(nondirected) Colimit in the category of rings
Jose Capco
This is something that bugs me every now and then. I haven't tried
giving it a serious thought. But I think there must be known reference
or construction on colimits in the category of rings that is not from a
directed diagram. I know exactly how I can construct directed colimit,
I have seen them in books.. but with arbitrary colimits I get confused.
In books the typical way of constructing colimit is to take the direct
sum of the group obtained from the ring (in our diagram) with addition
and then take a residue modulo some subgroup (here is when the
directedness of the diagram is used.. the residue namely takes away an
element difference with the directed diagram mappings of it) and then
define a multiplication of this group to make a ring which will turn
out to be a directed colimit.
As for the nondirected colimits, the category theory books I have just
say that the category of rings is cocomplete (meaning that there is a
colimit), but an exact construction is not available. Probably using
tensor product is a natural way of constructing it?
Sincerely,
Jose Capco
martin dowd
models of any equational theory. I'm not sure where else you can find
this, but it's in my self-published book, q.v. see www.hyperonsoft.com;
and "Higher Type Categories", Mathematical Logic Quarterly 39 (1993),
251--254.
The construction can undoubtedly be specialized to particular theories,
but this is at least a one day job.
- Martin Dowd
Arturo Magidin
Jose Capco <cliom...@kriocoucke.mailexpire.com> wrote:
>Dear NG,
>
>This is something that bugs me every now and then. I haven't tried
>giving it a serious thought. But I think there must be known reference
>or construction on colimits in the category of rings that is not from a
>directed diagram. I know exactly how I can construct directed colimit,
>I have seen them in books.. but with arbitrary colimits I get confused.
>In books the typical way of constructing colimit is to take the direct
>sum of the group obtained from the ring (in our diagram) with addition
>and then take a residue modulo some subgroup (here is when the
>directedness of the diagram is used..
You are confusing colimits and limits.
Perversely, the "direct limit" is a kind of colimit; and the "inverse
limit" is a kind of limit. You are describing inverse limits. So you
must be talking about limits.
(Here's how I keep them straight: the product involves projections
from the product to the components and the universal property gives
you a map into the product; the CO-product involves maps from the
components and the universal property gives you a map from the
co-product. Likewise, the limit involves maps to the components and
the universal property gives you a map into the limit, and the colimit
has maps from the components into the colimit and the universal
property gives you maps from the colimit. If you think about the usual
definitions of inverse and direct limit, you'll see that you have maps
from the inverse limit to the components, but not from the direct
limit to the components).
>the residue namely takes away an
>element difference with the directed diagram mappings of it) and then
>define a multiplication of this group to make a ring which will turn
>out to be a directed colimit.
>
>As for the nondirected colimits, the category theory books I have just
>say that the category of rings is cocomplete (meaning that there is a
>colimit), but an exact construction is not available. Probably using
>tensor product is a natural way of constructing it?
No, see: the thing is that if it is co-complete then it has colimits,
but you are thinking about limits, not co-limits.
The limits are taken by the ->other<- construction: take the disjoint
union modulo an equivalence relation.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
mag...@math.berkeley.edu
Jose Capco
>
> Perversely, the "direct limit" is a kind of colimit; and the "inverse
> limit" is a kind of limit. You are describing inverse limits. So you
> must be talking about limits.
No sorry, I meant directed limit not directed colimit (sometimes I
confuse the terminology, but I know what Im talking about). Thus here I
mean colimit
>
> (Here's how I keep them straight: the product involves projections
> from the product to the components and the universal property gives
> you a map into the product; the CO-product involves maps from the
> components and the universal property gives you a map from the
> co-product. Likewise, the limit involves maps to the components and
> the universal property gives you a map into the limit, and the colimit
> has maps from the components into the colimit and the universal
> property gives you maps from the colimit. If you think about the usual
> definitions of inverse and direct limit, you'll see that you have maps
> from the inverse limit to the components, but not from the direct
> limit to the components).
