If f were
not surjective, then g and h could do something different for the
results not produced by f, so f wouldn't be an epimorphism. On the
other hand, if f is surjective, then it exercises all possible inputs to
g and h. If g and h evaluate to the same thing for all inputs then they
are equal, so f is an epimorphism.

Are there books I need to have read before reading this one? Is this the
right group to ask a question like this?

This group is probably fine. sci.math could work, too, if you're
willing to ignore the 90% of trollish posts about the Cantorian
conspiracy to take over the world and crush "brilliant" amateur
mathematicians.

--
Chris Smith
.

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