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?
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
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.
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?
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
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.
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.
It seems to me that Isbell's notion of "dominions" are something like a precise way of saying "a complete set of generators" for the case of epimorphisms in Cat. Look at "Epimorphisms and Dominions III" by John Isbell, American Journal of Mathematics, 1968. That paper has references to earlier papers about the case for semigroups and other categories. Charles Wells
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.
-- Charles Wells abstract math website: http://www.abstractmath.org/MM//MMIntro.htm professional website: http://www.cwru.edu/artsci/math/wells/home.html personal website: http://www.abstractmath.org/Personal/index.html genealogical website: http://familytreemaker.genealogy.com/users/w/e/l/Charles-Wells/ NE Ohio Sacred Harp website: http://www.abstractmath.org/fasola/index.html
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?
participants (6)
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Agnes Boskovitz -
Charles Wells -
David Karapetyan -
Lawrence Stout -
Michael Barr -
Steve Vickers