>
>
> No, see: the thing is that if it is co-complete then it has colimits,
> but you are thinking about limits, not co-limits.
>
> The limits are taken by the ->other<- construction: take the disjoint
> union modulo an equivalence relation.
>
No, I do really mean the colimit :) ... like I said direct limit is
easy to construct, but not the colimit in general (i.e. a colimit that
is not from a directed diagram). I am quite aware of the diference
between inverse limits and direct limits. Inverse limit is just the
limit that comes from a directed diagram, and direct limit is the
COlimit that comes from a directed diagram. I just sometimes mistype
and write "directed colimit" which is of course wrong. But here I meant
the colimit. Thank nevertheless :)
Sincerely,
Jose Capco
Jose Capco
Indeed.. this is in fact one of the things everybody knows, and by
which the construction isn't explicitly stated in most reference I come
up with.
Thanks, I'll take a look at your book. I am fearing that I will need
some model theory or a lot of logic for this, I was only hoping for a
ring theoretic construction and not for some abstract abelian category.
My supervisor hinted me on doing the construction with some kind of
"iterative" tensor product that will result into a diagram that is
directed (and since the construction of direct limits are known then I
could use this afterwards). But I can imagine how complicated this can
be. I just wonder, who first made this construction, I cant imagine
that a model theorist would have been the first person who constructed
colimits for rings. But I can imagine that the directed limits where
already constructed before the introduction of category theory itself.
Sincerely,
Jose Capco
martin dowd
> some model theory or a lot of logic for this, I was only hoping for a
> ring theoretic construction and not for some abstract abelian category.
The book is entirely self-contained. One only needs a few basic facts,
about the term algebra of a first order language, and about concrete
categories.
The general construction can be "particularized" by using the axioms to
reduce quotients of the term algebra to more concrete objects. For
example, the coproduct of rings is the free Z-algebra with basis the
words, where no two consectuive letters are from the same ring. Any
colimit is a
- Martin Dowd
Marc Olschok
> Dear NG,
>
> This is something that bugs me every now and then. I haven't tried
> giving it a serious thought. But I think there must be known reference
> or construction on colimits in the category of rings that is not from a
> directed diagram. I know exactly how I can construct directed colimit,
> I have seen them in books.. but with arbitrary colimits I get confused.
> In books the typical way of constructing colimit is to take the direct
> sum of the group obtained from the ring (in our diagram) with addition
> and then take a residue modulo some subgroup (here is when the
> directedness of the diagram is used.. the residue namely takes away an
> element difference with the directed diagram mappings of it) and then
> define a multiplication of this group to make a ring which will turn
> out to be a directed colimit.
In fact you can construct the directed colimit at the level of Sets
and equip it with all the ring operations. There is no need for any
intermediate step (through groups).
>
> As for the nondirected colimits, the category theory books I have just
> say that the category of rings is cocomplete (meaning that there is a
> colimit), but an exact construction is not available. Probably using
> tensor product is a natural way of constructing it?
You can approach it in two steps:
(1) in almost every book on category theory there is a statement, that
in order for a category C to be complete, it suffices that C has equalizers
and coproducts. The proof usually proceeds by constructing for a given
functor D : I --> C its limit via two products and one equalizer.
In fact the index sets of the products used are Objects(I) and Arrows(I).
It is not difficult to dualize the statement and proof which amounts in
constructing a colimit via coproducts and coequalizers.
For rings, coequalizers can be described via the kernels of the corresponding
quotient maps, so this leaves the question of describing coproducts over
possibly infinite index sets.
(2) if a category C already has _finite_ coproducts and _directed_ colimits,
then it has arbitrary coproducts. Here for a functor D : I --> C one
replaces I with J := { X subset I | X finite }, considers the functor
E : J --> C with E(X) = Coproduct( D(i) | i in X ) and shows that
Colim E (which exists because J is directed) indeed serves as a colimit
for D.
