Re: monic epics

Re: monic epics

[Lawrence Stout]

The Goguen category of L-fuzzy sets on a lattice L (Objects are pairs
(A,\alpha) where \alpha:A\to L and morphisms are functions f:A\to B
such that \beta{f(a)) >= \alpha(a)) has all functions whose
underlying set function is an isomorphism both epic and monic, but
not, in general, isomorphisms, which must preserve the lattice valued
membership on the nose.  Since these monic, epic maps are the ones
which give the right subobjects to consider for fuzzy logic they are
of interest.  They do not determine a "folding" like the one you
describe.


On Mar 6, 2007, at 7:11 PM, David Karapetyan wrote:

>  So does every monic, epic arrow determine such a
> "folding" or are there monic, epics that can't be characterized in
> such
> a way?
>
>



Lawrence Stout
Professof of Mathematics
Illinois Wesleyan University

Re: monic epics

[Agnes Boskovitz]

Hi

You might be interested in the Masters thesis I wrote in 1980 called
"Epimorphisms in Algebraic and Some Other Categories", which might have
some relevant information in it for you.  You can get it from the McGill
University library, or from Library and Archives Canada, or I can email
you a copy if you wish.

Agnes Boskovitz

David Karapetyan wrote:
> Hi, I've been trying to learn some category theory and I came upon the
> example of a monic, epic in the category of monoids given by the
> inclusion function of (N,0,+) into (Z,0,+). I know that in monoids every
> monic arrow is also an injective function but the inclusion function of
> N into Z provides a counterexample of every epic arrow being a
> surjective function. I noticed that N is just a "folded" version of Z,
> where by "folded" I mean take Z and throw away all the inverses of the
> natural numbers. So does every monic, epic arrow determine such a
> "folding" or are there monic, epics that can't be characterized in such
> a way?
>
>

Re: monic epics

[David Karapetyan]

Michael Barr wrote:
> Category theory is too abstract for any such statement to be true (or
> even make sense).  For example, in the category denoted . ---> . (with
> two objects and one non-identity map), that map is monic and epic for
> want of any test maps.  More concretely, the inclusion of Z into R is
> both in the category of commtative rings.  In fact the following a
> characterization of monic/epics in commutative rings is this: a
> subring R \inc S is epic iff every element of of S can be written s =
> vAw where for some n, v is an n-dimensional row vector, w is an
> n-dimensional column vector and A is an n x n matrix of elements of S
> such that the entries of A, vA, and Aw all belong to R.  In general,
> very little can be said about monic/epics.
>
ok i got it. in all the examples given the subobjects given by the
monics are "generators" for the object, where by "generators" i mean the
elements of the subobject in some way determine the elements of the
bigger object. so how about this then: any time we have the situation
described above the monic arrow will also be epic.

Re: monic epics

[David Karapetyan]

Steve Vickers wrote:
> Dear David,
>
> There are more complicated examples. Here's one.
>
> Take A to be the semigroup {0,a,b,c} in which x0=0x=0 for all x, aa=a,
> cc=c, ab = bc = b, and all other products are 0. (This of this as
> being derived from a category with two objects and three morphisms, so
> a and c represent the two identities and b is a morphism between the
> two objects. 0 takes care of products of non-composable pairs.)
>
> Take B = A u {d}, with cd = da = d, bd=a, db=c and all other binary
> products involving d give 0. (Think of adjoining an inverse to b in
> the category.)
>
> Now the inclusion A -> B is a semigroup epi. To see this, suppose f: A
> -> C is a semigroup homomorphism, and x in C satisfies
>
>    xf(a) = f(c)x = x
>    f(b)x = f(a)
>    xf(b) = f(c)
>
> Then x is the unique such. For if x' is another then
>
>    x' = x'f(a) = x'f(b)x = f(c)x = x
>
> If g: B -> C is a semigroup homomorphism agreeing with f on A, then
> g(d) does satisfy those equations for x, and so any two such g's must
> be equal.
>
> This semigroup example can be easily turned into monoids by adjoining
> a unit element.
>
> There is still the same idea that B is got by adjoining inverses to
> elements of A, but they are not inverses in the monoid sense and it is
> not clear to me in general how one would formalize the idea that they
> are inverses when embedded in some category.
>
> There is a related epi in rings: the inclusion of upper triangular 2x2
> matrices (over any ring) into all 2x2 matrices.
>
> Regards,
>
> Steve Vickers.
>
i like the example of the semigroups. i think in some sense the addition
of d does not add enough information to our monoid so if we have two
semigroup homomorphisms that agree on A then by using the properties of
semigroup homomorphisms we are forced to define how they behave on d in
only one way. i'm still trying to incorporate the inclusion of the 2x2
upper triangular matrices into all 2x2 matrices and Michael Barr's
example into this framework.