So your guess is quite right. The tensor product (over Z) comes into play
in step (2) as it gives the finite coproducts of rings.
Marc
James Dolan
marc olschok <inv...@nowhere.com> wrote:
|So your guess is quite right. The tensor product (over Z) comes into
|play in step (2) as it gives the finite coproducts of rings.
i haven't been following this thread that closely, but i thought that
tensor product is generally only the co-product in the category of
commutative rings, whereas the question here was about the category of
rings in general.
i'm not sure that i've ever thought much about general co-limits in
the category of rings. as a wild guess, is the co-product of x and y
in the category of non-unital non-commutative rings maybe given by the
direct sum of x, y, x#y, y#x, x#y#x, y#x#y, and so forth, where # is
tensor product in the category of abelian groups? then for unital
rings replace the direct sum by the appropriate directed colimit?
--
[e-mail address jdo...@math.ucr.edu]
Arturo Magidin
Jose Capco <cliom...@kriocoucke.mailexpire.com> wrote:
>> You are confusing colimits and limits.
>>
>> Perversely, the "direct limit" is a kind of colimit; and the "inverse
>> limit" is a kind of limit. You are describing inverse limits. So you
>> must be talking about limits.
>
>No sorry, I meant directed limit not directed colimit (sometimes I
>confuse the terminology, but I know what Im talking about). Thus here I
>mean colimit
Hmmm... You seem to be trouncing terminology again.
"Directed limit" is a colimit over a directed set (i.e., take the
directed set as a category D, let F:D -> C be the functor that gives
you the objects and "structure mappings", and you take the colimit of
this functor). I do not believe "directed colimit" is standard
terminology. "Inverse limit" is a limit over an inversely directed set
(or a directed set if you use a contravariant functor).
What you described as the "standard" was the inverse limit (take the
direct sum, and consider the subobject of "consistent tuples"). this
is a LIMIT.
The direct limit for rings, by contrast, is constructed by taking the
disjoint union of the rings (say by considering the union of R_i x {i}
with i ranging over your directed set), and then defining an
equivalence relation by letting (a,i) be equivalent to (b,j) if and
only if there exists k such that i,j<=k and f_ik(a)=f_jk(b). Then you
consider the QUOTIENT of the union of R_i x {i} modulo the equivalence
relation and define an addition and multiplication on it. This is a
COLIMIT.
I have never seen "directed colimit" used; I've seen "limit" and
"colimit"; and I've seen "inverse limit" and "directed limit" (or
"direct limit") for the respectively special kinds.
So, which ones do you mean? Colimits, or limits? You described a kind
of limit, but spoke about colimits.
>> The limits are taken by the ->other<- construction: take the disjoint
>> union modulo an equivalence relation.
>
>No, I do really mean the colimit :) ... like I said direct limit is
>easy to construct, but not the colimit in general (i.e. a colimit that
>is not from a directed diagram).
But the ->description<- you gave was that of a ->limit<-: you said "take
the direct sum" and then consider a subobject. That's ->not<- how you
construct the colimit over a directed diagram!
Lee Rudolph
>I have never seen "directed colimit" used
Well, neither had I (and I was just as happy), though naturally I
wouldn't. However, not only does Google turn up 168 hits, only
some of which refer to the present thread, but MathSciNet finds
44: the oldest there is a review by Alex Heller of a 1968 paper by
Barry Mitchell with reference to work on the subject by Barbara
Osofsky; 16 papers of those 44 have appeared since 2000; and
one of the reviews is by sci.math irregular Michael Barr. So
I would say there are enough users, including some very heavy
hitters in the categorical world, to make the term unassailably
standard.
Lee Rudolph
James Dolan
arturo magidin <mag...@math.berkeley.edu> wrote:
|In article <1148099992....@g10g2000cwb.googlegroups.com>,
|Jose Capco <cliom...@kriocoucke.mailexpire.com> wrote:
|>> You are confusing colimits and limits.
|>>
|>> Perversely, the "direct limit" is a kind of colimit; and the
|>> "inverse limit" is a kind of limit. You are describing inverse
|>> limits. So you must be talking about limits.
|>
|>No sorry, I meant directed limit not directed colimit (sometimes I
|>confuse the terminology, but I know what Im talking about). Thus
|>here I mean colimit
|
|Hmmm... You seem to be trouncing terminology again.
there's a lot of confusion here; i don't know if it's too big a mess
to straighten out. it's true that (probably mostly by historical
accident) there are some perverse terminology clashes here that have
become somewhat standard across some subcultures and sometimes even
within the same subculture, but it seems that the two of you are
compounding the problem by careless reading and careless writing.
(and i'm not that optimistic that i'm being careful enough here
myself.)
one problem seems to be confusion between the words "direct" and
"directed". various approaches might be taken in order to try to
clear up this confusion (for example trying to prohibit the use of one
or the other, or prohibit the use of either of them except in
circumstances where they accidentally imply the other one as well) but
perhaps the best approach is to be aware of and to carefully
distinguish between their very different meanings.
"direct" here has the intuitive sense of "forwards" in contrast to
"inverse" in the sense of backwards, but neither "direct" nor
"inverse" has a technical meaning in isolation in this context; only
the phrases "direct limit" and "inverse limit" have technical
meanings. "direct limit" is a synonym for "colimit" while "inverse
limit" is a synonym for "limit". (questions about what "inverse
inverse limit" might mean are beyond the dumbness level of even this
post.)
"directed" is a modifier which can modify either "limit" (or its
synonym "inverse limit") or "colimit" (or its synonym "direct limit").
"directed" means that the diagram scheme has an ideal "confluent
direction" that all of the arrows "flow towards" (either in a sensible
"forwards" way in the case of a directed directed limit, or in a
"backwards" way in the case of a directed inverse limit.)
unfortunately the word "directed" is sometimes (but sometimes not)
implcitly assumed to have been ellipsized from in front of both
"direct limit" and "inverse limit", in part because before the
conceptual unity between directed limits (and colimits) and undirected
limits (and colimits) was recognized, the terminology "limit" was
reserved for the directed case (where the diagram scheme has a
confluent direction, thus reminiscent of the concept of "limit" in
classical analysis).
|"Directed limit" is a colimit over a directed set
i doubt whether any significant subculture is perverse enough to use
the terminology _that_ way, though it could happen for all i know.
you probably meant to say "direct limit" here rather than "directed
limit". lots of people would instead say "directed colimit", and i
personally would sometimes say "directed direct limit" in an attempt
to teach by contrastive emphasis that both "directed" and "direct" can
occur here but mean very different things.
|(i.e., take the
|directed set as a category D, let F:D -> C be the functor that gives
|you the objects and "structure mappings", and you take the colimit of
|this functor). I do not believe "directed colimit" is standard
|terminology. "Inverse limit" is a limit over an inversely directed
|set (or a directed set if you use a contravariant functor).
|
|What you described as the "standard" was the inverse limit (take the
|direct sum, and consider the subobject of "consistent tuples"). this
|is a LIMIT.
this is where i think you significantly misread the original post,
which said something about taking a direct sum and _modding out_ by a
subgroup. a direct sum is a colimit (direct but non-directed, just as
it should be!) in a somewhat relevant category here. an inverse limit
is _not_ the consistent tuples in a direct sum; rather it's the
consistent tuples in a cartesian product (unfortunately also called a
"direct product" even though it's an inverse limit rather than a
direct limit). if your subculture really does call the cartesian
product the "direct sum" while calling the _true_ direct sum the
"coproduct" then it's probably not so easy to come up with any good
excuse for that.