Re: monic epics

[Michael Barr]

Category theory is too abstract for any such statement to be true (or even
make sense).  For example, in the category denoted . ---> . (with two
objects and one non-identity map), that map is monic and epic for want of
any test maps.  More concretely, the inclusion of Z into R is both in the
category of commtative rings.  In fact the following a characterization of
monic/epics in commutative rings is this: a subring R \inc S is epic
iff every element of of S can be written s = vAw where for some n, v is
an n-dimensional row vector, w is an n-dimensional column vector and A is
an n x n matrix of elements of S such that the entries of A, vA, and Aw
all belong to R.  In general, very little can be said about monic/epics.

On Tue, 6 Mar 2007, David Karapetyan wrote:

> Hi, I've been trying to learn some category theory and I came upon the
> example of a monic, epic in the category of monoids given by the
> inclusion function of (N,0,+) into (Z,0,+). I know that in monoids every
> monic arrow is also an injective function but the inclusion function of
> N into Z provides a counterexample of every epic arrow being a
> surjective function. I noticed that N is just a "folded" version of Z,
> where by "folded" I mean take Z and throw away all the inverses of the
> natural numbers. So does every monic, epic arrow determine such a
> "folding" or are there monic, epics that can't be characterized in such
> a way?
>
>

-- 
Any society that would give up a little liberty to gain a little
security will deserve neither and lose both.

            Benjamin Franklin

Re: monic epics

[Steve Vickers]

Dear David,

There are more complicated examples. Here's one.

Take A to be the semigroup {0,a,b,c} in which x0=0x=0 for all x, aa=a,
cc=c, ab = bc = b, and all other products are 0. (This of this as being
derived from a category with two objects and three morphisms, so a and c
represent the two identities and b is a morphism between the two
objects. 0 takes care of products of non-composable pairs.)

Take B = A u {d}, with cd = da = d, bd=a, db=c and all other binary
products involving d give 0. (Think of adjoining an inverse to b in the
category.)

Now the inclusion A -> B is a semigroup epi. To see this, suppose f: A
-> C is a semigroup homomorphism, and x in C satisfies

    xf(a) = f(c)x = x
    f(b)x = f(a)
    xf(b) = f(c)

Then x is the unique such. For if x' is another then

    x' = x'f(a) = x'f(b)x = f(c)x = x

If g: B -> C is a semigroup homomorphism agreeing with f on A, then g(d)
does satisfy those equations for x, and so any two such g's must be equal.

This semigroup example can be easily turned into monoids by adjoining a
unit element.

There is still the same idea that B is got by adjoining inverses to
elements of A, but they are not inverses in the monoid sense and it is
not clear to me in general how one would formalize the idea that they
are inverses when embedded in some category.

There is a related epi in rings: the inclusion of upper triangular 2x2
matrices (over any ring) into all 2x2 matrices.

Regards,

Steve Vickers.


David Karapetyan wrote:
> Hi, I've been trying to learn some category theory and I came upon the
> example of a monic, epic in the category of monoids given by the
> inclusion function of (N,0,+) into (Z,0,+). I know that in monoids every
> monic arrow is also an injective function but the inclusion function of
> N into Z provides a counterexample of every epic arrow being a
> surjective function. I noticed that N is just a "folded" version of Z,
> where by "folded" I mean take Z and throw away all the inverses of the
> natural numbers. So does every monic, epic arrow determine such a
> "folding" or are there monic, epics that can't be characterized in such
> a way?
>
>

Re: monic epics

[David Karapetyan]

> Yes.  Another common example of a morphism that is both a monomorphis and
> an
> epimorphism but not an isomorphism is the inclusion of the rational
> numbers
> into the real numbers in the category of topological spaces.

i understand that such arrows exist and i'm trying to get an intuitive feel
for why they are epic. one way i think of a surjective function is that it
is a map that entirely covers the codomain so any two function that agree
on all of the codomain must be the same. it is not the case with epic
arrows that they cover the entire codomain as set functions but that is i
think because most categories are much more structured than the category of
sets so it is enough to cover certain parts of the codomain and the rest of
the structure can be recovered. i think that is what happens with the
inclusion of the rationals into the reals because the reals are defined as
equivalence classes of sequences of rationals so if two functions agree on
the rationals and they are continuous then they automatically agree on the
reals.

monic epics

[David Karapetyan]

Hi, I've been trying to learn some category theory and I came upon the
example of a monic, epic in the category of monoids given by the
inclusion function of (N,0,+) into (Z,0,+). I know that in monoids every
monic arrow is also an injective function but the inclusion function of
N into Z provides a counterexample of every epic arrow being a
surjective function. I noticed that N is just a "folded" version of Z,
where by "folded" I mean take Z and throw away all the inverses of the
natural numbers. So does every monic, epic arrow determine such a
"folding" or are there monic, epics that can't be characterized in such
a way?