--
[e-mail address jdo...@math.ucr.edu]
martin dowd
The coproduct is the free Z-module with the words specified in post 4
as basis, and obvious multiplication of words (concatenate and reduce
by multiplications in a summand ring). This is not a free Z-algebra.
The definitions of limit and colimit were I believe given in Eilenberg
and Maclane's 1945 paper. In a limit (colimit) of objects Ri, the
arrows of a cone go to (from) the Ri and the induced arrow goes to
(from) the limit (colimit) object. I'm not sure when the terms
"direct" and "inverse" system were introduced, but they are completely
standard, and authors occasionally remark that the terminology is
perverse in that a direct limit is a colimit and an inverse limit is a
limit.
That the above ring R is the coproduct of the Ri is readily verified.
There is an obvious cone from the Ri to R; and a cone from the Ri to
any ring Q are readily verified to induce a unique homomorphism from R
to Q. As mentioned in an earlier post, the construction can be arrived
at by particularizing a general construction for models of any
equational theory, although it can certainly be arrived at more
directly.
The construction for finite coproducts is the same, so nothing is
gained by doing this first and then using the direct system of finite
subsets. One can see, however, that this yields the same result. The
same remarks apply for the models of an equational theory.
The coproduct of a pair of rings is not the tensor product. The only
tensor product which makes sense is that of the Z-algebras, and this is
not the same (it is the coproduct of the Z-algebras though).
To obtain any colimit, one uses the construction as a coequalizer of a
pair of maps between two coproducts. The coequalizer can be
constructed for the models of an equational theory; for rings it is
well known.
By sorting through the construction, one can undoubtedly give a more
explicit description.
- Martin Dowd
martin dowd
martin dowd wrote:
> The coproduct of a pair of rings is not the tensor product. The only
> tensor product which makes sense is that of the Z-algebras, and this is
> not the same (it is the coproduct of the Z-algebras though).
This is another mistake; Z-algebras = rings. The tensor product is the
coproduct in the category of commutative R-algebras over a commutative
ring. If the algebras are allowed to be noncommutative, the tensor
product is universal in the cones Ri->S, where
f1(x1)f2(x2)=f2(x2)f1(x1) ; this is in Jacobson. If the cones are
allowed to be arbitrary, to get a universal cone one must take
arbitrary words rather than just pairs as in the tensor product.
Lee Rudolph
Jose Capco
Arturo Magidin wrote:
> Hmmm... You seem to be trouncing terminology again.
>
> "Directed limit" is a colimit over a directed set (i.e., take the
> directed set as a category D, let F:D -> C be the functor that gives
> you the objects and "structure mappings", and you take the colimit of
> this functor). I do not believe "directed colimit" is standard
> terminology. "Inverse limit" is a limit over an inversely directed set
> (or a directed set if you use a contravariant functor).
>
> What you described as the "standard" was the inverse limit (take the
> direct sum, and consider the subobject of "consistent tuples"). this
> is a LIMIT.
>
> The direct limit for rings, by contrast, is constructed by taking the
> disjoint union of the rings (say by considering the union of R_i x {i}
> with i ranging over your directed set), and then defining an
> equivalence relation by letting (a,i) be equivalent to (b,j) if and
> only if there exists k such that i,j<=k and f_ik(a)=f_jk(b). Then you
> consider the QUOTIENT of the union of R_i x {i} modulo the equivalence
> relation and define an addition and multiplication on it. This is a
> COLIMIT.
Arturo and the rest, sorry for the confusion. I wanted to describe the
construction given for DIRECT LIMITs in the book of Ernst Kunz
"Einführung in die algebraische Geometrie" (page 99-100, I can put up
the translation of the description if this is absolutely necessary). I
did it in a handwaving manner, and maybe that lead into a confusion (or
maybe I messed up with my wordings). In any case, I am questioning
colimits. I hope that rests the case :)
But regarding the terminology "direct limit", when I was beginning to
study category theory. It came rather confusing to me too. Every now
and then I just blurt "direct(ed) colimit" to mean "direct limit",
which is probably not correct way of saying it.
Sincerely,
Jose Capco
Jose Capco
apparently an authority in category theory) and here I see that he used
"directed colimit" to mean what Arturo and most people would call
"direct limit". So I guess I didnt blurt the words "directed colimit"
by chance (although sometimes it mutates to "direct colimit" which isnt
really meaningful).
Ok.. lets not talk about this confusing matter any longer :)
Sincerely,
Jose Capco
Marc Olschok
> Aha.. I was just taking a look at the works of Adámek (who is
> apparently an authority in category theory) and here I see that he used
> "directed colimit" to mean what Arturo and most people would call
> "direct limit".
Not most people. By now perhaps not even Arturo any more. See the first post
of James explaining the terminology.
See also 11.28 (4) (beginning of page 203) of the Adamek-Herrlich-Strecker
book (available at <http://katmat.math.uni-bremen.de/acc> .
Marc
Marc Olschok
> in article <4d9248F...@news.dfncis.de>,
> marc olschok <inv...@nowhere.com> wrote:
>
> |So your guess is quite right. The tensor product (over Z) comes into
> |play in step (2) as it gives the finite coproducts of rings.
>
> i haven't been following this thread that closely, but i thought that
> tensor product is generally only the co-product in the category of
> commutative rings, whereas the question here was about the category of
> rings in general.
Yes, upon seeing "tensor product" in the original posting I just
assumed (perhaps wrongly) that the OP had meant commutative rings only.
>
> i'm not sure that i've ever thought much about general co-limits in
> the category of rings. as a wild guess, is the co-product of x and y
> in the category of non-unital non-commutative rings maybe given by the
> direct sum of x, y, x#y, y#x, x#y#x, y#x#y, and so forth, where # is
> tensor product in the category of abelian groups? then for unital
> rings replace the direct sum by the appropriate directed colimit?
That will probably do. It is clear that that in a possible coproduct
A --> A+B <-- B elements coming from different parts must not commute.
So one could start with all alternating words over A and B and form the
free abelian group over the resulting monoids. This step is more or less
the above sum. Then an appropriate equivalence relation has to be introduce
to take care of products like (ua)(a'v) or (ua)(1_B)(av) etc. and also
of sums like uav + ua'v. Perhaps this can even be decomposed nicely so
that a coproduct of A and B can be seen as quotient of Z[M] where the
monoid M is the coproduct of the multiplicative monoids of A and B.
Marc
Jose Capco
books and I think maybe I have it right now,
Let
D: I --> R
be a small diagram into the category of commutative rings with unity
Then for every t in Mor(I), I can write
t: d(t)-->c(t)
And I have the canonical morphisms
w_i : R_i --> (X)_{i in Obj(I)) R_i
where (X) is the tensor product with respect to Z.
then we have the parallel morphisms,
w_d(t), w_c(t)(Dt) : R_d(t) --> (X)_{i in Obj(I)) R_i
we also have the parallel morphisms,
(x)_{t in Mor(I)} w_d(t), (x)_{t in Mor(I)} w_c(t)(Dt) :
(X)_{t in Mor(I)) R_d(t) --> (X)_{i in Obj(I)) R_i
and we have coequalizers in the category of rings, that brings this
parallel morphism
to R, then R is the colimit of D and the colimit cone is the canonical
morphism
R_i -->(X)_{t in Mor(I)) R_d(t) ===> (X)_{i in Obj(I)) R_i ---> R
for all i in Obj(I)
whew... :)
Sincerely,
Jose Capco
PS: All tensor products here are with respect to the whole numbers, Z.
Marc Olschok
As far as I could follow the notation, this is the usual construction
via coproducts and coequalisers _except_ that you have to be careful
with using the tensor product. It is not clear to me why the coproduct
of _infinitely_ many commutative rings should be their tensor product.
Marc